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Phase Transitions and Critical P h e n o m e n a
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Phase Transitions and Critical P h e n o m e n a Volume 18
Edited by C. Domb
Department of Physics, BarI/an University, RamatGan, Israel
and
J. L. Lebowitz
Department of Mathematics and Physics, Rutgers University, New Brunswick, New Jersey, USA
ACADEMIC PRESS A HarcourtScienceand TechnologyCompany San Diego San Francisco London Sydney Tokyo
New York
Boston
This book is printed on acidfree paper. Copyright (~) 2001 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Explicit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press A Harcourt Science and Technology Company Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK http://www.academicpress.com Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 921014495, USA http://www.academicpress.com ISBN 0122203186
A catalogue record for this book is available from the British Library
Typeset by Focal image Ltd, London Printed and bound in Great Britain by MPG Books Ltd, Cornwall, UK 00 01 02 03 04 05 MP 9 8 7 6 5 4 3 2 1
Contributors M. J. ALAVA, Laboratory of Physics, Helsinki University of Technology, PO Box 1100, HUT 02015, Finland E M. DUXBURY, Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University, East Lansing, M148824, USA
H.O. GEORGII, Mathematisches Institut, Universitiit, D80333 Miinchen, Germany
LudwigMaximilians
O. HAGGSTROM, Department of Mathematics, Chalmers University of Technology, S412 96 Goteborg, Sweden C. MAES, Instituut voor Theoretische Fysica, K. U. Leuven, B3001 Leuyen, Belgium C. E MOUKARZEL, Instituto de Fisica, Fluminense, 24210340 Niteroi, R J, Brazil
Universidade
Federal
H. RIEGER, Institut fiir Theoretische Physik, Universitiit des Saarlandes, 66041 Saarbrucken, Germany
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General Preface This series of publications was first planned by Domb and Green in 1970. During the previous decade the research literature on phase transitions and critical phenomena had grown rapidly and, because of the interdisciplinary nature of the field, it was scattered among physical, chemical, mathematical and other journals. Much of this literature was of ephemeral value, and was rapidly rendered obsolete. However, a body of established results had accumulated, and the aim was to produce articles that would present a coherent account of all that was definitely known about phase transitions and critical phenomena, and that could serve as a standard reference, particularly for graduate students. During the early 1970s the renormalization group burst dramatically into the field, accompanied by an unprecedented growth in the research literature. Volume 6 of the series, published in 1976, attempted to deal with this new literature, maintaining the same principles as had guided the publication of previous volumes. The number of research publications has continued to grow steadily, and because of the great progress in explaining the properties of simple models, it has been possible to tackle more sophisticated models which would previously have been considered intractable. The ideas and techniques of critical phenomena have found new areas of application. After a break of a few years following the death of Mel Green, the series continued under the editorship of Domb and Lebowitz, Volumes 7 and 8 appearing in 1983, Volume 9 in 1984, Volume 10 in 1986, Volume 11 in 1987, Volume 12 in 1988, Volume 13 in 1989 and Volume 14 in 1991. The new volumes differed from the old in two new features. The average number of articles per volume was smaller, and articles were published as they were received without worrying too much about the uniformity of content of a particular volume. Both of these steps were designed to reduce the time lag between the receipt of the author's manuscript and its appearance in print. The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It is not longer an area of specialist interest, but has moved into a central place in
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General Preface
condensed matter studies. The editors feel that there is ample scope for the series to continue, but the major aim will remain to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments. CYRIL DOMB JOEL L. LEBOWITZ
Preface to Volume 18 The two review articles in this volume complement each other in a remarkable way. They both deal with what might be called the modern geometric approach to the properties of macroscopic systems. The first one by Georgii et al., is primarily analytical. It describes in a rigorous, yet generally accessible, mathematical way recent advances in the application of geometric ideas, such as percolation, to visualizing the structure present in a typical configuration of the spins or atoms making up the microscopic constituents of a macroscopic system. This leads to a better understanding of pure phases and phase transitions in equilibrium systems. The authors illustrate these ideas by carrying out an in depth analysis of some of the basic models in statistical mechanics. These include the Ising model (both ferromagnetic and antiferromagnetic), the Potts model, the WidomRowlinson model, etc. Typical of the geometric ideas discussed here are those underlying the behavior of the twodimensional Ising ferromagnet with nearest neighbor interactions at zero magnetic field. For temperatures above the Onsager critical temperature neither the pluses nor the minuses percolate while below it there is percolation of pluses (minuses) and only pluses (minuses) in the plus (minus) phase. The review goes, however, much beyond such classical results to bring the reader right up to date on this exciting topic. The second article in this volume by Alava et al., also focuses on geometrical aspects of manybody systems. It does so in a handson way going beyond abstract theory to obtain practical answers. This requires the use of computers, but not just their blind use. Computing power alone is simply not enough. One also needs a deep understanding of the physics and cleverness of programming. This is, in fact, what this article is all about. This article, the first one in this series in which the computer is what might be called the star of the show, focuses on geometrical aspects of the use of computers in statistical mechanics. It provides, to quote the authors, "an introduction to combinatorial optimization algorithms and reviews their applications to groundstate problems in disordered systems." This covers an astonishingly large class of problems of current interest ranging from
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Preface to V o l u m e 18
the random field Ising model to elastic media and rigidity percolation. It is fair to say that the review brings together the most recent advances in computer science which are useful for solving problems of interest to statistical mechanicians and to material scientists. To end this preface on a historical note we mention that computers have played a role in statistical mechanics for more than 50 years now. One of the earliest works was that of Fermi, Pasta and Ulam who used the computer to solve Newton's equations of motion for an anharmonic chain consisting of 32 particles and discovered an apparent lack of (or at least an extremely slow) equipartition of energy in that system. That led to what is now known as molecular dynamics. Another landmark work was the introduction of Monte Carlo sampling techniques by Metropolis, Rosenbluth, Rosenbluth, Teller and Teller for evaluating equilibrium properties of large systems. Other firsts included the use of computers by the King's College group of Domb and associates to help evaluate coefficients in the high and lowtemperature expansions of different spin systems which led to the first evaluation of critical exponents and the notions of universality. It is our expectation that the combination of computers and geometrical ideas described in this volume will play a major role in the development of statistical mechanics in the twentyfirst century. CYRIL DOMB JOEL L. LEBOWITZ
Contents Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G e n e r a l Preface
v
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Preface to Volume 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
1 The Random Geometry of Equilibrium Phases H .  O . GEORGII, O. HAGGSTR()M AND C. MAES 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 E q u i l i b r i u m phases
6
3 Some models
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 C o u p l i n g and stochastic d o m i n a t i o n 5 Percolation
. . . . . . . . . . . . . . . . . . .
14 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
6 R a n d o m  c l u s t e r representations . . . . . . . . . . . . . . . . . . . . . .
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7 U n i q u e n e s s and exponential mixing from n o n  p e r c o l a t i o n . . . . . . . .
72
8 Phase transition and percolation
. . . . . . . . . . . . . . . . . . . . .
91
. . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 R a n d o m interactions 10 C o n t i n u u m m o d e l s
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
2 Exact Combinatorial Algorithms: Ground States of Disordered Systems M. J.
ALAVA,P.
1 Overview
M.
DUXBURY,C.
F. MOUKARZEL AND H. RIEGER
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Basics o f graphs and algorithms
145
. . . . . . . . . . . . . . . . . . . . .
148
3 Flow algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
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4 5 6 7 8 9 10
Contents
M a t c h i n g algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical programming . . . . . . . . . . . . . . . . . . . . . . . . Percolation and m i n i m a l path . . . . . . . . . . . . . . . . . . . . . . . R a n d o m Ising m a g n e t s . . . . . . . . . . . . . . . . . . . . . . . . . . Line, vortex and elastic glasses . . . . . . . . . . . . . . . . . . . . . . Rigidity theory and applications . . . . . . . . . . . . . . . . . . . . . C l o s i n g remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Subject index
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175 181 186 196 234 264 295 297 297 318
Contents of Volumes 117 Contents of Volume 1 (Exact Results) t Introductory Note on Phase Transitions and critical phenomena. C. N. YANG. Rigorous Results and Theorems. R. B. GRIFFITHS. Dilute Quantum Systems. J. GINIBRE. C*Algebra Approach to Phase Transitions. G. Emch. One Dimensional ModelsShort Range Forces. C. J. THOMPSON. Two Dimensional Ising Models. H. N. V. TEMPERLEY. Transformation of Ising Models. I. SYOZI. Two Dimensional Ferroelectric Models. E. H. LIEB and E Y. Wu. Contents of Volume 2 t Thermodynamics. M. J. BUCKINGHAM. Equilibrium Scaling in Fluids and Magnets. M. VICENTINIMISSONI. Surface Tension of Fluids. B. WIDOM. Surface and Size Effects in Lattice Models. P. G. WATSON. Exact Calculations on a Random Ising System. B. McCoY. Percolation and Cluster Size. J. W. ESSAM. Melting and Statistical Geometry of Simple Liquids. R. COLLINS. Lattice Gas Theories of Melting L. K. RUNNELS. Closed Form Approximations for Lattice Systems. D. M. BURLEY. Critical Properties of the Spherical Model. G. S. JOYCE. Kinetics of Ising Models. K. KAWASAKI. Contents of Volume 3 (Series Expansions for Lattice Models) t Graph Theory and Embeddings. C. DOMB. Computer Enumerations. J. L. MARTIN. Linked Cluster Expansions. M. WORTIS. ~ Out of print.
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Contents of Volumes 117
Asymptotic Analysis of Coefficients. D. S. GAUNT and A. J. GUTTMAN. Ising Model. C. DOMB Heisenberg Model. G. A. BAKER, G. S. RUSHBROOKE and R W. WOOD. Classical Vector Models. H. E. STANLEY. Ferroelectric Models. J. E EAGLE. X  Y Model. D. D. BETTS. Contents of Volume 4 t
Theory of Correlations in the Critical Region. M. E. FISHER and D. JASNOW. Contents of Volume 5at
Scaling, Universality and Operator Algebras. Leo R KADANOFF. Generalized Landau Theories. Marshall Luban. Neutron Scattering and Spatial Correlation near the Critical Point. JENS ALSNIELSEN. Mode Coupling and Critical Dynamics. KYOZI KAWASAKI. Contents of Volume 5b t
Monte Carlo Investigations of Phase Transitions and Critical Phenomena. K. BINDER. Systems with Weak LongRange Potentials. R C. HEMMER and J. L. LEBOWITZ. Correlation Functions and Their Generating Functionals: General Relations with Applications to the Theory of Fluids. G. STELE. Heisenberg Ferromagnet in the Green's Function Approximation. R. A. TAHIRKHELI. Thermal Measurements and Critical Phenomena in Liquids. A. V. VORONEL. Contents of Volume 6 (The Renormalization Group and its Applications) t
Introduction. K. G. WILSON. The Critical State, General Aspects. F. J. WEGNER. Field Theoretical Approach. E. BREZIN, J. C. LE GUILLOU and J. ZINNJUSTIN. The 1/n Expansion. S. MA. The eExpansion and Equation of State in Isotropic Systems. D. J. WALLACE. Universal Critical Behaviour. A AHARONY. Renormalization: Isinglike Spin Systems. TH. NIEMEUER and J. M. J. VAN LEEUWEN Renormalization Group Approach. C. DI CASTRO and G. JONALASINIO.
Contents of Volumes 117
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Contents of Volume 7 t
DefectMediated Phase Transitions. D. R. NELSON. Conformational Phase Transitions in a Macromolecule: Exactly Solvable Models. E W. WIEGEL. Dilute Magnetism. R. B. STINCHCOMBE. Contents of Volume 8
Critical Behaviour at Surfaces. K. BINDER. FiniteSize Scaling. M. N. BARBER. The Dynamics of First Order Phase Transitions. J. D. GUNTON, M. SAN MIGUEL and P. S. SAHNI. Contents of Volume 9 t
Theory of Tricritical Points. I. D. LAWRIE and S. SARBACH. Multicritical Points in Fluid Mixtures: Experimental Studies. C. M. KNOBLER and R. L. SCOTT. Critical Point Statistical Mechanics and Quantum Field Theory. G. A. BAKER, JR. Contents of Volume 10
Surface Structures and Phase TransitionsExact Results. D. B. ABRAHAM. FieldTheoretic Approach to Critical Behaviour at Surfaces. H. W. DIEHL. Renormalization Group Theory of Interfaces. D. JASNOW. Contents of Volume 11
Coulomb Gas Formulation of TwoDimensional Phase Transitions. B. NIENHUIS. Conformal Invariance. J. L. C ARDY. LowTemperature Properties of Classical Lattice Systems: Phase Transitions and Phase Diagrams. J. SLAWNY. Contents of Volume 12 t
Wetting Phenomena. S. DIETRICH. The Domain Wall Theory of TwoDimensional CommensurateIncommensurate Phase Transitions. M. DEN NUS. The Growth of Fractal Aggregates and their Fractal Measures. P. MEAKIN.
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Contents of Volumes 117
Contents of Volume 13
Asymptotic Analysis of PowerSeries Expansions. A. J. GUTTMANN. Dimer Models on Anisotropic Lattices. J. E NAGLE,. S. O. YOKOI and S. M. BHATTACHARJEE. Contents of Volume 14
Universal CriticalPoint Amplitude Relations. V. PRIVMAN,P. C. HOHENBERG and A. AHARONY. The Behaviour of Interfaces in Ordered and Disordered Systems. G. FORGACS, R. LIPOWSKY and TH. M. NIEUWENHUIZEN Contents of Volume 15
Spatially Modulated Structures in Systems with Competing Interactions. W. SELKE. The Largen Limit in Statistical Mechanics and the Spectral Theory of Disordered Systems. A. M. KHORUNZHY, B. A. KHORUZHENKO, L. A. PASTUR and M. V. SHCHERBINA. Contents of Volume 16
SelfAssembling Amphiphilic Systems. G. GOMPPER and M. SCHICK. Contents of Volume 17
Statistical Mechanics of Driven Diffusive Systems. B. SCHMITTMANN and R. K. P. ZIA.
The Random Geometry of Equilibrium Phases HansOtto Georgii Mathematisches Institut, LudwigMaximiliansUniversit#t, D80333 MSnchen, Germany
Email" g e o r g i i @ r z , m a t h e m a t i k , u n i  m u e n c h e n , de
Olle H~iggstr6m Department of Mathematics, Chalmers University of Technology, S412 96 G6teborg, Sweden
Email: o l l e h @ m a t h , c h a l m e r s , se
Christian Maes Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, B3001 Leuven, Belgium
Email: C h r i s t i a n . M a e s @ f y s . k u l e u v e n , ac .be
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Equilibrium phases
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2.1
T h e lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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R a n d o m fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The Hamiltonian
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Gibbs measures
PHASE TRANSITIONS
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Copyright ('6) 2001 Academic Press Limited
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Phase transition and phases . . . . . . . . . . . . . . . . . . . . . . . . .
3 Some models
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3.1
The ferromagnetic Ising model . . . . . . . . . . . . . . . . . . . . . . .
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3.2
The antiferromagnetic Ising model . . . . . . . . . . . . . . . . . . . . .
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3.3
The Potts model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4
The hardcore lattice gas model . . . . . . . . . . . . . . . . . . . . . . .
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3.5
The W i d o m  R o w l i n s o n lattice model
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4 Coupling and stochastic domination
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4.1
The coupling inequality . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2
Stochastic domination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3
Applications to the Ising model . . . . . . . . . . . . . . . . . . . . . . .
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4.4
Application to other models . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Percolation 5.1
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Bernoulli percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.2
Dependent percolation: the role of the density . . . . . . . . . . . . . . .
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Examples of dependent percolation . . . . . . . . . . . . . . . . . . . . .
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5.4
The number of infinite clusters . . . . . . . . . . . . . . . . . . . . . . .
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6 Randomcluster representations . . . . . . . . . . . . . . . . . . . . . . . . . .
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Randomcluster and Potts models
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Infinitevolume limits . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Phase transition in the Potts model . . . . . . . . . . . . . . . . . . . . .
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6.4
Infinite volume randomcluster measures . . . . . . . . . . . . . . . . . .
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An application to percolation in the Ising model . . . . . . . . . . . . . .
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Cluster algorithms for computer simulation
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Randomcluster representation of the W i d o m  R o w l i n s o n model
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7 Uniqueness and exponential mixing from nonpercolation . . . . . . . . . . . . 7.1
Disagreement paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Stochastic domination by randomcluster measures
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Exponential mixing at low temperatures
8 Phase transition and percolation
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8.1
Agreement percolation from phase coexistence
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Plusclusters for the Ising ferromagnet . . . . . . . . . . . . . . . . . . .
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Constantspin clusters in the Potts model . . . . . . . . . . . . . . . . . .
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Further examples of agreement percolation
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Percolation of groundenergy bonds
9 R a n d o m interactions
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Diluted and random Ising and Potts ferromagnets
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Mixing properties in the Griffiths regime . . . . . . . . . . . . . . . . . . ................................
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10 Continuum models
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1 Random geometry of equilibrium phases
10.1 Continuum percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The continuum WidomRowlinson model . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
Equilibrium statistical mechanics describes and explains the macroscopic behavior of systems in thermal equilibrium in terms of the microscopic interaction between their very many constituents. As a typical example, let us take some ferromagnetic material like iron: the constituents are then the spins of elementary magnets at the sites of some crystal lattice. Or we may think of a lattice approximation to a real gas, in which case the constituents are the particle numbers in the elementary cells of any partition of space. The central object is the Hamiltonian describing the interaction between these constituents. This interaction determines the relative energies between configurations that differ only microscopically. The equilibrium states with respect to the given interaction are described by the associated Gibbs measures. These are probability measures on the space of configurations which have prescribed conditional probabilities with respect to fixed configurations outside of finite regions. These conditional probabilities are given by the Boltzmann factor, the exponential of the inverse temperature times the relative energy. This allows one to compute, at least in principle, equilibrium expectations and spatial correlation functions following the standard Gibbs formalism. The socalled extremal Gibbs measures are very important since they describe the possible macrostates of our physical system. In such a state, macroscopic observables do not fluctuate while the correlation between local observations made far apart from each other decays to zero. Since the early days of statistical mechanics, geometric notions have played a role in elucidating certain aspects of the theory. This has taken many different forms. Arguably, the thermodynamic formalism, as first developed by Gibbs, already admits some geometric interpretations primarily related to convexity. For example, entropy is a concave function of the specific energy, the pressure is convex as a function of the interaction potential, the LegendreFenchel transformation relates various fundamental thermodynamic quantities to each other, and the set of Gibbs measures for an interaction is a simplex with vertices corresponding to the physically realized macrostates, the equilibrium phases. Here, however, we will not be concerned with this kind of convex geometry which is described in detail e.g. in the books by Israel (1979) and Georgii (1988). Rather, the geometry considered here is a way of visualizing the structure in the typical realizations of the system's constituents. To be more specific, let
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us consider for a moment the case of the standard ferromagnetic Ising model on the square lattice. At each site we have a spin variable taking only two possible values, +1 and  1 . The interaction is nearestneighbor and tends to align neighboring spins in the same direction. By the ingenious arguments first formulated by Peierls (1936) (see also Dobrushin, 1965; Sinai, 1982; Georgii, 1988), the phase transition in this model can be understood from looking at the typical configurations of contours, i.e., the broken lines separating the domains with plus resp. minus spins. The plus phase (the positively magnetized phase) is realized by an infinite ocean of plus spins with finite islands of minus spins (which in turn may contain lakes of plus spins, and so on). On the other hand, above the Curie temperature (first computed by Onsager) there is no infinite path joining nearest neighbors with the same spin value. So, for this model the geometric picture is rather complete (as we will show later). In general, however, much less is known, and much less is true. Still, certain aspects of this geometric analysis have wide applications, at least in certain regimes of the phase diagram. These "certain regimes" are, on the one hand, the hightemperature (or, in a lattice gas setting, lowdensity) regime and, to the other extreme, the lowtemperature behavior. At high temperatures, all thermodynamic considerations are based on the fact that entropy dominates over energy. That is, the interaction between the constituents is not effective enough to enforce a macroscopic ordering of the system. As a result, every constituent is more or less free to behave at random, not much influenced by other constituents which are far apart. So, the system's behavior is almost like that of a free system with independent components. This means, in particular, that in the center of a large box we will typically encounter more or less the same configurations no matter what boundary conditions outside this box are imposed. That is, if we compare two independent realizations of the system in the box with different boundary conditions outside then, still at high temperatures, the difference between the boundary conditions cannot be felt by the spins in the center of the box; specifically, there should not exist any path from the boundary to the central part of the box along which the spins of the two realizations disagree. This picture is rather robust and can also be applied when the interaction is random (see Sections 7 and 9). At low temperatures, or large densities (when the interaction is sufficiently strong), the above picture no longer holds. Rather, the specific characteristics of the interaction will come into play and determine the specific features of the lowtemperature phase. In many cases, the lowtemperature behavior can be described as a random perturbation of a ground state, i.e., of a fixed configuration of minimal energy. Then we can expect that at low temperatures, and sometimes even up to the critical temperature, the equilibrium phases are realized as a deterministic ground state configuration, perturbed by finite random islands on which the configuration disagrees with the ground state. This means that the ground
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Random geometry of equilibrium phases
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state pattern can percolate through the space to infinity. One prominent way of confirming this picture is provided by the socalled PirogovSinai theory which has been described in detail by Zahradnfk (1996). In Section 8 we will discuss some other techniques of establishing the same geometric picture. It is evident from the above that percolation theory will play an important role in this text. In fact, we will mainly be concerned with dependent percolation, but one can say that independent percolation stands as a prototype for the study of statistical equilibrium properties in geometric terms. In independent percolation, the model is extremely simple: the components are binaryvalued and independent from each other. What is hard is the type of question one asks, namely the question of existence of infinite paths of l's and their geometry. We will introduce percolation below but refer to other publications (e.g. Grimmett, 1999) for a systematic account of the theory. Percolation will come into play here on various levels. Its concepts like clusters, open paths, connectedness etc. will be useful for describing certain geometric features of equilibrium phases, allowing characterizations of phases in percolation terms. Examples will be presented where the (thermal) phase transition goes hand in hand with a phase transition in an associated percolation process. Next, percolation techniques can be used to obtain specific information about the phase diagram of the system. For example, equilibrium correlation functions are sometimes dominated by connectivity functions in an associated percolation problem which is easier to investigate. Finally, representations in terms of percolation models will yield explicit relations between certain observables in equilibrium models and some corresponding percolation quantities. In fact, the resulting percolation models, like the randomcluster model, have some interest in their own right and will also be studied in some detail. This text is supposed to be selfcontained. Therefore we need to introduce various concepts and techniques which are well known to some readers. On the other hand, important related issues will not be discussed when they are not explicitly needed. For more complete discussions on the introductory material we will refer the interested reader to other sources. More seriously, we will not include here a discussion of some important geometric concepts developed in the 1980s for the study of critical behavior in statistical mechanical systems, namely random walk expansions or random current representations. Fortunately, we can refer to an excellent book (Fernandez et al., 1992) where the interested reader will find all the relevant results and references. Important steps in this context include (Aizenman, 1982; Aizenman and Barsky, 1987; Aizenman et al., 1987; Aizenman and Fernandez, 1986; Brydges et al., 1982, 1983) and the references therein. Finally, to avoid misunderstanding, the random geometry in the title of this work should not be confused with stochastic geometry (or geometric probability) which, as a branch of integral geometry, provides a very interesting toolbox
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al.
for the discussion of morphological characteristics of random fields appearing in statistical physics and beyond, see Klain and Rota (1997); Adler (2000) and Mecke (1998).
2 2.1
Equilibrium phases The lattice
Our object of study are physical systems with many constituents, spins or particles, which will be located at the sites of a crystal lattice s The standard case is when s = Z d, the ddimensional hypercubic lattice. In general, we shall assume that s is the vertex set of a countable graph. That is, s is an at most countable set and comes equipped with a (symmetric) adjacency relation. Namely, we write x ~ y if the vertices x, y E s are adjacent, and this is visualized by drawing an edge between x and y. In this case, x and y are also called neighbors, and the edge (or bond) between x and y is denoted by (xy). We write B for the set of all edges (bonds) in s A complete description of the graph is thus given by the pairs (s ~ ) or (s B). In the case s = Z d, the edges will usually be drawn between lattice sites of distance one; hence x ~ y whenever Ix  Yl = 1. Here, I 9I stands for the sumnorm, i.e. Ixl  ~/d_l Ixil whenever x  (xl . . . . . xd) C Z d. This choice is natural because then Ix  Y l coincides with the graphtheoretical (or lattice) distance, viz. the length of the shortest path (consisting of consecutive edges) connecting x and y. For convenience, we sometimes use the same notation in the case of a general graph. On Z d we will occasionally distinguish between the standard metrics d l ( x , y)   Z i Ixi  Yil, d2(x, y) = [zid=l ( x i  yi)2)] 1/2 and de~(x, y) = max/Ixi  Y i l . Given any metric d on s we write d ( A , A) = infxeA,y~zX d(x, y) for the distance of two subsets A, A C s We will always assume that the graph (s ~ ) is locally finite, which means that each x 6 s has only a finite number Nx of nearest neighbors. In other words, Nx is the number of edges emanating from x. Nx is also called the degree of the graph at x. In many cases we will even assume that (s ~ ) is of bounded degree, which means that N  SUPx~z; Nx < e~. Common examples of such graphs, besides Z d, are the triangular lattice in two dimensions, and the regular tree Td (also known as the Cayley tree or the Bethe lattice), which is defined as the (unique) infinite connected graph containing no circuits in which every vertex has exactly d + 1 nearest neighbors. A region of the lattice, that is a subset A C s is called finite if its cardinality IAl is finite. We write E for the collection of all finite regions. The complement of a region A will be denoted by A c = s \ A. The boundary 0 A of A is the set of all sites (vertices) in A c which are adjacent to some site of A.
1 Random geometry of equilibrium phases
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On some occasions we will need the notion of thermodynamic (or infinite volume) limit, and we need to describe in what sense a region A ~ ,5' grows to the full lattice s For our purposes, it will, in general, be sufficient to take an arbitrary increasing sequence (An) with [.Jn>__l An = / ~ . In the case/2 = Z a, we will often make the standard choice An  [  n , n] a N Z a, the lattice cubes centered around the origin. As ,5' is a directed set ordered by inclusion, we will occasionally also consider the limit along E. In each of these cases we will use the notation A 1" s
2.2 Configurations The constituents of our systems are the spins or particles at the lattice sites. So, at each site x 6 s we have a variable cr (x) taking values in a nonempty set S, the state space or singlespin space. In a magnetic setup (to which we mostly adhere for simplicity), cr (x) is interpreted as the spin of an elementary magnet at x. In a lattice gas interpretation, there is a distinguished vacuum state 0 6 S representing the absence of any particle, and the remaining elements correspond to the types and/or the number of the particles at x. Unless stated otherwise, we will always assume that S is finite. Elements of S will typically be denoted by a, b . . . . . A configuration is a function cr : s + S which assigns to each vertex x 6 s a spin value cr(x) 6 S. In other words, a configuration cr is an element of the product space S2 = S s f2 is called the configuration space and its elements are usually written as cr, r/, ~ . . . . . (It is sometimes useful to visualize a, b . . . as colors. A configuration is then a coloring of the lattice.) A configuration cr is constant if for some a ~ S, cr (x)  a for all x ~ / 2 . Two configurations cr and r/ are said to agree on a region A C s written as "or _= q on A", if or(x) = q(x) for all x c A. Similarly, we write "cr  77 off A" if cr(x) = O(x) for all x r A. We also consider configurations in regions A C s These are elements of S A, again denoted by letters like cr, 77, ~ . . . . . Given o, ~ 6 f2, we write crA r/Ac for the configuration ~ 6 f2 with ~(x) = or(x) for x 6 A and ~(x) = O(x) for x 6 A c. Then, obviously, ~  cr on A. The cylinder sets A/'A (cr) = {~ ~ S2 :~  cr on A}, A 6 E, form a countable neighborhood basis of cr 6 f2; they generate the product topology on f2. Hence, two configurations are close to each other if they agree on some large finite region, and a diagonalsequence argument shows that f2 is compact in this topology. We will often change a configuration cr ~ f2 at just one site x ~ / 2 . Changing or(x) into a prescribed value a E S we obtain a new configuration written crx,a. In particular, for S  {  1 , + 1} we write cry (y) _ [ cr (y) ! or(x)
for y r x for y  x
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for the configuration resulting from flipping the spin at x. We will also deal with automorphisms of the underlying lattice (s .~). Each such automorphism defines a measurable transformation of the configuration space f2. The most interesting automorphisms are the translations of the integer lattice/2 = zd; the associated translation group acting on f2 is given by Ox~r(y) = cr(x + y), y ~ Z d. In particular, any constant configuration is translation invariant. Similarly, we can speak about periodic configurations which are invariant under Ox with x in some sublattice of Z a. Later on, we will also consider configurations which refer to the lattice bonds rather than the vertices. These are elements 7/of the product space {0, 1}t3, and a bond b 6 13 will be called open if q(b) = 1, and otherwise closed. The above notations apply to this situation as well.
2.3
Observables
An observable is a real function on the configuration space which may be thought of as the numerical outcome of some physical measurement. Mathematically, it is a measurable real function on ~. Here, the natural underlying ofield of measurable events in f2 is the product ~ralgebra Y  (70) c, where .To is the set of all subsets of S..T" is defined as the smallest aalgebra on f2 for which all projections X ( x ) : f2 + S, X ( x ) ( o ) = or(x) with cr ~ f2 and x c s are measurable. It coincides with the Borel aalgebra for the product topology on f2. We also consider events and observables depending only on some region A C /2. We let .T'A denote the smallest subafield of Y containing the events A/'A(or) for o e S A and A c ,f with A C A. Equivalently, YA is the oalgebra generated by the projections X (x) with x 6 A. YA is the aalgebra of events occurring in A. An event A is called local if it occurs in some finite region, which means that A 6 .T'A for some A 6,5'. Similarly, an observable f :S2 + R is called local if it depends on only finitely many spins, meaning that f is measurable with respect to YA for some A 6 ,5'. More generally, an observable f is called quasilocal if it is (uniformly) continuous, i.e., if for all ~ > 0 there is some A 6 E such that I f ( a )  f ( $ ) l < ~ whenever ~ = cr on A. The set C ( ~ ) of continuous observables is a Banach space for the supremum norm Ilfll = sup~ If(~r)l, and the local observables are dense in it. The local events and observables should be viewed as microscopic quantities. On the other side we have the macroscopic quantities which only depend on the collective behavior of all spins, but not on the values of any finite set of spins. They are defined in terms of the tail oralgebra T = (]A~E YA c, which is also called the o'algebra of all events at infinity. Any tail event A 6 T and any T measurable observable is called macroscopic.
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As a final piece of notation we introduce the indicator function IA of an event A; it takes the value 1 if the event occurs (IA (or) = 1 if cr 6 A) and is zero otherwise.
2.4
Random fields
As the spins of the system are supposed to be random, we will consider suitable probability measures/z on (f2, .T'). Each such # is called a random field. Equivalently, the family X = (X(x), x ~ E) of random variables on the probability space (S2, 9t, #) which describe the spins at all sites is called a random field. Here are some standard notations concerning probability measures. The expectation of an observable f with respect to # is written as # ( f ) = f f d # . The probability of an event A is # ( A )  lZ(IA) = fA dlz, and we omit the set braces when A is given explicitly. For example, given any x 6 /2 and a c S we write # (X (x) = a) for # (A) with A = {or 6 f2 : o (x) = a }. Covariances are abbreviated as # ( f ; g) = # ( f g )  # ( f ) # ( g ) . Whenever we need a topology on probability measures on f2, we shall take the weak topology. In this (metrizable) topology, a sequence of probability measures #n converges to #, denoted by #n ~ #, if #n(A) + #(A) for all local events A E U A~s f A . This holds if and only if lZn(f) + # ( f ) for all local, or equivalently, all quasilocal functions f . In applications, #n will often be an equilibrium state in a finite box An tending to s as n + c~, and we are interested in whether the probabilities of events occurring in some fixed finite volume have a welldefined thermodynamic (or bulk) limit. That is, we observe what happens around the origin (via the local function f ) while the boundary of the box in which we realize the equilibrium state receeds to infinity. As there are only countably many local events, one can easily see by a diagonalsequence argument that the set of all probability measures on f2 is compact in the weak topology.
2.5
The Hamiltonian
We will be concerned with systems of interacting spins. As usual, the interaction is described by a Hamiltonian. As the spins are located at the sites of a graph (s ~ ) , it is natural to consider the case of homogeneous neighbor potentials. (We will deviate from homogeneity in Section 9 when considering random interactions.) The Hamiltonian H then takes the form H(o) = ~ x~y
U(o(x), ~r(y)) + ~
V(cr(x))
(2.5.1)
x
with a symmetric function U : S • S + R U {~}, the neighborinteraction, and a selfenergy V : S + R. The infinite sums are formal; the summation index
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x ~ y means that the sum extends over all bonds (xy) ~ 13 of the lattice. U thus describes the interaction between spins at neighboring sites, while V might come from the action of an external magnetic field. In a lattice gas interpretation when S = {0, 1 } (the value 1 being assigned to sites which are occupied by a particle), V corresponds to a chemical potential. To make sense of the formal sums in (2.5.1) we compare the Hamiltonian for two different configurations ~r, r/ 6 g2 which differ only locally (or are "local perturbations" or "excitations" of each other), in that cr = r/off some A 6 g. For such configurations we can define the relative Hamiltonian H(cr It/)  ' ~ [ U ( c r ( x ) ,
cr (y)) 
U(rl(x), r/(y))] + Z [ V ( c r ( x ) )
x"~y
 V(r/(x))]
x
(2.5.2) in which the sums now contain only finitely many nonzero terms: the first part is over those neighbor pairs (xy) for which at least one of the sites belongs to A, and the second part is over all x E A.
2.6
Gibbs measures
Gibbs measures are random fields which describe our physical spin system when it is in macroscopic equilibrium with respect to the given microscopic interaction at a fixed temperature. Here, macroscopic equilibrium means that all parts of the system are in equilibrium with their exterior relative to the prescribed interaction and temperature. So it is natural to define Gibbs measures in terms of conditional probabilities. Definition 2.1 A probability measure lz on the configuration space f2 is called a Gibbs measure for the Hamiltonian H in (2.5.1) or (2.5.2) at inverse temperature ~ 1 / T iffor all A E g and all cr E f2, # ( X = o" on AIX ~ r/off A) /Z~,A(Cr)
(2.6.1)
for #almost all q c f2. In the above, #~,A (or) is the BoltzmannGibbs distribution in A f o r fi and H, which is given by
U~,A (or) 
I{~_~ off A} exp[fiH(alq)]. ZA (fi, ~)
(2.6.2)
Here, ZA (13, q) is a normalization constant making lz~, A a probability measure, and the constraint that cr has to coincide with rl outside A is added because we want to realize these probability measures immediately on the infinite lattice. Note that/zfi,A~ only depends on the restriction of rl to A c.
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So, # is a Gibbs measure if it has prescribed conditional distributions inside some finite set of vertices, given that the configuration is held fixed outside, and these conditional distributions are given by the usual BoltzmannGibbs formalism. This definition goes back to the work of Dobrushin (1968a) and Lanford and Ruelle (1969), whence Gibbs measures are often called DLRstates. By this work, equilibrium statistical physics and the study of phase transitions made firm contact with probability theory and the study of random fields. A thermodynamic justification of this definition can be given by the variational principle, which states that (in the case s = Z d) the translation invariant Gibbs measures are precisely those translation invariant random fields which minimize the free energy density (Lanford and Ruelle, 1969; Ellis, 1985; Georgii, 1988). For better distinction, the Gibbs distributions/z~,A are often called finite volume Gibbs distributions, whereas the Gibbs measures are sometimes specified as infinite volume Gibbs measures. We write G(fiH) for the set of all Gibbs measures with given Hamiltonian H and inverse temperature ft. In the special case U  0 of no interaction, there is only one Gibbs measure, namely the product measure with onesite marginals /z(X(x)  a)  eflV(a)/ Y~bES e~V(b). In general, several Gibbs measures for the same interaction and temperature can coexist. This is the fundamental phenomenon of nonuniqueness of phases which is one of our main subjects. We return to this point in Section 2.7 below. First we want to emphasize an important consequence of our assumption that the underlying interaction U involves only neighbor spins. Due to this assumption, the Gibbs distribution/zt~,A~ only depends on the restriction r/~A of r/to the boundary OA of A, and this implies that each Gibbs measure/z E G(fiH) is a Markov random field. By definition, this means that for each A 6 ~c and a c S A /z(X = a on AI~A~) : / z ( X ~ a on AI~0A)
(2.6.3)
/zalmost surely. This Markov property will be an essential tool in the geometric arguments to be discussed in this review. There is, in fact, an equivalence between Markov random fields and Gibbs measures for nearest neighbor potentials, see Grimmett (1973); Averintsev (1975) or Georgii (1988). As an aside, let us comment on the case when the interaction of spins is not nearestneighbor but only decays sufficiently fast with their distance. The BoltzmannGibbs distributions in (2.6.2), and therefore also Gibbs measures, can then still be defined, but the Gibbs measures fail to possess the Markov property (2.6.3). Rather their local conditional distributions/z~,A~ satisfy a weakening of the Markov property called quasilocality or almostMarkov property: for every A c ~c and A c .TA,/z~,A (A) is a continuous function of r/. So, in this case, Gibbs measures have prescribed continuous versions of their local conditional probabilities. To obtain a sufficiently general definition of Gibbs measures including this
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and other cases, one introduces the concept of a specification G = (GA, A 6 s This is a family of probability kernels GA from (~, 9rAc) to (f2, .T). GA(, 0) stands for any distribution of spins with fixed configuration r/Ac 6 S s outside A; the standard case is the Gibbs specification GA(., r/)  #~,A A Gibbs measure is then a probability measure # on f2 satisfying #(AlgAe)  GA (A,) /zalmost surely for all A c ,5" and A 6 ~ ; this property can be expressed in a condensed form by the invariance equation/zGA = / z . In order for this definition to make sense the specification G needs to satisfy a natural compatibility condition for pairs of volumes A C A' expressing the fact that if the system in A' is in equilibrium with its exterior, then the subsystem in A is also in equilibrium with its own exterior. It is easy to see that the Gibbs distributions in (2.6.2) are compatible in this sense. Details and further discussion can be found in many books and articles dealing with mathematical results in equilibrium statistical mechanics, (Preston, 1974a; Ruelle, 1978; Israel, 1979; Sinai, 1982; Georgii, 1988; van Enter et al., 1993). Liggett (1985) explains the relation between Gibbs measures and the condition of detailed balance (reversibility) in certain stochastic dynamics. The state of Statistical Mechanics just before the introduction of Gibbs measures is described in Ruelle (1969). Finally, we mention an alternative and constructive approach to the concept of Gibbs measures. Starting from the finitevolume Gibbs distributions/z~,A, one might ask what kind of limits could be obtained if r/is randomly chosen and A increases to the whole lattice s (This slightly older but still important approach was suggested by Minlos (1967)). To make this precise we consider the measures /z~,A  f/z~,AP(do) , where p is any probability measure on f2 describing a "stochastic boundary condition". Any such/z/~,AP is called a (finite volume) Gibbs distribution with respect to H at inverse temperature/3, and their collection is denoted by (TA(fill). The set of all (infinite volume) Gibbs measures is then equal to
G(fiH)  ("] ~A(flH). Equivalently, a probability measure # on f2 is a Gibbs measure for the Hamiltonian/3 H if it belongs to the closed convex hull of the set of limit points of #/~,A asA ?s One important consequence is that G(fiH) (= 0. This is because each GA (fl H) is obviously nonempty and compact. Equivalently, to obtain an infinite volume Gibbs measure one can fix a particular configuration r/ and take it as boundary condition. By compactness, we obtain an infinite volume Gibbs measure #~ by taking the limit of (2.6.2) as A 1' s at least along suitable subsequences; for details see Preston (1974a) or Georgii (1988). We remark that, in general, there is no unique limiting measure #~; rather there may be several such limiting measures obtained as limits along different subsequences.
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Fortunately, however, this is not the case for a wide class of models, either at low temperatures (/~ large) when ~ is a ground state configuration (in the realm of the PirogovSinai theory) (Pirogov and Sinai, 1976), or at high temperatures when g is small. We conclude this subsection with a general remark. As all systems in nature are finite, one may wonder why we consider here systems with infinitely many constituents. The answer is that sharp results for bulk quantities can only be obtained when we make the idealization to an infinite system. The thermodynamic limit eliminates finite size effects (which are always present but which are not always relevant for certain phenomena) and it is only in the thermodynamic limit of infinite volume that we can get a clean and precise picture of realistic phenomena such as phase transitions or phase coexistence. This is a consequence of the general probabilistic principle of large numbers. In this sense, infinite systems serve as an idealized approximation to very large finite systems.
2.7
Phase transition and phases
As pointed out above, in general there may exist several solutions # to the DLRequation (2.6.1) for given U, V and g, which means that multiple Gibbs measures exist. The system can then choose between several equilibrium states. (In a dynamical theory this choice would depend on the past; but here we are in a pure equilibrium setting.) The phenomenon of nonuniqueness therefore corresponds to a phase transition. In fact, it is then possible to construct different Gibbs measures as infinite volume limits of Gibbs distributions with different choices of boundary conditions (Georgii, 1973, 1988). Since any two Gibbs measures can be distinguished by a suitable local observable, a phase transition can be detected by looking at such a local observable which is then called an order parameter. Varying the external parameters such as temperature or an external magnetic field (which can be tuned by the experimenter) one will observe different scenarios; these are collected in the socalled phase diagram of the considered system. As we have indicated in the introduction, the phase transition phenomenon is of central interest in equilibrium statistical mechanics. When phase transitions occur and when they do not is also one of the primary questions (although we will encounter many others) that we will try to answer with the geometric methods to be developed in subsequent sections. If multiple Gibbs measures for a given interaction exist, the structure of the set ~(g H) of all Gibbs measures becomes relevant. We only state here the most basic results; a detailed exposition can be found in Georgii (1988), for example. The basic observation is that G ( g H ) is a convex set. Its extremal elements, the extremal Gibbs measures, have a trivial tail ofield 7 (which means that all events in 7" have probability 0 or 1). Equivalently, all macroscopic observables
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are almost surely constant. In addition, the tail triviality can be characterized by an asymptotic independence (or mixing) property. On the other hand, any Gibbs measure # can be decomposed into extremal Gibbs measures; therefore every configuration which is typical for # is in fact typical for some extremal Gibbs measure. This shows that the extremal Gibbs measures correspond to what one can really see in nature as far as large systems in equilibrium are concerned. The extremal Gibbs measures therefore correspond to the physical macrostates, whereas nonextremal Gibbs measures only provide a limited description when the system's precise state is unknown. For all these reasons, the extremal Gibbs measures are called (equilibrium) phases. The central subject of this review is the geometric analysis of their typical configurations, and thereby the analysis of the phase diagram giving the variation in the number and the nature of the phases as one changes various control parameters (coupling, temperature, external fields, etc.). Often it is natural to consider automorphisms of the graph (/2, ~). For example, if s  Z d we consider the translation group (Ox)xcZd . A homogeneous phase is then an extremal Gibbs measure which is also translation invariant. On the other hand, we can regard the extremal points of the convex set of all translation invariant Gibbs measures. These are ergodic, which means that they cannot be decomposed into distinct translation invariant probability measures, and are trivial on the aalgebra of all translation invariant events. However, these extremal translation invariant Gibbs measures need not be homogeneous phases; they are only ergodic. Yet, ergodic measures # satisfy a law of large numbers: for any observable f and any sequence of increasing cubes A, 1
lim ~" f o Ox  # ( f ) al, za IAI
/zalmost surely.
Hence, ergodic Gibbs measures are suitable for modelling macrostates in equilibrium if one limits oneself to measuring certain bulk observables or macroscopic quantities with additivity properties. Notice, however, that there exists a certain nonuniformity in the literature concerning the nomenclature. Sometimes these ergodic Gibbs measures are called (pure) phases. It is then argued that it might happen that two phases (as defined above) for a system can by no means be macroscopically distinguished (for example if one is a translation of the other). We do not wish to enter into a detailed discussion of these points.
3
S o m e models
In this section we discuss briefly the phase transition behaviour of some prototypical examples of Gibbs systems. Although these examples are fairly standard and
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wellknown to most of our readers, we need to include them here to set the stage. They will be studied in detail in the later sections. An account of phase transition phenomena in more general lattice models can be found in many other sources, including Sinai (1982); Georgii (1988); Simon (1993); MiracleSole (1994) and Koteck2~ (1995).
3.1 The ferromagnetic Ising model The Ising model was introduced by Wilhelm Lenz (1920) and his student Ernst Ising (1925) as a simple model for magnetism and, in particular, ferromagnetic phase transitions. Each site x 6 /2 can take either of two spin values, +1 ("spin up") and  1 ("spin down"), so that the state space is equal to S  {1, +l}. The Hamiltonian is given by (2.5.1) with U ( ~ ( x ) , ~ ( y ) )   ~ ( x ) ~ ( y ) and V ( ~ ( x ) )   h c ~ ( x ) . The parameter h 6 R describes an external field. The finite volume Gibbs distribution in a box A with external field h at inverse temperature/3 > 0 with boundary condition r/ is thus the probability measure #~,r on ~2  {1, + 1}s which to each ~ 6 ~2 assigns probability proportional to
x~AoryEA
For /3  0 ("infinite temperature") the spin variables are independent under lZh,~, A , ~ but as soon as/3 > 0 the probability distribution starts to favour configurations with many neighbor pairs of aligned spins. This tendency becomes stronger and stronger as/3 increases. In the case h  0 of no external field, the model is symmetric under interchange of the spin values  1 and + 1, so that there is an equal chance of having many pairs of plus spins or having many pairs of minus spins. This dichotomy gives rise to the following interesting behavior. Suppose that s  Z a, d > 2. If /3 is sufficiently small (i.e., in the high temperature regime), the interaction is not strong enough to produce any long range order, so that the boundary conditions become irrelevant in the infinite volume limit and the Gibbs measure is uniquely determined. By ergodicity and the t symmetry, the limiting fraction of plus spins will almost surely be 1/2 under this unique Gibbs measure. In contrast, when/3 is sufficiently large (in the lowtemperature regime), the interaction becomes so strong that a longrange order appears: the bias towards neighbor pairs of equal spin then implies that Gibbs measures prefer configurations with either a vast majority of plus spins or a vast majority of minus spins, and this preference even survives in the infinite volume limit. The system thus undergoes a phase transition which manifests itself in a nonuniqueness of Gibbs measures. Specifically, there
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exist two particular Gibbs measures # + and tz, obtained as infinite volume limits with respective boundary conditions 77 =_ § 1 and r/ =  1 , which can be distinguished by their overall density of + 1s: the density is greater than 1/2 under # + and (by symmetry) less than 1/2 under #  . This is the spontaneous magnetization phenomenon that Lenz and Ising were looking for but were discouraged by not finding it in one dimension. In higher dimensions, the uniqueness regime and the phase transition regime are separated by a sharp critical value/3c, as is summarized in the following classical theorem (Peierls, 1936; Dobrushin, 1965, 1968b): Theorem 3.1 For the ferromagnetic Ising model on the integer lattice Z a of dimension d > 2 at zero external field, there exists a critical inverse temperature ~c E (0, cx~) (depending on d) such that for ~ < ~c the model has a unique Gibbs measure while for ~ > ~c there are multiple Gibbs measures. A stochasticgeometric proof of this result will be given in Section 6. The result (as well as its proof) holds for any graph (s ,~) in place of Z a, except that/~c may then take the values 0 or cx~. For instance, on the onedimensional lattice Z 1 we have/~c  cx~, which means that there is a unique Gibbs measure for all /~. For Z 2, the critical value has been found to be/~c  89log(1 + ~/2). This calculation is a remarkable achievement which began with Onsager (1944). An account of various (algebraic and/or combinatorial) methods can be found in Thompson (1972) and McCoy and Wu (1973). Let us also mention the work done by Abraham and MartinL6f (1973) relating these exact computations to the real magnetization in the appropriate Gibbs measures; it also gives the result that there is a unique Gibbs measure at the critical value/~ =/~c. A rigorous calculation of the critical value in higher dimensions is beyond current knowledge. It is believed that uniqueness holds at criticality in all dimensions d > 2, but so far this is only known for d  2 and d > 4 (Aizenman and Fernandez, 1986). The case of a nonzero external field h ~ 0 is less interesting, in that one finds a unique Gibbs measure for all/~ and d. The intuitive explanation is that for h ~: 0 there is no 4 symmetry which could be broken; depending on the sign of h, the system is forced to prefer either § l s or  l s . This comes from the fact that the magnetic field acts on the whole volume, whereas the influence of a boundary condition is of smaller order as the volume increases. In contrast, a phase transition for h ~ 0 does occur when Z d is replaced by certain nonamenable graph structures for which the boundary of a volume is of the same order of magnitude as the volume itself (which makes them physically perhaps less realistic)  an example is the regular tree Td with d >_ 2; we refer to Spitzer (1975); Georgii (1988) and Jonasson and Steif (1999). A phase transition can also occur for a nonzero external field for the Ising model on a halfspace where it is due to the socalled Basuev phenomenon (Basuev, 1984, 1987).
1 Random geometry of equilibrium phases
17
Because of the simplicity of its model assumptions, the standard Ising model has inspired a variety of techniques for analyzing interacting random fields. Its ferromagnetic structure suggests various monotonicity properties which can be checked by the coupling methods to be described in Section 4, and the assumption of neigbor interaction implies the spatial Markov property (2.6.3) which plays a fundamental role in the geometric analysis of typical configurations. Many techniques which were developed on this testing ground also turned out to be fruitful in more general cases.
3.2 The antiferromagnetic Ising model The Ising antiferromagnet is defined quite similarly to the ferromagnetic case, except that U(a(x), a(y)) is taken to be +a(x)a(y) rather than  a ( x ) a ( y ) . This means that neighboring sites now prefer to take opposite spins. Suppose that h = 0 and that the underlying graph is bipartite. This means that/2 can be partitioned into two sets l~even and 12odd such that sites in ~2even only have edges to sites in ~2odd, and vice versa. Clearly, Z a is an example of a bipartite graph. In this situation, we can reduce the antiferromagnetic Ising model to the ferromagnetic case by a simple spinflipping trick: The bijection a ~ 6 of f2 defined by
6(x)
[ a(x) I  a (x)
if x E ~,even, if x ~ s
(3.2.1)
maps any Gibbs measure for the antiferromagnetic Ising model to a Gibbs measure for the ferromagnetic Ising model with the same parameters, and vice versa. As a consequence, a phase transition in the antiferromagnetic model is equivalent to a phase transition in the ferromagnetic model with the same parameters. Hence, Theorem 3.1 immediately carries over to the antiferromagnetic case. The model becomes more interesting (or, at least, more genuinely antiferromagnetic) if either h ~: 0 or the graph is taken to be nonbipartite. Suppose first that h r 0 but still/2 = Z d. If Ihl is small and fl sufficiently large, we have the same picture as in the case h  0: there exist two distinct phases, one having a majority of plus spins on the even sublattice and a majority of minus spins on the odd sublattice, the other one having a majority of plus spins on the odd sublattice and a majority of minus spins on the even sublattice. We will show this in Section 8.5, Example 8.17; see also Dobrushin (1968b) and Georgii (1988). Note that this phase transition is somewhat different in flavor compared to that in the ferromagnetic Ising model: whereas in the Ising ferromagnet the phase transition produces a breaking of a statespace symmetry, the phase transition in the Ising antiferromagnet instead breaks the translation symmetry between the sublattices ~even and 12odd.
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To see what happens in the case of a nonbipartite graph we consider the triangular lattice which can be obtained by taking the usual square lattice Z 2 and adding an edge between each vertex x and its northeast neighbor x + (1, 1). In this case, one expects uniqueness when h  0, and existence of three distinct phases when Ihl # 0 is small and fl is large. Phase transitions in these models were studied by Dobrushin (1968b) and Heilmann (1974), for example.
3.3
The Potts model
A natural generalization of the ferromagnetic Ising model is the (ferromagnetic) Potts model (Potts, 1952), in which each spin may take q >_ 2 (rather than only two) different values. The state space is then S = {1, 2 . . . . . q}, and the pair interaction is given by U(cr(x),
~(y))
= 1  2I{,~(x)=cr(y)}.
We confine ourselves to the case of zero external field, so that V(cr (x))  0. Taking q = 2 and identifying the state space {1, 2} with {1, + 1} we reobtain the ferromagnetic Ising model with zero external field. Just as in the latter case, the Potts interaction favours configurations where many neighbor pairs agree, and Theorem 3.1 can be extended to the Potts model as follows, as we will show in Section 6.3. Theorem 3.2 For the qstate Potts model on Z d, d > 2, there exists a critical inverse temperature fie ~ (0, e~) (depending on d and q) such that f o r fl < fie the model has a unique Gibbs measure while f o r fl > fie there exist q mutually singular Gibbs measures. In the same way as Theorem 3.1, this theorem also holds on general graphs provided we allow fie to be 0 or ec. The Potts model differs from the Ising model in that, for q large enough, there are also multiple Gibbs measures at the critical value fi  fie, as demonstrated by Koteck3) and Shlosman (1982); an outline of a proof will be given in Example 8.21. The Onsager critical value for the twodimensional Ising model is believed to extend to the Potts model on Z 2 through the formula fie(q)  89log(1 + x/if); see Welsh (1993), for example. This has so far only been established when q is sufficiently large (Lannait et al., 1991).
3.4
The hardcore lattice gas model
The hardcore lattice gas model (or hardcore model for short) describes a gas of particles which can only sit on the lattice sites but are so large that adjacent
1
Random geometry of equilibrium phases
19
sites cannot be occupied simultaneously. The state space is S  {0, 1}, the pair interaction oo i f a ( x ) = a ( y ) = 1 U ( a ( x ) , a ( y ) )  0 otherwise, describes the hard core of the particles, and the chemical potential is V ( a ( x ) ) = (log)~) a ( x ) . Here Z > 0 is the socalled activity parameter. The hardcore model shows some similarities to the Ising antiferromagnet in an external field and can be obtained from it by a limiting procedure (/3 + c~, h + 2d, fl(h  2d) = const., (Dobrushin et al., 1985)). Since U is either 0 or c~, the inverse temperature/3 is irrelevant and will thus be fixed as 1, and we can vary only the parameter ~.. Finite volume Gibbs distributions can then be thought of as first letting all spins be independent, taking values 0 and 1 with respective 1 )~ and then conditioning on the event that no two l s sit probabilities 1~ and 1~, next to each other anywhere on the lattice. The phase transition behavior of the hardcore model on Z a, d > 2, is as follows. For ~. sufficiently close to 0, the particles are spread out rather sparsely on the lattice, and we get a unique Gibbs measure, just as in the Ising antiferromagnet at high temperatures. When ~. increases, the particle density also increases, and the system finally starts looking for optimal packings of particles. There are two such optimal packings, one where all sites in l~even a r e occupied and those in s are empty, and one vice versa. We denote these configurations by r]even and florid, respectively. (These chessboard configurations look similar to those favoured in the Ising antiferromagnet.) For sufficiently large ,k, the infinite volume construction of Gibbs measures with these two choices of boundary condition produces different Gibbs measures, so we get a phase transition (Dobrushin, 1968b). T h e o r e m 3.3 For the hardcore model on Z d, d >_ 2, there exist two constants 0 < )~c < Uc < cx~ (depending on d) such that f o r )~ < )~c the model has a unique Gibbs measure while f o r )~ > )~'c there are multiple Gibbs measures.
This result will be proved in Section 6.7. From a computerassisted proof (Radulescu, 1997) we know that ~.c > 1.507 62. It is widely believed that one should be able to take ~.c  Uc in this result, which would mean that the occurrence of phase transition is increasing in ~. Such a result, however, would (unlike Theorems 3.1 and 3.2) not extend to arbitrary graph structures; some counterexamples were recently provided by Brightwell et al. (1999). The hardcore model analogue of introducing an external field in the Ising model on Z d is obtained by replacing the single activity parameter )~ by two different activities ~.even and )~odd, one for sites in s and the other for sites By analogy with the Ising model, one would expect to have a unique in s Gibbs measure as soon a s ~,even ~ ~.odd; this was conjectured by van den Berg and Steif (1994) and proved for the case d  2 by H~iggstr6m (1997a).
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The WidomRowlinson lattice model
The WidomRowlinson model is another lattice gas model, where this time there are two types of particles, and two particles are allowed to sit on neighboring sites only if they are of the same type. Actually, Widom and Rowlinson (1970) originally introduced it as a continuum model of particles living in Rd; see Section 10.2 below. The lattice variant described here was first studied by Lebowitz and Gallavotti (1971). The state space is S = {  l, 0, + 1}, where  1 and + 1 are the two particle types, and 0s correspond to empty sites. The pair interaction is given by zx3 if ~ ( x ) ~ ( y )   1 , U(~(x), ~(y)) 0 otherwise, and the chemical potential by
logT._ V (t7 (x)) =
0 log)~+
if o ( x )   1 , if if
~ (x)  0, ~(x)+l.
Here k_, k+ > 0 are the activity parameters for the two particle types  1 and + 1. As in the hardcore model, we fix the inverse tempreature/3 = 1 and only vary the activity parameters. Gibbs measures can then be thought of as first picking all spins independently, taking values  l, 0 or + 1 with probabilities proportional to X_, 1, and )~+, and then conditioning on the event that no two particles of different type sit next to each other in the lattice. We are mainly interested in the symmetric case 7._  )~+  k, where the phase transition behavior on Z d, d >_ 2 is similar to the Ising model. For )~ small, there is a unique Gibbs measure in which the overall density of plus particles is almost surely equal to that of the minus particles. For 7. sufficiently large, the system wants to pack the particles so densely that the tl symmetry is broken. As for the Ising model, one can construct two particular Gibbs measures # + and #  using boundary conditions 7/ + 1 and r/  l ' f o r small )~ we get # + = # _ whereas for large 7. the two measures are different (and distinguishable through the densities of the two particle types), producing a phase transition. T h e o r e m 3.4 For the WidornRowlinson model on Z d, d >_ 2, with activities )~_  )~+ = 7, there exist 0 < )~c )~tc there are multiple Gibbs measures. As in the hardcore model, we expect that one should be able to take ~.c  ~/c, but such a monotonicity is not known. Examples of graph structures where the desired monotonicity fails can be found in Brightwell et al. (1999). We furthermore expect that the asymmetric WidomRowlinson model on Z d with 7._ ~= )~+ always has a unique Gibbs measure (similarly to the Ising model with a nonzero external field), but this also is not rigorously known.
1 Random geometry of equilibrium phases
4
21
Coupling and stochastic domination
Geometry alone will not be sufficient for our analysis of equilibrium phases. We also need some probabilistic tools which allow us to compare different configurations and different probability measures. So we need to include another preparatory section describing these tools and their basic applications to our setting. Coupling is a probabilistic technique which has turned out to be immensely useful in virtually all areas of probability theory, and especially in its applications to statistical mechanics. The basic idea is to define two (or more) stochastic processes jointly on the same probability space so that they can be compared realizationwise. This direct comparison often leads to conclusions which would not be easily available by considering the processes separately. Although an independent coupling is sometimes quite useful (as we will see in Section 7.3, for example), it is usually more efficient to introduce a dependence which relates the two processes in an efficient way. One such particularly good relationship is that one process is pointwise smaller than the other in some partial order. This case is related to the central concept of stochastic domination, via Strassen's theorem (Theorem 4.6) below. We will confine ourselves to those parts of coupling theory that are needed for our applications; a more general account can be found in the monograph by Lindvall (1992).
4.1 The coupling inequality In this and the next subsection of general character, 12 will be an arbitrary finite or countably infinite set. As the notation indicates, we think of the standard case that 12 is the lattice introduced in Section 2.1, but the following results will also be applied to the case when 12 is replaced by its set B of bonds. We consider again the product space S2 = S z;, where for the moment S is an arbitrary measurable space. Suppose X and X' are random elements of f2, and let # and #I be their respective distributions. We define the (half) total variation distance lilt  #'11 between # and #I by
I1~ ~'11 =
sup I#(A)  #'(A)I
(4.1.1)
ACf2
where A ranges over all measurable subsets of f2. The coupling inequality below provides us with a convenient upper bound on this distance. To state it we first need to define what we mean by a coupling of X and X'. Definition 4.1 A coupling P of two Qvalued random variables X and X', or of their distributions tt and lz t, is a probability measure on f2 x f2 having marginals
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H.O. Georgii et al.
# and #~, in that for every event A C f2 P((~, ~ ' ) ' ~ E A)  #(A)
(4.1.2)
P((~, ~') 9~' E A) = #t(A).
(4.1.3)
and
We think of a coupling as a redefinition of the random variables X and X' on a new common probability space such that their distributions are preserved. Sometimes it will be convenient to keep the underlying probability space implicit, but in general, as in (4.1.2) and (4.1.3), we make the canonical choice, which is the product space t2 • t2; X and X' are then simply the projections on the two coordinate spaces. With this in mind, we write P ( X E A) and P(X' E A) for the lefthand sides of (4.1.2) and (4.1.3), respectively. In the same spirit, P (X = X') is a shorthand for P((~, ~') : ~ = ~').
Proposition 4.2 (The coupling inequality) Let P be a coupling of two f2valued random variables X and X ~, with distributions # and #'. Then
II~ ~'ll ~ P(X #
(4.1.4)
X').
Proof: For any A E ~2, we have #(A)  #'(A)

P(X E A)
=
P(XEA,
P ( X ' E A)
X'r
tEA)
_ ~ p ( a N ) ~ p ( B N ) > O. [] Proof:
Note that the proof above applies to the much broader class of all measures with positive correlations (recall Definition 4.10), rather than only the Bernoulli measures. Next we ask whether both possibilities in Proposition 5.2 really occur, that is, if 0 < Pc < 1. For, only in this case we really have a nontrivial critical phenomenon at Pc. The answer depends on the graph. For/2  Z d with dimension d >_ 2, the threshold Pc is indeed nontrivial, as is stated in the theorem below. This nontriviality of Pc is a fundamental ingredient of many of the stochasticgeometric arguments employed later on. On the other hand, it is easy to see that pc  1 for/2  Z1. 5.3 The critical value Pc d > 2, satisfies the inequalities
p c ( d ) f o r site percolation on s 
Theorem
1
2d
1
6 < Pc c~} we have Nk _ 1 for each k, whence
O(p) < ~pp(Nk).
(5.1.3)
The number of all paths of length k starting at 0 is at most 2d(2d  1)kl, and each path is open with probability pk. Hence
7zp(Nk) < 2 d ( 2 d  1)klp k which tends to 0 as k + cxz whenever p < 1 / ( 2 d  1). In combination with (5.1.3) this implies that O(p) = 0 for p < 1/(2d  1), and the first half of (5.1.2) is established. The second half of (5.1.2) only needs to be proved for d = 2; this is because Z 2 c a n be embedded into Z d for any d >_ 2, so that pc(d) 2/3, then an infinite cluster exists with positive /zprobability. Imagine the following allocation of mass to the edges of T2. Originally every edge receives mass 1. Then the mass is redistributed, or transported, as follows. If an edge e is open and is contained in a finite open cluster, then it distributes all its mass equally among those closed edges that are adjacent to the open cluster containing e. If e is open and contained in an infinite open cluster, then it keeps its mass. Closed edges, finally, keep their own mass and happily accept any mass that open edges decide to send them. The expected mass at each edge before transport is obviously 1, and one can show B this is an instance of the masstransport principle (Benjamini et al., 1999) B that the expected mass at a given edge is also 1 after the transport. Suppose now, for contradiction, that p(/z) > 2/3 and that all open clusters are finite/za.s. Then all open edges have mass 0 after transport. Furthermore, since each open cluster containing exactly n edges has exactly n + 3 adjacent closed edges (as is easily shown by induction ~ it is here that the tree structure is used), the mass after transport at a closed edge adjacent to two open clusters of sizes n l and n2 has mass nl n2 1~ ~ Pc, and the absence of infinite clusters will follow if the model at hand is stochastically dominated by the Bernoulli model for some p < Pc. Let us demonstrate this technique for the Ising model on Z d.
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Consider percolation of plus spins in the plus measure #h,t~' + defined in Section 4.3. If we keep I3 fixed then Proposition 4.16 tells us that #h,~+ is stochastically increasing in h. Consequently, both the probability of having an infinite cluster of plus spins, as well as the probability that a given vertex is in such an infinite cluster, are increasing in h. Furthermore, as Z d is of bounded degree N 2d, the remarks after the same proposition imply that, for any given p e (0, 1) + stochastically dominates the Bernoulli measure and fl, the Ising m e a s u r e ~h,fl lpp when h is large enough, and is dominated by Op for h below some bound. We may combine this observation with Proposition 5.1 to deduce the following critical phenomenon: Theorem 5.10 For the Ising model on Z d, d > 2, at a fixed temperature fi, there exists a critical value hc ~ R (depending on d and fl) f o r the external field, such that
#h,r + (3 an infinite cluster of plus spins)  [ 01 ifif hh >< hc.hC As we shall see later in Theorem 8.2, we have hc = 0 when d > 2 and fl > tic. Higuchi (1993b) has shown that the percolation transition at hc is sharp, in that the connectivity function decays exponentially when h < hc, and that the percolation probability is continuous in (fl, h), except on the critical halfline h = O, fl > tic. In Section 6 below, we will make a similar use of stochastic comparison arguments for randomcluster measures, cf. Proposition 6.11. The stochastic domination approach works also in the framework of lattice gases with attractive potential; see Lebowitz and Penrose (1977). In the rest of this subsection we shall give some examples of strongly dependent systems where other approaches to the question of percolation are needed. Typically in these examples, the probability that all vertices in a finite region A are open (or closed) fails to decay exponentially in the volume of A, and as a consequence, the random field neither dominates nor is stochastically dominated by any nontrivial Bernoulli model. The geometry of level heights of a random field forms an important object of study both from the theoretical and the applied side. For example, it relates to the presence of hills and valleys on a rough surface, or to the random location of potential barriers in a doped semiconductor. To fix the ideas we consider a random field X  ( X ( x ) , x e Z d) with values X ( x ) e S C R which are not necessarily discrete. It is often interesting to divide S into two parts $1 and So and to define a new discrete random field Y via Y ( x )  I{x(x)eS~}. For S  R one typically considers S1 = [g, cx~) for some level g e R. In this way we obtain a coarsegrained description of a system of continuous spins. One question is to which extent one can reconstruct the complete image from this information. We consider here a different question: what is the geometry of the random set {x ~ Z d 9Y ( x )  1}? This set is called the excursion or exceedance set when it
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Random geometry of equilibrium phases
41
corresponds, as in the example above, to the set on which the original random field exceeds a given level. For a recent review of this subject we refer to Adler (2000). We now give four examples of equilibrium systems with continuous spins where one can show (the absence of) percolation of an excursion (exceedance) set. Here we only state the results. Some hints on the proofs will be given later in Section 8 via Theorem 8.1. Details can be found in the paper by Bricmont et al. (1987). E x a m p l e 5.11 Consider a general model of realvalued spins ( a ( x ) , x c Z d) with ferromagnetic nearestneighbor interaction. The formal Hamiltonian is given by (5.3.1) H ( a )   Z a ( x ) a ( y ) . x,~y
The reference (or singlespin) measure ~, ~: 30 on R is assumed to be even and to decay fast enough at q  ~ so that the model is well defined. Then, for any Gibbs measure # relative to (5.3.1) with # ( s g n ( a ( 0 ) ) ) > 0, there will be percolation of all sites x with a (x) >_ 0. Such Gibbs measures always exist at sufficiently low temperatures when d > 2. E x a m p l e 5.12 Consider again a spin system ( a ( x ) , x Hamiltonian has the 'massless' form H(a)   ~
c z d ) , where now the
7r(a(x)  a ( y ) )
x,~y
with a (x) 6 R or Z and ~ an even convex function. The singlespin measure ~ is either a Lebesgue measure on R or a counting measure on Z. The case a (x) 6 Z and ~ ( t )  It[ corresponds to the socalled solidonsolid (SOS) model of a ddimensional surface in z d + l ; the choice a ( x ) 6 R and 7r(t)  t 2 gives the harmonic crystal. Let # be a Gibbs measure which is obtained as infinite volume limit of finite volume Gibbs distributions with zero boundary condition. (In the continuousspin case, such Gibbs measures exist for any temperature when d > 3 and ~ ( t )  ottz+4~(t), where ot > 0 and ~b is convex (Brascamp and Lieb, 1976).) Then, for any s < 0, there is percolation of the sites x 6 Z d with a (x) _> s E x a m p l e 5.13 Consider next a model of twocomponent spins a (x) 6 R 2, x E Z d, a (x)  (rx cos dpx, rx sin ~bx), with formal Hamiltonian H ( a )   ~
a(x). a(y)
x~,y
and some rotationinvariant and suitably decaying reference measure )~ on R 2. Then, for any Gibbs measure # with #(cos 4~0) > 0, there is percolation of the sites x c Z d with cos qSx > 0. Such Gibbs measures exist at low temperatures if d>3.
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H.O. Georgii e t al.
Example 5.14 Consider again the massless harmonic crystal of Example 5.12 above (with 7z(t)  t 2) in d  3 dimensions. There exists a value gc < cx~ so that for all g > gc there is no percolation of sites x 6 Z d with a (x) _ ~. Finally, we give an example of a strongly correlated system, sharing some properties with the harmonic crystal of Example 5.14, where at present there is no proof of a percolation transition. The model is one of the simplest examples of an interacting particle system. What makes the problem difficult is that the random field is not Markov (not even Gibbsian) and not explicitly described in terms of a family of local conditional distributions. Example 5.15 The voter model is a stochastic dynamics in which individuals (voters) sitting at the vertices of a graph update their position (yes/no) by randomly selecting a neighboring vertex and adopting its position, see Liggett (1985) for an introduction. Using spin language and putting ourselves on Z 3, the time evolution of this voter model is specified by giving the rate c(x, ~) for a spin flip at the site x when the spin configuration is cr 6 {+1,  1 }z3, 1
Z ('y~x
There is a oneparameter family of extremal invariant measures #p each obtained asymptotically (in time) from taking the Bernoulli measure ~p with density p as the initial condition. These stationary states #p are strongly correlating. The spinspin correlations decay as the inverse 1/r of the spindistance r on Z 3. It is an open question whether for p sufficiently close to 1 the plus spins percolate, and whether for sufficiently small p there is no percolation. Simulations by Lebowitz and Saleur (1986) indicate that there is indeed a nontrival percolation transition with critical value Pc ~ O. 16. The same problem may be considered for d _> 4. For d  1, 2, however, the problem is not interesting because in these cases #p is known to put mass p on the "all + 1" configuration and mass 1  p on the "all  1 " configuration. Alternatively, one may consider the same model with Z 3 replaced by Td" Theorem 5.9 can then be applied to show that the plus spins do percolate for p > d+l 2d" Other interesting examples of dependent percolation with remarkable properties can be found in Molchanov and Stepanov (1983). 
5.4

The number of infinite clusters
Once infinite clusters have been shown to exist with positive probability in some percolation model, the next natural question is: How many infinite clusters can exist simultaneously ? For Bernoulli site or bond percolation on Z d, Aizenman et
1 Random geometry of equilibrium phases
43
al. (1987) obtained the following, now classical, uniqueness result: with probability 1, there exists at most one infinite cluster. Simpler proofs were found later by Gandolfi et al. (1988) and by Burton and Keane (1989). The argument of Burton and Keane is not only the shortest (and, arguably, the most elegant) so far. It also requires much weaker assumptions on the percolation model, namely: translation invariance and the finiteenergy condition below, which is a strong way of stating that all local configurations are really possible. Its significance for percolation theory had been discovered before by Newman and Schulman (1981).
Definition 5.16 A probability measure lz on {0, 1}s with s a countable set, is said to have finite energy if f o r every finite region A C ~, /z(X~7?onAlX~offA)>0 for all 7? ~ {0, 1}A and #a.e. ~ ~ {0, 1}AC.
Theorem 5.17 (The BurtonKeane uniqueness theorem) Let # be a probability measure on {0, 1}zd which is translation invariant and has finite energy. Then, lza.s., there exists at most one infinite open cluster.
Sketch proof: Without loss of generality we can assume that # is ergodic with respect to translations. For, one can easily show that the measures in the ergodic decomposition of # admit the same conditional probabilities, and thus inherit the finiteenergy property. Since the number N of infinite clusters is obviously invariant under translations, it then follows that N is almost surely equal to some constant k c {0, 1. . . . . ~ } . In fact, k ~ {0, 1, zx~}. Otherwise, with positive probability each of the k clusters would meet a sufficiently large cube A; by the finiteenergy property, this would imply that with positive probability all these clusters are connected within A, so that in fact k = 1, in contradiction to the hypothesis. (This part of the argument goes back to Newman and Schulman (1981).) We thus only need to exclude the case k  ~ . In this case, # ( N > 3) = 1, and the finiteenergy property implies again that # ( A x ) = 3 > 0, where Ax is the event that x is a triple point, in that there exist three disjoint infinite open paths with starting point x and that these paths would fall in three different components if the vertex x were removed. By the (norm) ergodic theorem, for any sufficiently large cubic box A we have ~(IA]I Z
lAx >_ 3 / 2 ) >
1/2.
(5.4.1)
xEA
On the other hand, for geometrical reasons (which are intuitively obvious but need some work when made precise), there cannot be more triple points in A than points in the boundary 0 A of A. Indeed, each of the three paths leaving a triple
44
H.O. Georgii et aL
point meets 0 A, which gives three boundary points associated to each triple point in A. If one identifies these boundary points successively for one triple point after the other one sees that, at each step, at least one of the boundary points must be different from those obtained before. Hence,
IAI~ ~ IAx ~ IAI]IOAI < 3/2 xEA
when A is large enough. Inserting this into (5.4.1) we arrive at the contradiction # (0) > 1/2, and the theorem is proved. For more details we refer to the original paper by Burton and Keane (1989). [] We stress that the last argument relies essentially on the amenability property of Z d discussed in Section 5.2. The finiteenergy condition is also indispensable: In another paper, Burton and Keane (1991) construct, for any k E {2, 3 . . . . . cx~}, translation invariant percolation models on Z 2 for which finite energy fails and which have exactly k infinite open clusters. For example, we have k = cx~ in Example 5.7. Fortunately, the finiteenergy condition holds in most of the dependent percolation models which show up in stochasticgeometric studies of Gibbs measures. The situation becomes radically different when Z d is replaced by the nonamenable tree Td. Instead of having a unique infinite cluster, supercritical percolation models on Td tend to produce infinitely many infinite clusters. It is not hard to verify that this is indeed the case for supercritical Bernoulli site or bond percolation (except in the trivial case when the retention probability p is 1), and a corresponding result for automorphism invariant percolation on Td can be found in H~iggstr6m (1997b). On more general nonamenable graph structures, the uniqueness of the infinite cluster property can fail in more interesting ways than on trees; see e.g. Grimmett and Newman (1990) and H~iggstr6m and Peres (1999). Let us next consider the particular case of (possibly dependent) site percolation on Z 2. We know from Theorem 5.17 that under fairly general assumptions there is almost surely at most one infinite open cluster. Under the same asumptions there is almost surely at most one infinite closed cluster (i.e., at most one infinite connected component of closed vertices). In fact, the proof of Theorem 5.17 even shows that almost surely there is at most one infinite closed ,cluster. (Here, a closed ,cluster is a maximal set C of closed sites which is ,connected, in that any two x, y E C are connected by a ,path in C; ,paths were introduced in the proof of Theorem 5.3. Any closed cluster is part of some closed ,cluster.) But perhaps an infinite open cluster and an infinite closed , cluster can coexist? Theorem 5.18 below asserts that under reasonably general circumstances this cannot happen. Under slightly different conditions (replacing the finiteenergy assumption by separate ergodicity under translations in the two coordinate directions), it was proved by Gandolfi et al. (1988).
1 Random geometry of equilibrium phases
45
Theorem 5.18 Let lz be an automorphism invariant and ergodic probability measure on {0, 1}Z2 with finite energy and positive correlations. Then
#(3 infinite open cluster, 3 infinite closed ,cluster) = 0. Note that automorphism invariance in the Z2case means that, in addition to translation invariance, # is also invariant under reflection in and exchange of coordinate axes. Under the conditions of the theorem, we have in fact some information on the geometric shape of infinite clusters. If an infinite open cluster exists and thus all closed ,clusters are finite, each finite box of Z 2 is surrounded by an open circuit, and all these circuits are part of the (necessarily unique) infinite open cluster. Hence the infinite open cluster is a sea, in the sense that all "islands" (i.e., the ,clusters of its complement) are finite. Similarly, if a closed ,cluster in Z 2 exists, it is necessarily a sea (and in particular unique). The corresponding result is false in higher dimensions. To see this, consider Bernoulli site percolation on Z 3. The critical value Pc for this model is strictly less than 1/2 (see Campanino and Russo (1985)), whence for p = 1/2 there exist almost surely both an infinite open cluster and an infinite closed cluster. The proof of Theorem 5.18 below is based on a geometric argument of Yu Zhang who gave a new proof of Harris' (1960) classical result that the critical value Pc for bond percolation on Z 2 is at least 1/2. (Recall that this bound is actually sharp.) Zhang's proof appeared first in (the first edition of) Grimmett (1999) and was exploited later in other contexts by H~iggstr6m (1997a) and H~iggstr6m and Jonasson (1999). Proof of Theorem 5.18: Let A be the event that there exists an infinite open cluster, let B be the event that there exists an infinite closed ,cluster, and assume by contradiction that #(A n B) > 0. Then, by ergodicity, /z (A N B) = 1. (This is the only use of ergodicity we make, and ergodicity could clearly be replaced by tail triviality or some other mixing condition.) Next we pick n so large that lz(An) > 1  10 3 and # ( B n ) > 1  l0 3,
where An (resp. Bn) is the event that some infinite open cluster (resp. some infinite closed ,cluster)intersects An  [  n , n] 2 n Z 2. Let AnL (resp. Ann, An~ and AnB) be the event that some vertex in the left (resp. right, top and bottom) side of the squareshaped vertex set An \ An1 belongs to some infinite open path which contains no other vertex of An, and define B~, Bnn, Bn:r and Bn8 analogously. Then An  A L u ARn U A T u ABn .
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Since all four events in the righthand side are increasing and /z has positive correlations,
/zCAn)

/z(ALn U Aft U A T U Aft)
=
1  / z (   , A L n,Aft n,A T n,Aft)
lZ~,q(X(x) =i)#~,q(X(y)  i).
Proof: The
measure
/Z~,q is invariant under permutation of the spin set
{1 . . . . . q}, so that 1
q
1 Random geometry of equilibrium phases
51
We therefore need to show that lZ~,q(X(x) = i, X ( y )  i) >
1
q2"
We may now think of X as being obtained as in Corollary 6.3 by first picking an edge configuration Y E {0, 1}t3 according to the randomcluster measure ~bP6, q and then assigning i.i.d, uniform spins to the connected components. Given Y, the conditional probability that X ( x ) = X ( y ) = i is 1/q if x and y are in the same connected component of Y, and 1/q 2 if they are in different connected components. Hence, for some ~ E [0, 1], lzGq(X(x)e,

1 1 1 i, X ( y )  i)  or + (1  oe) > q q2~"
An easy modification of the above proof shows that if G is connected and/5 > 0, then the correlation between I{x(x)=i} and I{x(y)=i} is strictly positive. Note that the relation between the randomcluster model and the Potts model depends crucially on the fact that all spins in {1 . . . . . q } are a priori equivalent. This is no longer the case when a nonzero external field is present in the Ising model. Several attempts to find useful randomcluster representations of the Ising model with external field have been made, but progress has been limited. Perhaps the recent duplication idea of Chayes et al. (1998) represents a breakthrough on this problem.
6.2
Infinitevolume limits
In this subsection we will exploit some stochastic monotonicity properties of randomcluster distributions on finite subgraphs of Z d. This will give us the existence of certain limiting randomcluster distributions, and also the existence of certain limiting Gibbs measures for the Potts model. The basic observation is stated in the lemma below which follows directly from definitions. L e m m a 6.6 Consider the randomcluster model with parameters p and q on a finite graph G with edge set 13. For any edge e = (xy) E 13, and any configuration /7 E {0, 1}t~\{e}, we have if x and y are connected via open edges in q q5G (e is openlq)  { p P,q
P p+(1p)q
otherwise. (6.2.1)
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H.O. Georgii et al.
For q > l, Lemma 6.6 means in particular that the conditional probability in (6.2.1) is increasing in 7/ (and also in p). This allows us to use Holley's theorem and the FKG inequality to prove the following very useful result. We write qSp G for Bernoulli bond percolation on G with parameter p. Corollary 6.7 For a finite graph G and the randomcluster p 6 [0, 1]andq > 1, we have
measure
qbpGq with
(a) (~pGqis monotone, and therefore it has positive correlations, (b) ~ 6p,q _._79 ~G
P p+(1p)q
9
Furthermore, for 0 < Pl < P2 < 1 and q > 1, we have (d) (DpG,q "_ 1, the limiting measures 4)b
,q
=
lim ~bp,q,A, A],Z d
b E {0, l}
exist and are translation invariant. This convergence result has consequences for the convergence of Gibbs distributions for the Potts model, as we will show next. Let q 6 {2, 3 . . . . }, and for i { 1 . . . . . q} and any finite region A in Z d let #},q,h denote the Gibbs distribution in A for the Potts model at inverse temperature/3 with boundary condition r/ i on A c. For i = 0, let #~,q,A be the corresponding Gibbs distribution with free boundary condition, which is defined by letting/2 = A in (2.6.2), i.e., by ignoring all sites outside A; we think of #~,q,A as a probability measure on the full configuration space {1 . . . . . q }zd by using an arbitrary extension. Still for i = 0, Theorem 6.2 shows t h a t / ~ , q , a and q~0,q,h admit an EdwardsSokal coupling (on A) when p = 1  e 2~. A similar EdwardsSokal coupling is possible for i 6 {1,.. ., q } when A is simply connected. Indeed, let Pp,q,Ai be the probability measure on {1. . . . . q}s x {0, 1}B corresponding to picking a random siteandbond configuration according to the following procedure. (1) Assign to each vertex of A c value i, and to all edges of B \ B1A value 1.
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(2) Assign to each vertex in A a spin value chosen from {1. . . . . q } according to uniform distribution, assign to each edge in/31A value 1 or 0 with respective probabilities p and 1  p, and do this independently for all vertices and edges. (3) Condition on the event that no two vertices with different spins have an edge with value 1 connecting them. It is now a simple modification of the proof of Theorem 6.2 to check that the vertex i and d/)p,q,A' 1 respectively. (Note that by and edge marginals of pip,q, A are #~,q,A the simple connectedness of A there is always a unique component containing Ac.) Analogues of Corollaries 6.3 and 6.4 follow easily. This leads us to the following result extending Proposition 4.14 to the Potts model.
Proposition 6.9 For any i E {0, 1. . . . . q }, the limiting probability measure i
~fl,q

lim
i
A.~Zd~fl,q, A
on { 1. . . . . q }zd exists and is a translation invariant Gibbs measure for the qstate Potts model on Z d at inverse temperature ft.
Proof: In view of the general facts reported in Section 2.6, the limits are Gibbs measures whenever they exist. We thus need to show that #~,q,A i ( f ) converges as A 1" Z d, for any local observable f . For definiteness, we do this for i c { 1 . . . . . q }; the case i = 0 is completely similar. Fix an f as above, and let A C Z d be the finite region on which f depends. As shown above, for a simply connected A we may think of a {1. . . . . q} zdas arising by first picking an valued random element X with distribution #ri edge configuration Y 6 {0, 1}A according to q~pl,q,A (with p  1  e 2/3) and then assigning random spins to the connected components, forcing spin i to the (unique) infinite cluster. Forx, y c A, we write {x ~ y} for the event that x and y are in the same connected component in Y, and {x ~ ~ } for the event that x is in an infinite cluster. Clearly, the conditional distribution of f given Y depends only on the indicator functions (I{x~y})x,y~/X and ( l { x ~ } ) x c / X , since the conditional distribution of X on A is uniform over all elements of { 1 , . . . , q}A such that firstly X ( x ) = X ( y ) whenever x +~ y, and secondly X ( x ) : i whenever x ~ e~. Hence, the desired convergence of #~,q,A ( f ) follows if we can show that the joint distribution of (I{x~y})x,y~A and ( I { x ~ } ) x ~ A converges as n + cx~. This, however, follows from Lemma 6.8 upon noting that (I{x~y})x,yc/X and (l{x~e~})x~/X are increasing functions. E3
1
6.3
Random geometry of equilibrium phases
55
Phase transition in the Potts model
As promised, this subsection is devoted to proving Theorems 3.1 and 3.2, using randomcluster arguments. The original source for the material in this subsection is Aizenman et al. (1988); see also H~iggstr6m (1998) for a slightly different presentation. We consider the Potts model on Z a, d > 2. All the arguments to be used here, except those showing that the critical inverse temperature fie is strictly between 0 and c~, go through on arbitrary infinite graphs; we stick to the Z d case for definiteness and simplicity of notation. We consider the limiting Gibbs measures /X~,q obtained in Proposition 6.9. For i 6 {1 . . . . . q}, these play a role similar to that of the "plus" and "minus" measures #~ and # ~ for the Ising model. In fact, we have the following result which extends Theorem 4.15 to the Potts model and also gives a characterization of phase transition in terms of percolation in the randomcluster model.
T h e o r e m 6.10 Let fl > 0 and p 
1  e 2~. For any x E Z d and any i { 1 . . . . . q }, the f o l l o w i n g statements are equivalent.
(i) There is a unique Gibbs measure f o r the qstate Potts model on Z d at inverse temperature ft.
(ii) I~fl,q i (X (x)  i)  1/q. (iii) qS~,q(x +~ ec) = 0. As we will see in a moment, it is the percolation criterion (iii) which is most convenient to apply. In this context we note that 4); (x +~ o c ) = 'q
inf dl);,q,A(X ~
A,A
A c)  lim ~1 AtZ d
p,q,A (X +~ A c ),
(6.3.1)
where {x +~ A c} stands for the event that there exists an open path from x to some site in A c. This follows from (6.2.3) and the fact that {x ~+ A c } decreases to {x ~ ~ } as A 1" Z d. The usefulness of the percolation criterion is demonstrated by the next result which extends the scenario for Bernoulli percolation to the randomcluster model. Together with Theorem 6.10, this gives Theorem 3.2 with tic  89log(1  Pc).
Proposition 6.11 For the randomcluster model on Z d, d >_ 2, and any fixed q >_ 1, there exists a percolation threshold Pc E (0, 1) (depending on d and q) such that
C/);,q (X ~ (X3) {  0 for p < Pc, >0
f o r p > Pc.
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H.O. Georgii et al.
Proof: The statement of the proposition consists of the following three parts:
(i) 4~plq(x ~ cx~)  0 for p sufficiently small, (ii) ~bl,q (X +> ~ )
> 0 for p sufficiently close to 1, and
(iii) 4)P,q 1 (x +~ cx~) is increasing in p. We first prove (i). Suppose p < pc(Z d, bond), the critical value for Bernoulli bond percolation on Z d. For e > 0, we can then pick A large enough so that 4~p(0 ~ A ~) _< e. By Corollary 6.7(b), we have that the projection of q~p on {0, 1}B~ stochastically dominates the projection of q51p,q,A on {0, 1}~x for any A D A, so that
4~pl,q,A (O ~
A
c) < 4~1p,q,A (O ~
A c) < 6
for any A D A. Since e was arbitrary, we find lim 4) 1 Ac A'~Zd p,q,A(O ~ ) 0 which in conjunction with (6.3.1) implies (i). Next, (ii) can be established by a similar argument. Let p be such that p* = p / [ p + (1  p)q] > p c ( Z d, bond). Corollary 6.7(c) then shows that ~plp,q,A >7) Cpp. for every A, so that lim qSl,q,A (0 +~ ~ )
> 0,
A~'Z d
proving (ii). To check (iii) we note that Corollary 6.7(d) implies that, for any A, cPll ,q,A _ flplp , q , A ( C )
> 1 e
by Lemma 6.13 and the choice of A. Assuming without loss of generality that IIf II _< 1, we can therefore conclude that
]P(foXiY
E
C)/z$,q,A(f)l
(6.3.5)
< 28.
However, the conditional expectation on the left is an average of the conditional = F) with A C F C A, and these in turn are averages expectations P ( f o X I F ( Y ) of conditional expectations of the form P ( f o X IY = r/off BilL, Y = 0 on B~ \ / 3 ~ which, by construction of P, are equal to /z~,q,F(f)" and (6.3.5), we conclude that
Together with (6.3.4)
]/z~,q,A (f)  /zOl3,q(f)l y in Y, Y ( B j )  ~ ( B j )
+ Pn(Yn(Bj) ~= Y ( B j ) ) c
where A denotes symmetric difference. Since {x < > y} is the decreasing limit of the local events {x +> y in A} U {x ++ A c, y ++ A c} as A 1" s an analogue of (6.3.1) together with (6.4.6) shows that the last expression tends to zero as n + cx). It follows that "Bn,1
lim ~p,q (X
yl~(Bj))
"1 C qbp,q(X < > y ] ~ ( B j ) )
"Bn,1 which is at least 1  e by (6.4.5). But since qbp,q (e is open I~!) = P for each n
and a l l ~ ' ~ { x
y},weget p
"Bn,1 (e is open [~(Bj)) _< p. e < lim Cp,q n+oo
Hence,
p
"1 g O, r is a randomcluster measure f o r p and q if and only if it is a Crandomcluster measure f o r p and q.
1 Random geometry of equilibrium phases
65
This means that whenever "uniqueness of the infinite cluster" can be verified, the two types of randomcluster measures coincide. An example is obtained if we consider translation invariant randomcluster measures for Z d, since the BurtonKeane uniqueness theorem (Theorem 5.17) applies in this situation. For Z a, the "1 measures q~0,q a n d ~p,q are translation invariant, by Lemma 6.8. On the other hand, uniqueness of the infinite cluster typically fails on trees, leading to very different behavior for the two types of randomcluster measures; see H~iggstr6m (1996a, b) for a discussion. Proof of Proposition 6.19: For p = 0 or 1 the result is trivial, so we may assume that p 6 (0, 1). The conditional probabilities in (6.4.2) and its Ccounterpart differ only on the event C
Axy : {X < > y} \ {x +~ y}.
Hence if 4~ is a randomcluster measure but not a Crandomcluster measure (or vice versa), then Axy has to have positive 4~probability for some edge e  (xy) E /3. But then the event Axy O {e is closed} has positive 4~probability, and since this event implies the existence of at least two infinite clusters, we are done. n Much of the study of infinite volume randomcluster measures that has been done so far concerns the issue of uniqueness (or nonuniqueness) of randomcluster measures. A discussion of this issue would, however, lead us too far, so instead we advise the reader to consult Grimmett (1995), H~iggstr6m (1996a) and Jonasson (1999) to find out what is known and what is conjectured in this field.
6.5
An application to percolation in the Ising model
In Theorem 5.10 we have seen that the probability of percolation of plus spins in the Ising model is an increasing function of the external field. A much harder question is to determine monotonicity properties of percolation probabilities as fl (rather than h) is varied. An interesting open problem is to decide whether for G = Z d, d > 2, the probability a~(x
ec)
is increasing in/3. Here we write #~ for the plus phase in the Ising model at +
inverse temperature i3 with external field h  0, and {x ~. , ec} is the event that there exists an infinite path of plus spins starting at x. At first sight, one might be seduced into thinking that this would be a consequence of the connection between Ising and randomcluster models, and the stochastic monotonicity of randomcluster measures as p varies; see (6.3.2). However, such a conclusion
66
H.O. Georgii et al.
is unwarranted. For example, in the coupling of Theorem 6.2 the existence of an open path between x and y in the randomcluster representation is a sufficient but not necessary condition for x and y to be in the same spin cluster. In fact, H~iggstr6m (1996c) showed, by means of a simple counterexample and in response to a question of Cammarota (1993), that the probability that x and y are in the same spin cluster need not be increasing in fl, and similarly for the expected size of the spin cluster containing x. However, when the underlying graph is a tree, monotonicity in fl of the probability of plus percolation can be established: Theorem 6.20 For the lsing model on the regular tree Td, d >__ 2, with a dis+ tinguished vertex x, the percolation probability # ~ (x < > (x~) is increasing in
An interesting aspect of this result is that its proof, unlike those of the monotonicity results mentioned earlier in this section, is not based on stochastic domination between the probability measures in question. In fact, stochastic domination fails, i.e. it is not always the case (in the setting of Theorem 6.20) that #~1%Z~ # ~
(6.5.1)
when/31 < f12. An easy way to see this is as follows. Just as in Theorem 3.1, let/?c be the critical inverse temperature for nonuniqueness of the Gibbs measure. (It is straightforward to show, using either the randomcluster approach or the methods in Section 7, that fie > 0 for L; = Ta.) Pick fll < f12 in (0,/?c). By Theorem 4.15, we then have #~ (~ 9~ (y)  41)   ]/~fl2(~ " ~ (y) = + 1)  1/2 for every vertex y. If now (6.5.1) was true we would have #~1 = #~2 by Proposition 4.12. This, however, is impossible because the two measures have different conditional distributions on finite regions. Theorem 6.20 can be proved using the exact calculations for the Ising model on Td, which can be found in e.g. Spitzer (1975) and Georgii (1988). Here we present a simpler proof which does not require any exact calculation, but which exploits randomcluster methods. Proof of Theorem 6.20: As usual we write L; and B for the vertex and edge sets of Ta. Since lim
+
+ (x < > c x z )   #
A1,s ~fl, A
+
(x < > ~ )
for any fl in analogy to (6.3.1), it suffices to show that for/~l < f12 and any A, we have +
+
,A (x < > oo).
(6.5.2)
1 Random geometry of equilibrium phases
67
This we will do by constructing a coupling P of two {1, + 1}Cvalued random objects X1 and X2 with respective distributions/z~l ,A and/z~2, A and the property + that if x < > c~ in X1, then the same thing happens in X2. Recall the EdwardsSokal coupling of spin and edge configurations described ahead of Proposition 6.9. In the present case of the tree Td, this construction requires the Cversion of counting clusters, which corresponds to making the complement of A connected. Therefore we will work with the Crandomcluster distributions. Let Pl  1  e 2/3~ and P2  1  e 2/32, and let B /31 C / 3 be the set of edges incident to at least one vertex in A. We first let Y1 and Y2 be two {0, 1}t3valued random edge configurations distributed according to the random^B,1 and ~bp2,2, ^B,1 and such that P(Y1 < Y2)  1" this is possible cluster measures 4~p~,2 ^B,1 by the ~bp,zanalOgue of (6.3.2). X1 and X2 can now be obtained by assigning spins to the connected components of Y1 and Y2 in the usual way; these spin assignments are coupled as follows. First we must assign spin + 1 to all infinite clusters in Y1 and Y2. Then we let (Z(y))ycA be i.i.d, random variables taking values + 1 and  1 with probability 1/2 each, and assign to each finite cluster C of Y1 and Y2 the value Z(y), where y is the (unique) vertex of C that minimizes the distance to x. This defined X 1 and X2. A moment's thought reveals that the set of vertices that can be reached from x via spins in X1 is almost surely contained in the corresponding set for X2. Hence (6.5.2) is established, and we are done. [] Note that this proof did not use any property of Td except for the tree structure, so Theorem 6.20 can immediately be extended to the setting of arbitrary trees.
6.6
Cluster algorithms for computer simulation
An issue of great importance in statistical mechanics which we have not touched upon so far is the ability to perform computer simulations of large Gibbs systems. Many (most?) questions about phase transition behavior etc. can, with current knowledge, only be answered partially (or not at all) using rigorous mathematical arguments. Computer simulations are then important for supporting (or rejecting) heuristic arguments, or (in case not even a good heuristic can be found) to provide ideas for what a good conjecture might be. This topic is somewhat beside the main issue of our survey, but since randomcluster representations have played a key role in simulation algorithms for more than a decade we feel that it is appropriate to describe some of these algorithms. In fact, it was the need of efficient simulation which, in the late 1980s, sparked the revival of the randomcluster model (Swendsen and Wang, 1987) which up to then had raised only little interest since its introduction by Fortuin and Kasteleyn in the early 1970s. Consider for instance the Ising model with free boundary condition on a large cubic region A C Z d. Direct sampling from the Gibbs distribution ~h,/3,A with
68
H.O. Georgii et al.
free boundary condition is not feasible, due to the huge cardinality of the state space S2, and the (related) intractability of computing the normalizing constant for the Gibbs measure. The most widely used way to handle this problem is the Markov chain Monte Carlo method, which dates back to the paper by Metropolis et aL (1953). The idea is to define an ergodic Markov chain having as unique stationary distribution the target distribution #h,~,A. Starting the chain in an arbitrary state and running the chain for long enough will then produce an output with a distribution close to the target distribution. An example of such a chain is the singlesite heat bath algorithm, whose evolution is as follows. At each integer time, a vertex x ~ A is chosen at random, and the spin at x is replaced by a new value according to the conditional distribution (under #h,~,A) of the spin at x given the spins at its neighbors. It is immediate that #h,~,A is stationary for this chain, and ergodicity of the chain follows from elementary Markov chain theory upon checking that it is aperiodic and irreducible. The problem with this approach is that the time taken to come close to equilibrium may be very long. For example, let h  0. Then, for fi < fie (with fie defined as in Theorem 3.1), the time taken to come within a fixed small variational distance from the target distribution grows only like n log n in the size of the system (here n is the number of vertices in A) whereas in contrast the time grows (stretched) exponentially in the size of the system for/3 > fie; see Martinelli and Olivieri (1994) and Martinelli (1999). This means that simulation using this heat bath algorithm is computationally feasible even for fairly large systems provided that/3 < fi~, but not for fi > /~c. What happens for/~ > /3c is that if the chain starts in a configuration dominated by plus spins, then the plus spins continue to dominate for an astronomical amount of time, and similarly for starting configurations dominated by minus spins. The set of configurations where the fraction of plus spins is around 1/2 (rather than around the fractions predicted by the magnetization in the infinitevolume Gibbs measures # ; and # ~ ) has small probability and thus can be seen as a "bottleneck" in the state space, slowing down the convergence rate. A way to tackle the exceedingly slow convergence rate in the phase coexistence regime is to use the heat bath algorithm for the corresponding randomcluster model rather than for the Ising model itself, and only in the end go over to the Ising model by the random mapping described in Corollary 6.3. This has the disadvantage that the calculation of singlesite (or, rather, singleedge) conditional probabilities become computationally more complicated due to the possible dependence on edges arbitrarily far away (see Lemma 6.6). This disadvantage, however, seems to be by far outweighed by the fact that the convergence rate of the Markov chain (for fi > tic) appears to be very much faster than for the heat bath applied directly to the spin variables. The reason for this phenomenon is that the randomcluster representation "doesn't see any difference" between the plus state and the minus state. This approach can, of course, be used also for the q > 3
1 Random geometry of equilibrium phases
69
Potts model, and is due to Sweeny (1983). Later, Propp and Wilson (1996) built on this approach by coupling several such Markov chains (i.e. running them in parallel) in an ingenious way, producing an algorithm which runs for a random amount of time (determined by the algorithm itself) and then outputs a state which has exactly the target distribution. The running time of this algorithm turns out (from experiments) to be moderate except for the case of large q and fi close to the critical value. The ProppWilson approach, known as exact or perfect simulation, has received a vast amount of attention among statisticians during the last few years (see the annotated bibliography (Wilson, 1998)) and we believe that it also has interesting potential in physics. There is, however, another Markov chain which appears to converge even faster than those of Sweeny, Propp and Wilson. We are talking about the SwendsenWang (1987) algorithm, which runs as follows for Ising and Potts models on a graph with vertex set /2 and edge set /3: starting with a spin configuration X0 6 {1 . . . . . q}s a bond configuration Y0 6 {0, 1}t3 is chosen according to the random mapping defined in Corollary 6.4. Then another spin configuration X1 is produced from Y0 by assigning random spins to the connected components, i.e. by the random mapping of Corollary 6.3. This procedure is then iterated, producing a new edge configuration I11 and a new spin configuration X2, etc. By combining the two corollaries, we see that if X0 is chosen according to the target distribution, then the same holds for X1, and consequently for X2, X3 . . . . . In other words, the target distribution is stationary for the chain {Xk}~=0, and by the (easily verified) ergodicity of the chain we have a valid Markov chain Monte Carlo algorithm. Although it is not exact in the sense of the ProppWilson algorithm, it appears to converge much faster, thus in practice allowing simulation of systems that are orders of magnitude larger. Heuristically, the reason for this faster convergence is that large chunks of spins may flip simultaneously, allowing the chain to tunnel through any bottlenecks in the target distribution. However, rigorous upper and lower bounds on the time taken to come close to equilibrium are to a large extent lacking, although Li and Sokal (1989) have provided a lower bound demonstrating the phenomenon of "critical slowing down" as/~ approaches fie. The SwendsenWang algorithm has, since its introduction in 1987, become the standard approach to simulating Ising and Potts models. Interesting variants and modifications of this algorithm have been developed by Wolff (1989) and Machta et al. (1995); the last paper is an interesting attempt at combining the original approach of Swendsen and Wang with ideas from socalled invasion percolation (see Chayes et al., 1985) to get an algorithm specifically aimed at sampling from a Gibbs distribution at the critical inverse temperature fie, i.e. where the use of other algorithms have proved to be most difficult. Generalizations of the SwendsenWang algorithm for various models other than Ising and Potts models have also been obtained, see e.g. Campbell and Chayes (1998),
70
H.O. Georgii et aL
Chayes and Machta (1997, 1998), and H~iggstr6m et al. (1999).
6.7
Randomcluster representation of the WidomRowlinson model
The randomcluster model can be seen as a perturbation of Bernoulli bond percolation, where the probability measure is changed in favour of configurations with many (for q > 1) or few (for q < 1) connected components. A fairly natural question is what happens if we perturb Bernoulli site percolation in the same way. For lack of an established name, we call the resulting model the siterandomcluster model. Let G be a finite graph with vertex set/2 and edge set B. For a site configuration 7/6 {0, 1}s we write k(q) for the number of connected components in the subgraph of G obtained by deleting all vertices x with r/(x)  0 and their incident edges. 6 f o r G with parameters Definition 6.21 The siterandomcluster measure grP,q
p 6 [0, 1] and q > 0 is the probability measure on {0, 1}z; which to each q {0, 1 } s assigns probability
where Zp,q a is a normalizing constant.
Analogously to the usual randomcluster model living on bonds, taking q = 1 gives the ordinary Bernoulli site percolation Op, while other choices of q lead to dependence between vertices. Taking q  2 is of particular interest because it leads to a representation of the WidomRowlinson model which is similar to (and slightly simpler than) the usual randomcluster representation of the Ising model. Let #G be the Gibbs measure for the WidomRowlinson model with activity ~. on G, i.e. # ~ is the probability measure on {  1, 0, + 1}z; which to each ~ ~ {  1, 0, + 1}z; assigns probability proportional to
H I{~(x)~(y)#l} II ~l~(x)l. (xy)~13 xcs The following analogues of Corollaries 6.3 and 6.4 are trivial to check. Proposition 6.22 Let p 
x and suppose we pick a random spin configurai42'
tion X ~ {  1, O, + 1 }s as follows.
(1) Pick Y ~ {0, l} z; according
to ~rpG2.
(2) Set X ( x ) = Of o r each x ~ E such that Y ( x )  O.
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71
(3) For each open cluster C of Y, flip a fair coin to decide whether to give spin + 1 or  1 in X to all vertices of C. Then X is distributed according to the WidomRowlinson Gibbs measure # G. )~
Proposition 6.23 Let p = V4~' and suppose we pick a random spin configuration Y E {0, l} s as follows. (1) Pick X ~ {  1, 0, + 1}s according to #G.
(2) Set Y ( x )  I X ( x ) l f o r each x E •. Then Y is distributed according to the siterandomcluster m e a s u r e
~pG2.
We remark that for q 6 {3, 4 . . . . }, these results extend in the obvious way to a connection between ~pGq and the generalized WidomRowlinson model whith q types of particles rather than just 2 (and strict repulsion between all particles of different type). Many of the arguments applied to Ising and Potts models in Section 6.3 can now be applied to the WidomRowlinson model in a similar manner. To apply Theorem 4.8, we need to calculate the conditional probability in the siterandomcluster model that a given vertex is open given the status of all other vertices. For x ~ s and 77 ~ {0, 1}s we get pqlx(x,o)
gt 6 (x is open It/) P'q

pq
1K(x,r/)
+ 1
_
p
(6.7.1)
where x ( x , ~) is the number of open clusters of 0 which intersect x's neighborhood {y 6 /2 9 y ~ x}. If the degree of the vertices in G is bounded by N, say, then 0 < x ( x , rl) < N for any x 6 s and 17 6 {0, 1}s For fixed q and any p* 6 (0, 1) , we can thus apply Theorem 4.8 to show that ~G p , q stochastically dominates ~p, for p sufficiently close to 1, and is dominated by ~pp, for p small enough. The arguments of Section 6.3 leading to a proof of Theorem 3.1, with the randomcluster model replaced by the siterandomcluster model, therefore go through to show Theorem 3.4. One thing that does not go through in this context, however, is the analogue of (6.3.2). The reason for this is that, in contrast to (6.2.1), the conditional probability in (6.7.1) fails to be increasing in ~, so that Theorem 4.8 is not applicable for comparison between siterandomcluster measures with different values of p. In fact, the analogue of (6.3.2) for siterandomcluster measures sometimes fails, and moreover the occurrence of phase transition for the WidomRowlinson model on certain graphs fails to be increasing in )~, as demonstrated by Brightwell et al. (1999).
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al.
Another consequence of the failure of the conditional probability in (6.7.1) to be increasing is that the FKG inequality (Theorem 4.11) cannot be applied to ~rpGq. AS a consequence, the proof of Theorem 6.10 cannot be adapted to the case of the multitype (q > 3) WidomRowlinson model. Such a WidomRowlinson analogue of Theorem 6.10 is known to be false, as shown by Runnels and Lebowitz (1974); see also Chayes et al. (1997) and Nielaba and Lebowitz (1997).
7
Uniqueness and exponential mixing from nonpercolation
In Section 6 we saw examples where phase transition in one system was equivalent to the existence of infinite clusters in another, suitably defined, system. In this section we shall discuss various approaches where conclusions about the phase transition behavior can only be drawn from nonexistence (and not from existence) of infinite clusters. On the other hand, these approaches typically apply to a much wider range of models. We address two problems: the uniqueness of the Gibbs measure, and the decay of correlations for a given Gibbs measure. The general theme of this section can be stated as: to what extent can a given spin be influenced by a configuration far away? If such an influence disappears in the limit of infinite distance, it follows (depending on the setting) that either there is no longrange influence of boundary conditions at all (implying uniqueness of the Gibbs measure), or that a specific low temperature phase exhibits some mixing properties. In both cases, the decreasing influence comes from the absence of infinite clusters of suitable type which could transport a dependence between spins. So, both uniqueness and mixing will appear here as a consequence of nonpercolation. First we will address the problem of uniqueness. We will encounter conditions which not only imply the uniqueness of the Gibbs measure, but also lead us into a regime where "all good things" happen, i.e., where the unique Gibbs measure exhibits nice exponential mixing properties and the free energy depends analytically on all relevant parameters. (In general, the uniqueness of the Gibbs measure does not imply the absence of other critical phenomena, which might manifest themselves as singularities of the free energy or other thermodynamic quantities. For example, in Section 9 we will see that in the socalled Griffiths regime of a disordered system there is a unique Gibbs measure, but the free energy is not analytic.) The "nice regime" above is usually referred to as the high temperature, or weak coupling, low density, or also analytic regime, and is usually studied by high temperature cluster expansions. Dobrushin and Shlosman (1985b, !987) developed a beautiful and general theory describing a regime of "complete analyticity" by various equivalent properties. One of these ranks at the top of a hierarchy
1 Random geometry of equilibrium phases
73
of mixing properties. While complete analyticity makes precise what actually the "nice regime" is, and applies mainly to high temperatures or large external fields, it is not limited to this case (van Enter et al., 1997). The relationships between this and related notions and also with dynamical properties have been studied in many papers. Although some of these have an explicit geometric flavor, we do not discuss them here because of limitations of space. We rather refer to the sources (Dobrushin and Shlosman, 1985b, 1987; Stroock and Zegarlinski, 1992; Martinelli and Olivieri, 1994; Martinelli, 1999) and also to the references following condition (9.2.2). In Section 7.3 we shall discuss an application of the percolation method to the low temperature regime, and see how percolation estimates for the covariance between two distant observables, combined with contour estimates, give rise to exponential mixing properties.
7.1 Disagreementpaths Let (/2, ~) be an arbitrary locally finite graph, and suppose we are given a neighbor interaction U : S x S + R N {cx~}and a selfpotential V : S ~ R. Consider the associated Gibbs distributions/z~,Ao introduced in (2.6.2). More generally, we could consider an arbitrary Markov specification (GA)Aes in the sense of Section 2.6. Such specifications appear, in particular, if we have an interaction of finite range R, say on Z d, and draw edges between all sites of distance at most R. However, for definiteness and simplicity we stick to the setting described by the Hamiltonian (2.5.1). We will often consider the inverse temperature ~ as fixed and then simply write/z~ instead of/z~,A" If A is a singleton, we use the shorthand x for {x}. We look for a condition implying that there is only one Gibbs measure/z for the Hamiltonian (2.5.1), i.e., a unique probability measure on fla = S s satisfying # (. IX = r/off A) = # ~
for/zalmost all r/ E fla.
Since this property needs only to be checked for singletons A = {x} (cf. Theorem 1.33 of Georgii (1988)), it is sufficient to look for conditions on the singlespin Gibbs distributions #x~ with x 6 s Intuitively, we want to express that /z~x(X(x) = a) depends only weakly on 77 (which can be expected to hold for small fl). This dependence can be measured by the maximal variation Px = max II~x~ r/,r/tEf2

~ x ~'
IIx,
(7.1.1)
where
IIvlIA =
sup Iv(A)l A~rA
(7.1.2)
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is the total variation norm on the suboalgebra 3cA of events which depend only on the spins in A. We write p as a shorthand for the family (Px)x~Z;. Given two configurations ~, ~' 6 f2, a path in s will be called a path of disagreement (for ~ and ~,) if ~(x) r ~'(x) for all its vertices x. For each finite region A C 12 and any two configurations r;, r / o n A c we will construct a coupling P of/z~x and # ~ describing the difference of these measures in terms of paths of disagreement running from the boundary 0A into the interior of A. Intuitively, these paths of disagreement indicate the range of influence of the r boundary conditions. We write {A < ~ 0A} for the event in S A • S A that there exists a path of disagreement from some point of a set A C A to some point of 0A. Although the coupling P to be constructed is not best suited for direct use, it has a useful special feature: its disagreement distribution is stochastically dominated by a Bernoulli measure. This will allow us to conclude that absence of percolation for the latter implies uniqueness of the Gibbs measure for the Hamiltonian (2.5.1). We write gro for the Bernoulli measure on {0, 1}s with 7zp(X(x)  1) = Px for all x E s and 7zp,A for the analogous product measure on {0, 1}A. As in Section 4, we use the notation X (x) and X' (x) for the projections from S2 x f2 to S. The following theorem is due to van den van den Berg and Maes (1994). Theorem 7.1 For each finite A C 17, and each pair rl, rl' E ~2 there exists a
coupling P = PA,o,o' ~ lz~ and lz~ having the following properties: (i) For eachx c A , {X(x) C X t ( x ) }   { x
# < ~ OA}Pa.s.
(ii) For the distribution "~ of (l{(x(x)#X'(x)})x~A under P, we have P 2 Z3p 1/fp, A 9
(iii) For each A C A,
I I ~  ~A77! IIzx~ P ( A ~
0A) < ~ p ( A +  >  o q A ) .
(7.1.3)
Proof: We construct a coupling (X, X') of # ~ and # ~ by the following algorithm. In a preparatory step we introduce an arbitrary linear ordering on A , set A = A, and define X ( x ) = rl(x), X'(x) = r/(x) for x ~ A c. For fixing the main iteration step, suppose that (X, X') is already defined on the complement of a nonempty set A C A and is realized as a pair (~, ~,) off A, where (~, ~') = (r;, r/t) off A. Consider the Gibbs distributions # ~ and # ~ obtained by conditioning # ~ and # ~ on X = ~ resp. ~t off A. If ~ = ~, on OA then # ~ 
# ~ on ~A by the Markov property, so that we can take the
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obvious optimal coupling for which X = X' on A, and we are done. Otherwise we pick the smallest vertex x = x(~, ~') 6 A for which there exists some vertex y 6 A c with y ~ x and ~(y) # ~'(y). Consider the single vertex distributions /z~, x  # ~ (X (x)  .) and #~,x  # ~ (X (x)  .) on S. Conditionally on (X, X') = (~, ~') off A, we then let (X (x), X' (x)) be distributed according to an optimal coupling (as in Definition 4.3) of #~,x and #~,x" The coupling (X, X') is then defined on the set x U A c, so that we can replace A by A \ x and repeat the preceding iteration step. It is clear that the algorithm above stops after finitely many iterations and gives us a coupling of #~x and #7(" Property (i) is evident from the construction, since disagreement at a vertex is only possible if a path of disagreement leads from this vertex to the boundary. For (ii), we note that the measures #~,x and #~,x are mixtures of the Gibbs distributions #x~ with suitable boundary conditions or, by the consistency of Gibbs distributions. Hence
By construction, this means that in each iteration of the main step we have
P ( X ( x ) r X ' ( x ) l ( X , x ' )  (~, ~')off A) _< p~ for x  x(~, ~'), so that (ii) follows by induction. Finally, (iii) follows directly from (i) and (ii) because for each A C A 7] ! II#~x ~AI A ~ P(X(x) # X'(x)for some x c A)
by the coupling inequality (4.1.4). The proof is therefore complete. Although the algorithm in the proof above is quite explicit, it is not easy to deal with directly. In particular, it is not clear in which way the coupling depends on the chosen ordering, because the site x to be selected in each step depends on (~, ~') and is therefore random. Nevertheless, if the Gibbs distributions are monotone (in the sense of Definition 4.9), we get some extra properties. R e m a r k Suppose S is linearly ordered and the conditional distributions #~x are stochastically increasing in ~. Then, if 77 _ r/', the coupling P of Theorem 7.1 can be chosen in such a way that, in addition to properties (i) to (iii), X _~ X' Pa.s. and, for each x c A, , r (7.1.4) This is because in each step of the algorithm proving Theorem 7.1 we can achieve X (x) _< X' (x), and for the second inequality in (7.1.4) it is sufficient to note that
P(x
#
.~ O A ) 
P(X(x) < X'(x)) _< Z acS\{m}
[P(X(x) 2. Recall the bound (5.1.4) for the percolation threshold pc when d  2, and the largedimensions asymptotics of Pc in (5.1.5). E x a m p l e 7.3 The Isingferromagnet. Let fl > 0 be any inverse temperature and h an external field. Then, for any x, we obtain from (4.3.1) by a short computation +  # h , ~,x Ilx = [ t a n h ( f l ( h + 2 d ) )  t a n h ( / ~ ( h  2 d ) ) ] / 2 . Px = II#h,~,x
Hence, the Gibbs measure is unique when h  0 and tanh(2dfl) < pc, or if Ih[ > 2d is so large that 2d < pc cosh2(fi(Ihl  2d)), for example. E x a m p l e 7.4 The hardcore lattice gas. Setting fl = 1, we see that Px = )~/(1 + k) for any x, so that uniqueness of the Gibbs measure follows for k < p c / ( 1  p c ) . (This can also be obtained by using the product coupling mentioned above, cf. van den Berg and Steif (1994).) E x a m p l e 7.5 The WidomRowlinson lattice gas. We take again fl = 1 and set ~+ = ~._ = )~. It turns out that the maximum in equation (7.1.1) is attained for the boundary conditions r / = 0 and r / e q u a l to + 1 and  1 on (at least) two different neighbors of x, whence px = 2~/(1 + 2~) for any x. It follows that the Gibbs measure is unique when ~. < pc~(2(1  Pc)). It is interesting to compare the uniqueness condition of Theorem 7.2 with the celebrated Dobrushin uniqueness condition, cf. Georgii (1988)and the original papers (Dobrushin, 1968b, 1972). This condition reads sup~ x
y
max
q ~ r f offy
II~x~ ~x~'lix < 1.
(7.1.5)
The constraint "r/ = r/' off y" means that the configurations r/, r/' differ only at the vertex y. For systems with hardcore exclusion or in certain antiferromagnetic
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models it often happens that, for every y ~ Ox, the maximum in (7.1.5) is actually the same as that in (7.1.1), see van den Berg and Maes (1994). Dobrushin's uniqueness condition then takes the form SUPx IOxlPx < 1. For L  Z d and px = Po independently of x, this means that P0 < 1/(2d), while Theorem 7.2 only requires P0 < Pc, and it is known that Pc > 1 / ( 2 d  1) f o r d > 1. However, if the constrained maximum in Dobrushin's condition is much smaller than the unconstrained maximum in (7.1.1), then Dobrushin's condition will be weaker than that of Theorem 7.2. For example, for the Ising ferromagnet on Z d with external field h  0, Dobrushin's condition requires that 2d tanh/5 < 1 which, in view of (5.1.5), is less restrictive than the condition obtained in Example 7.3. Thus, roughly speaking, Theorem 7.2 works best for "constrained" systems with strong repulsive interactions and lowdimensional lattices (or graphs with small 10xls) for which reasonable lower bounds of the critical probability Pc are available. Examples are the hardcore lattice gas and the WidomRowlinson lattice gas on Z 2 considered above. There is also another reason why Theorem 7.2 is useful. Namely, its condition of nonpercolation is a global condition: the absence of percolation does not depend on the value of px at any single site x. In particular, Px could be large or even be equal to 1 for all xs in an infinite subset (say, a periodic sublattice) of L; once the pxS are sufficiently small on the complementary set, there is still no infinite open cluster. This can be applied to nontranslation invariant interactions where, in general, it is impossible to obtain uniform small bounds on the pxS (or on the strength of the interaction, as would be required by the Dobrushin condition or for some standard clusterexpansion argument). We will return to this point in Section 7.3.
7.2
Stochastic domination by randomcluster measures
Recently, Alexander and Chayes (1997) introduced a variant of the randomcluster technique that applies to a substantially greater class of systems than those considered in Section 6. This approach involves a socalled graphical representation of the original system. The graphical representation is stochastically dominated by a randomcluster model, and absence of infinite clusters in this randomcluster model implies uniqueness of the Gibbs measure for the original spin system. The price to pay for the greater generality is that the implication goes only one way: percolation in the randomcluster model does not, in general, imply nonuniqueness of Gibbs measures. We assume that the state space S is a finite group with unit element 1; the inverse element of a 6 S is denoted by a  l , so that a  l a  aa 1  1. For simplicity we assume that the underlying graph is/2  Z d (although this will not really matter). We consider the Hamiltonian (2.5.1) for a pair potential U and with
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79
no selfenergy, V = 0. By adding some constant to U (which does not change the relative Hamiltonian) we can arrange that U _< 0. The basic assumption is that U is leftinvariant, so that U(a, b)  u ( a  l b ) (7.2.1) for all a, b E S and the even function u  U(1, .) < 0. Note that this setting includes the qstate Potts model for which S = Zq and u  21{0}. For any finite A C Z d we consider the Gibbs distribution
,)
(xy)EBA
at inverse temperature fl with boundary condition r] E ~. Here we write ~A for the set of all bonds b E 13 with at least one endpoint in A. The graphical representation of #~,A will be based on bond configurations co E {0, 1 }t~A. Each such co will also be viewed as a subset of BA, and the bonds in co will be called open. The key idea of this representation is taken from the classical high temperature expansion. For fixed fl > 0 and any a e S we introduce the difference
Ra  e fu(a)  1 >_ 0.
(7.2.2)
With this notation we can write
#f,A (or)

l{~=n offA} V I (1 + Ra(x)l~r(y)) ZA(fl,/7) (xy)E/3h
Z II R(r(x)'cr(Y)" l{aO off A} ZA(fl, 77) o~e{o,1}~A(zy)e~o This shows that #r
is the first marginal distribution of a probability measure
Pfl, A on I2 • {0, 1 }t3A, namely
P;,A ((7, co) : l{an off A} V I R~(x) 1 ZA (fl, 17) (xy)~w  (~(Y)' cr E s
co E {0, 1 }t3A. The second marginal distribution of P;,A is equal to
•
(OJ)  W~, A (~o)
ZA (f, n),
where W~, A(co) =
Z
II Rcr(x)'cr(y)
o'1/ off A (xy)Ew
(7.2.3)
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H.O. Georgii e t al.
is the "graphical weight" of any co c {0, 1}I3A. The probability measure Y~O,A on {0, 1}I3A is called the graphical distribution or the grey measure (since it ignores the spins which are considered as colors). The graphical representation of #~,A thus obtained is analoguous to the randomcluster representation of the Potts model and can be summarized as follows. L e m m a 7.6 In the setup described above, the Gibbs distribution lz3, o A can be derived from the graphical distribution Yfl,A by means of the conditional probabilities
P;,A ((9"[co) = W~,A(o9)1
H
Rcr(x)lcr(Y)"
b=<xy>~w That is,
fl,A oJ~{O,1}~A For the Potts interaction u = 21{0} with state space S  Zq, the graphical representation above is easily seen to coincide with the randomcluster representation studied in Section 6. One important feature is that the graphical weights factorize into cluster terms. Indeed, each bond configuration co divides A into connected components called open clusters (which may possibly consist of isolated sites). The set of bonds belonging to an open cluster C is denoted by wc. Writing C(co) for the set of all open clusters we then obtain
W~,A (co) =
H W~,A (C, coc) CcC(co)
(7.2.4)
with
~'r;, A(C' coC) 
Z
H
cr~SC:a=rl onCAOA (xy)~wc
ecr(x)l~r(Y)"
(We make the usual convention that the empty product is equal to 1; hence W~,A (C, ~oc)  ISI if C is an isolated site.) Together with Lemma 7.6, equation (7.2.4) shows that the spins belonging to disjoint open clusters are conditionally independent. In particular, we can simulate the spin system by first drawing a bond configuration co with weights (7.2.3) and then obtain in each open cluster a spin configuration according to Pfl,A (or Io9). Suppose we knew that there is no percolation in the graphical representation, in the sense that max0 Y;,A (0 ++ 0A) ~ 0 as A 1' s The conditional independence of spins in different open clusters would then suggest that there is only one Gibbs measure for the spin system. Unfortunately, this is not known (though weaker statements have been established by Chayes and Machta (1997)). However, one can make a stochastic comparison of the graphical distributions
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with wired randomcluster distributions (Lemma 7.7 below), and the absence of percolation in the dominating randomcluster distribution will then guarantee that the original system has a unique Gibbs measure. This will be achieved in Theorem 7.8 allowing to bound the dependence on boundary conditions in terms of the connectivity probability in a randomcluster model. To this end we also need to consider Gibbs distributions #f,A
with free bound
ary condition. These admit similar graphical representations Y~,A based on bond configurations inside A; that is, the bonds leading from A to A c are removed. In the following, the superscript f will refer to this case. The stochastic comparison with randomcluster distributions will be formulated using R * = m aa6S xRa,
~ _ IS1[ E
Ra,
p
 R* /(1 + R* ),
q =
R*/[~. (7.2.5)
aES Note that these quantities depend on/3 since the Ra in (7.2.2) do. In the case of the rstate Potts model when u  2I{0}, we have R*  1  e 2/~ and q  r; that is, in this case the parameters p and q are nothing but the standard parameters of the randomcluster representation. For p and q as above we consider now the wired (resp. free) randomcluster distribution r (resp. r in A as introduced in Section 6.2.
For any A ~ g, fl > 0 and p, q as above, Y;,A ~,q,A is thus given by
F(co) = W~,A (co)/(R*)l~
k(~
Since ~bl,q,A has positive correlations, the lemma will therefore be proved once
we have shown that F is a decreasing function of co. To this end we let co ___co' be such that col _ co t3 {b} for a bond b c BA \ co.
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We first consider the case when b  (xy) is not connected to 0A and joins two open clusters Cx, Cy e C(co). For each open cluster C let W(C) = W~,6 (C, coc) be as in (7.2.4). Suppose we stipulate that the spin cr (z) at any site z E C is equal to some a 6 S. It is then easy to see that the remaining sum in the definition of W(C) does not depend on a and thus has the value W(C)/[S[. Prescribing the values of o" (x) and cr (y) in this way we thus find that ~ V ( C x (J Cy (.) b) 
lfV(Cx)W(Cy)lS12
R~r(x)_lcr(y),
cr(x),cr (y) and therefore Wg, A (w')  / ~ W g , A (co). Since k(co', A)  k(co, A)  1 and
I~dl 
[co[ + 1, it follows that F (co') = F (co), proving the claim in the first case. If b links some cluster to the boundary which otherwise was separated from the boundary, then the argument above shows again that F (co~) = F (co). Next we consider the case when b = (xy) closes a loop in co but is still not connected to the boundary. Since clearly
Rcr(x)lcr(y) < max Ra aES


R*,
we find that Wg, A (co') _< Wg, A (w)R*. On the other hand, in this case we have [w'[ = Icol + 1 and k(co', A) = k(co, A), so that F(co') < F(co). As we are considering the wired randomcluster measure, this argument remains valid if b joins two clusters already attached to the boundary. [] We are now in a position to state the main result of Alexander and Chayes (1997), an estimate on the dependence of Gibbs distributions on their boundary condition in terms of percolation in the wired randomcluster distribution. Recall the notation (7.1.2) for the total variation norm on the sub~algebra .T'A of events in some A. T h e o r e m 7.8 Consider the spin system with pair interaction (7.2.1) at some in
verse temperature fl > O, and let p, q be given by (7.2.5). Then, for any A C A E C and any pair of boundary conditions rl, ~ E f2, /7
!

p,q,A ( A 0 A ) .
Proof: (This proof is different from the one that appeared in Alexander and Chayes (1997).) Let A be any event in .Yzx. From Lemma 7.6 we know that #~,A (A)  Z (.o
Yfl,A (o9)Pfl, A (A[w).
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To control the 7dependence of this probability we will proceed in analogy to the argument for the implication (ii) =r (iii) of Theorem 6.10. If o9 6 {A ~ 0A} then equation (7.2.4) shows that the conditional distribution Pfl,A (Alog) does not depend on 7. So we need to control the qdependence of yC/,A(A /> 0A). This, however, does not seem possible directly. So we will replace Y~,A by the 7independent ~blp,q,A by using a suitable coupling trick. By Lemma 7.7 and Strassen's theorem (Theorem 4.6) there exists a coupling (I?, I?') of ?'~,A and ~bl,q,A such that I? • I?' almost surely. If I?' 6 {A ~ 0 A }, there exists a largest (random) set F = F (I? ~) such that (a) A C F C A ,
and
(b) I?'(b) = 0 for all bonds connecting F with F c. For I?~ 6 {A +> 0 A} we set F = 0. Conditional on F, Lemma 7.7 and Strassen's theorem provide us further with a coupling (I?v, I?{.) of Y f r and r176 such that ~l f'r • Yr'.
It is then easy to see that the pair of random variables (Y, Y~) defined
by (I?r,, I?I~)(b)
(Y, Y')(b) 
(f', f")(b)
if b is contained in F, otherwise
is still a coupling of ?'fl,A and 4~pl,q,A such that Y ~_ Y' almost surely. (Notice that also I? (b) = 0 for all bonds from F to Fc.) We denote the underlying probability measure by Qo. Now we can write #~,A(A)

Qo(P~,A(AIY))
1 0A). We The first term in the last sum is at most Q0(F  0)   ~p,q,A(t claim that the second term does not depend on 7. Indeed, if 1' ~: 0 then, by only depends on the restriction YF of Y to the (7.2.4), p,7 /?,A (AIY)  Pfv(AIYr) ~_ set of bonds inside F. The second term can thus be written explicitly as
G#0
w in G
which is obviously independent of 7. The theorem now follows immediately.
[]
To apply the theorem we consider the limiting randomcluster measure 4)P,q 1 with arbitrary parameters p e]O, 1[ and q >_ 1 and wired boundary condition;
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H.O. Georgii et aL
recall from Section 6.2 that this limiting measure exists. By Corollary 6.7(d), it makes sense to define the percolation threshold Pc(q)  inf{P " q~lp,q(O +~ c~) > 0}.
We also consider the threshold Wc(q) for exponential decay of connectivities, which is defined as the supremum of all ps for which ~b;,q(O +~ OA) < Ce cd(O'OA) uniformly in A (or, at least, for A in a prescribed sequence increasing to/3) with suitable constants c > 0 and C < cx~. It is evident that We(q) < Pc(q); for large q it is known that we(q) = Pc(q) (van Enter et al., 1997). Theorem 7.8 then gives us the following conditions for hightemperature behavior; compare with Theorem 7.2. Corollary 7.9 Whenever ~ is so small that p < pc(q), there is a unique Gibbs measure f o r the Hamiltonian H with pair interaction (7.2.1). Furthermore, if in [act p < We(q) then the spin system is exponentially weakmixing in the sense that there are positive constants C < cx~, c > 0 such that f o r all A C A E g and all boundary conditions q, rlt E f2 !
IIz~,A  Zz~,AIIA ~ ClOAle cd(/x'Ac). In fact, Alexander and Chayes (1997) go a little further in their exploration of "nice" high temperature behavior, showing that for p < p c ( l ) the unique Gibbs measure satisfies the condition of "complete analyticity" (investigated by Dobrushin and Shlosman (1987), for example).
7.3
Exponential mixing at low temperatures
In the previous subsections we have seen how stochasticgeometric methods can be used to analyze the high temperature behavior of a spin system and, in particular, for establishing exponential decay of correlations. Here we want to demonstrate that similar percolation techniques can also be used in the lowtemperature regime in the presence of phase transition. We will present a method to show that, for a given phase #, the covariance # ( f ; g) of any two local observables f and g decays exponentially fast with the distance between their dependence sets. (The problem of phase transition at low temperatures will be addressed in Section 8.) As a matter of fact, the problem of exponential decay of covariances (or truncated correlation functions) arises in many physical situations. Correlation functions are related to interesting response functions or to fluctuations of specific
1 Random geometry of equilibrium phases
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order parameters. Exponential decay of covariances also provides estimates on higher order correlation functions, eventually providing infinite differentiability of the free energy with respect to an external field (von Dreifus et al., 1995). Motivated by these interests, a variety of techniques have been developed. The most familiar approach are cluster expansions which apply equally well to both the hightemperature (or lowdensity) regime and the lowtemperature (or highdensity) regime. Although they often employ geometric concepts, it seems useful to combine them with ideas of percolation theory to make geometry more visible. An example of this is the method to be described below which is taken from a paper by Burton and Steif (1995), where it was used to show that certain Gibbs measures exhibit a powerful mixing property called "quite weak Bernoulli with exponential rate". We consider a spin system on an arbitrary graph (/2, ~) with Hamiltonian (2.5.1). As before, the essential feature of this Hamiltonian is that only adjacent spins interact, so that the Gibbs distributions #~,A in finite regions A have the Markov property. The inverse temperature/~ > 0 does not play any role for the moment, so we set it equal to 1 and drop it from our notation. Our starting point is the following estimate on the qdependence in terms of disagreement paths for two independent copies of #~. This result is a weak version (and, in fact, a forerunner (van den Berg, 1993)) of Theorem 7.1. It is a pleasant surprise that although developed with hightemperature situations in mind, it also provides a useful alternative to some aspects of the standard lowtemperature expansions. P r o p o s i t i o n 7.10 For any A C A ~ 8 and q, ~ c f2,
,
,
r
For brevity let P  #~x x # ~ , and write X, X ~ for the two projections from f2 x f2 to f2. Then for any A E ~A we have
Proof:
Lt~A(A) # ~ ( A ) 
P(X E A )  P(X' E A).
We decompose the probabilities on the righthand side into the two contribu# tions according to whether the event {A ~ ~ 0A} occurs or not. In the latter case, there exists a random set F C A containing A such that X  X f on 01'. (The union of all disagreement clusters in 3, meeting A is such a set.) Let F be the maximal random subset of A with this property. Then for each G the event {F = G} only depends on the configuration outside G, and X = X f on OG. The Markov property therefore implies that, conditionally on {F = G} and ! ! (X6 c, XGc), X6 and X 6 are independent and identically distributed, and this
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# r shows that P ( X c A, A 1t+ OA)  P ( X ' c A, A y/+ OA). The proposition now follows immediately. [] What is gained with the disagreement estimate above? First, let us observe that this estimate provides bounds on covariances of local functions in terms of disagreement percolation. Corollary 7.11 Fix any A c g and rl ~ S2. Let f and g be any two local functions depending on the spins in two disjoint subsets A resp. A ~ of A. Then I # ~ ( f ; g)l < 3(f)a(g)#~a x # ~ ( A ", # , A' in A) where 6 ( f )  max~ f (~)  min~ f (~) is the total oscillation of f .
Proof: By rescaling and addition of suitable constants we can assume that 0 < f < 1 and 0 < g _< 1. Proposition 7.10 then shows that
< ~(f)~(g)
f
#~(d~)
f
# ~ ( d ~ ' ) # ~ \ A, x #~\A,(A .
'
> A in A)
,
because ~  ~'  ~ on 0A. By carrying out the last integration we obtain the result. [] The bounds above leave us with the task of estimating the probability of disagreement paths in a duplicated system. In contrast to the situation in Section 7.1, we are looking now for estimates valid at low temperatures. If a cluster expansion works, there is no need to look any further. For instance, a lowtemperature analysis and estimates of semiinvariants for the Ising model can be obtained using standard contour representations; see Dobrushin (1996). It needs to be emphasized, however, that the main step of cluster expansions consists in expanding the logarithm of the partition function. Only afterwards, by taking ratios of partition functions, does one obtain expressions for covariances and higher order correlation functions. Therefore, a point to appreciate is that the bound of Corollary 7.11 provides a direct geometric bound on covariances which avoids the machinery of cluster expansions and, in particular, the problems coming from taking logarithms. As a consequence, this estimate also applies to some cases where standard cluster expansions are doomed to fail. Alternative stochasticgeometric methods for estimating covariances in the absence of standard cluster algorithm arguments have been discussed in Bricmont and Kupiainen (1996, 1997).
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Let us illustrate this for the case of the Ising ferromagnet. This might not be the best example because other methods can also be applied to it. Nevertheless, it is useful to demonstrate the technique in this simple case. Afterwards we will discuss a case where cluster expansion techniques cannot be used equally easily. Consider the lowtemperature plus phase #~ of the ferromagnetic Ising model on the square lattice 12  Z 2 with zero magnetic field. By taking the infinite volume limit in Corollary 7.11 with r/=__ + 1 we obtain the estimate I # f ( f ; g)l _< 8 ( f ) 8 ( g ) # f x #~(A ~
A')
(7.3.1)
for the covariance of any two local functions f, g with disjoint dependence sets A, A'. Next we observe that the event {A < r > A' } is clearly contained in the event that there exists a path from A to A f along which (X, X') ~ (+, +). The > A' }. Burton and Steif (1995) have shown latter event will be denoted by {A # A') < Cmin{]0iA], ]OiA'l}e cd(A A'), #~ x #~(A +(+'+)
where Oi A = O(A c) is the inner boundary of A. Combining this theorem with (7.3.1) we obtain an exponential bound for the covariance of any two local observables in the lowtemperature Ising plus phase. While this result is well known, its proof below shows how one can proceed in more general cases.
Sketch proof of Theorem 7.12: It is sufficient to prove the statement with #~replaced by #~,A, where A is a sufficiently large square box containing A, A'. For brevity, let PA  #~,a + x #~,a" Suppose that [0A'I < IOAI , fix an arbitrary X E Oi At, and suppose that the event {x # 8m/9
and
jO
E
~ (j' O) > 8m/9
j =m
for sufficiently large m, and let 71 be defined as above. Then there are constants c > 0 and C < r (not depending on n) such that [#~,A, (X(0, 1); X(k, 1))] _< Ce clkl
whenever fi and [k] are sufficiently large and n > [kl. Sketch proof: We proceed as in the proof of Theorem 7.12. In dealing with the righthand side of (7.3.2) we must take into account that possibly 0C n 0A ~: 0. We therefore replace OC by OC \ 0A in the product term and also estimate the probabilities of intersections by conditional probabilities, yielding the upper bound +'~ (X =1 on D)#~+'uc (X =  1 o n OiC \ D) f~,8,c +,rl
for the summands on the righthand side of (7.3.2). Here, #~,c stands for the Gibbs distribution in C with boundary condition equal to +1 on OC N A and equal to r/on OC n 0A. To derive the theorem we need to replace the Peierls estimate (7.3.3) by a similar bound on the last product. The exponential decay of correlations then again follows by simple combinatorics and summations. To make the influence of the boundary condition r/explicit we exploit a contour representation leading to the estimate +,rl(X ##,c
~
lonD)#~+'~(X=lonOiC\D)
F , F t inside C F compatible with D F ! compatible with 0i C\D
y EF
y;EF ~
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The righthand side is defined as follows. For any configuration a E {+ 1, 1} a we draw horizontal resp. vertical lines of unit length between neighboring sites of opposite spins, as if the boundary spins were all plus. We then obtain a disjoint union of closed nonselfintersecting polygonal curves. Each of these curves is called a contour F, and a set F of contours arising in this way is called compatible. We thus have a onetoone correspondence between spin configurations o and compatible sets F of contours. If a   1 on D, then each component of D is surrounded by some contour y (i.e., belongs to the interior Inty of F); the smallest contours surrounding the components of D are collected into a set F of contours. Each set F arising in this way is called compatible with D. The probability that a given set F of contours occurs is not larger than 1I• w~(y), where
xEIntg
yEOA:y,~x
and [gl is the length of ?'; this can be seen by comparing the probability of a configuration containing 1" with the probability of the configuration obtained by flipping the spins in U g c r Inty. These observations establish the inequality (7.3.4). Note that the weight w,7(y) of a contour y depends on the boundary configuration ~; this is because we have chosen to draw the contours for plus boundary conditions rather than 9. It follows that w,l(y) does not necessarily tend to zero when Igl grows to infinity; this is in contrast with the case ~7 = 41. The standard lowtemperature expansion would therefore become much more complicated. However, if the density of plus spins in r/is sufficiently large, or if 0Intg AOA is rather small, the standard weight exp[2fllg l] of the Ising contours will dominate, and the righthand side of (7.3.4) can be estimated, as we will show now.
We unite the sets 1', F' in (7.3.4) into a single set of contours [" = F U 1". The contours in F can overlap, but a site of 0i C can only belong to the interior of at most two contours. On the other hand, every site of OiC is in the interior of at least one contour of 1', and IOiC] > k, the distance of the two spins considered. These ingredients allow us to control the sum on the fighthand side of (7.3.4). If k is so large that the density of plus spins in ~ between 0 and (k, 0) exceeds 8/9 then we find for any collection of contours [" = 1" U 1" as above Z
Z
~EI~ xEInt~
~ ( 1  r/(y)) _< 5/9 Z I~[. yEOA:y"~x ~'EF
This yields
H wo(Y)H
yEF
y~EF ~
wo(y')
0 for any local modification (or "excitation") cr of r/. In other words, 77is a ground state if, for any region A ~ g, the configuration 7/minimizes the energy in A when rlac is fixed. One should note in this context that, in the lowtemperature limit fi 1" oc, the finitevolume Gibbs distribution IX~,A from (2.6.2) tends to the equidistribution on the set of all configurations cr of minimal energy H(crlO). This suggests that, at least in some cases, the lowtemperature phase diagram is only a slight deformation of the zerotemperature phase diagram describing the structure of ground states. This is precisely the subject of PirogovSinai theory which provides sufficient conditions for this to hold, proposes a construction of lowtemperature phases as perturbations of ground states, and also shows that the size distribution of the deviation islands has exponential decay. Suppose next that the Gibbs measure IX is related to the ground state ~ in some way. For example, IX might be obtained as the infinite volume limit of the finite volume Gibbs distributions IX~,A with boundary condition r/, possibly along some subsequence. (Under the conditions of the PirogovSinai theory such a limit always exists.) In the case of a phase transition, when other phases than # exist and one is interested in characteristic properties of IX, one expects that the relationship between IX and ~ becomes manifest in a macroscopic pattern of the typical configurations, in that IX shows 0percolation. In short, we ask for the validity of the hypothesis IG(C3H)I > 1, Ix is extremal in G ( f i H ) and related to a ground state ~ 6 S'2 > # ( x < > ec) > 0
Vx~/2.
(8.1.2)
In the specific cases considered below it will always be clear in what sense # and rl are related; typically, # will be a limiting Gibbs measure with boundary condition rl. We emphasize that (8.1.2) does not hold in general; a counterexample can be constructed by combining many independent copies of the Ising ferromagnet to a layered system, see the discussion after Proposition 8.3 below. Also, even when (8.1.2) holds, it does not necessarily imply that the phase IX is uniquely characterized by the property of 0percolation. How can one establish (8.1.2)? In the context of the Ising model, Coniglio et al. (1976) and Russo (1979) developed a convenient citerion which is based on a multidimensional analog of the strong Markov property and thus can be used for general Markov random fields (Bricmont et al., 1987; Giacomin et al., 1995). One version is as follows. Theorem 8.1 Let (/2, ~) be a locally finite graph, IX a Markov field on f2  S s and tl ~ f2 any configuration. Suppose there exist a constant c E R and a local function f 9s + R depending only on the configuration in a connected set A,
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such that lz ( f ) > c but Ix(fiX
=~ ~ on OF) < c
(8.].3)
for all finite connected sets F ~ A and all ~ E f2 with so(~ ) =_ 0 on OF. Then /z(A
0, i.e., IZ exhibits agreement percolation for rl.
Proof: Suppose by contraposition that # ( A
0
we can then choose some f i n i t e A D A such t h a t # ( A
0. We are interested in agreement percolation for the constant configurations 77 = + 1 resp. r / =  1 , which are the only periodic ground states of the model. We write +
< ; resp. < ~ for the corresponding connectedness relation. Our first result shows that if there is a phase transition then there is pluspercolation for each
1 Random geometry of equilibrium phases
95
lo Gibbs measure except the minusphase/x~, that is, assertion (8.1.2) holds for 77  + 1. This result (due to Russo (1979)) is valid for an arbitrary locally finite graph (/2, "~) with finite critical inverse temperature/3c.
Theorem 8.2 Let Iz be an arbitrary Gibbs measure f o r the ferromagnetic Ising +
model with parameters ~ > O, h  O . I f lz (= lz~ then lz(x ~
~ c~) > Of o r all
+
x E s
In particular, ift~ > ~c then # ~ (x ~
~ c~) > O f or all x E s
Proof: By the sandwiching inequality (4.3.3) and Proposition 4.12, there exists a site x E /2 such that # ( X ( x )  1) > c  #  ~ ( X ( x )  1). On the other hand, the analogue of inequality (4.3.2) for the minus boundary condition shows that t z ( X ( x )  11X ~ 1 on o r )  / Z ~ , r ( X ( x )  1) 1. Using the ergodic decomposition, we can assume that # is ergodic with respect to this group of translations. By Lemma 8.7, there exists a pair ( ~ , 7(c) of halfplanes such that, with positive probability, both ~ and ~ c contain infinite clusters of spins of the same constant sign. For definiteness, suppose ~ is the upper halfplane, and the sign is plus. In view of the finite energy property, it then follows that #(A0) > 0, where for k E Z A~ = {(k, 0) ~
+
~ ec both in 7( and ~c}.
Let A be the event that A~ occurs for infinitely many k < 0 and infinitely many k > 0. The horizontal periodicity and Poincar6's recurrence theorem (or the ergodic theorem) then show that #(A0 \ A) = 0, and therefore # ( A ) > 0. Next, let B be the event that there exists an infinite minuscluster. We claim that # ( A n B)  0. Indeed, suppose # ( A N B) > 0. Since A is tail measurable and horizontally periodic, we can use the finite energy property and horizontal periodicity of # as above to show that the event C  A n {(k, 0) .~ ~ oc for infinitely many k < 0 and infinitely many k > 0} has positive probability. But on C there exist infinitely many minusclusters, which is impossible by the BurtonKeane theorem. To complete the proof, we note that # ( B ) < #(A c) < 1, and thus # ( B ) 0 by ergodicity. In view of Theorem 8.2, this means that #  #~. In the case considered, the proposition is thus proved. The other cases are similar; in particular, in the case of negative sign we find that # = # ~ . []
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H.O. Georgii et aL
Constantspin clusters in the Potts model
Consider the qstate Potts model on the lattice s  Z a introduced in Section 3.3, q, d > 2, and recall the results of Section 6.3 on the phase transition in this model. The periodic ground states are the constant configurations/7i ~ i, 1 < i < i q. We write < ~ for the agreement connectivity relation relative to r/i, and we consider the limiting Gibbs measure #~,q i at inverse temperature/3 associated to r/i, which exists by Proposition 6.9. As a further illustration of assertion (8.1.2), exhibits/percolation whenever there is a phase transition. we show that This is a Pottscounterpart of Theorem 8.2. For its proof, we use the randomcluster representation rather than Theorem 8.1 because for q > 2 there is no stochastic monotonicity available in the spin configuration.
/s
Theorem 8.8 For the Potts model on Z d at any inverse temperature fl with
i
1~(/3H)I > 1, we have [Z~,qi (X < ~ CXZ) > Of o r all x ~ Z d and i ~ {1 . . . . . q}.
Proof: By translation invariance we can choose x  0. In Theorem 6.10 we have seen thatq51p,q (0 ~ ec) = c > 0 f o r f l > tic , w h e r e p  1  e 2~ as usual. In view of (6.3.1), this means that qS~,q,a(0~ ~
A c) > c for all A ~
A1 0. But for the EdwardsSokal coupling pip,q,A o f /Z~,q,A and dpp,q,
before Proposition 6.9) we have {0 ~ A c} C {0
oo) > O f o r all x ~ /2, and activity )~ > O: I f IX x r lz~ dd then tx.even x (x < similarly with 'odd' in place o f 'even '.
This result is completely analogous to Theorem 8.11, and was conjectured by Hu and Mak (1989, 1990) from computer simulations. In these papers, the authors also discuss the case of hardcore particles on a triangular lattice, the hard hexagon model. While Theorem 8.12 does apply to the hardtriangle model on the hexagonal lattice (which is bipartite), the nonbipartite triangular lattice with nearestneighbor bonds is excluded. The results of Hu and Mak (1989, 1990) suggest that Theorem 8.12 still holds for the triangular lattice. A geometric proof of this conjecture would be of particular interest. The hardcore model on the square lattice Z 2 admits an analogue to Corollary 8.4, in that nonuniqueness of the Gibbs measure is equivalent to rlevenpercolation for the Gibbs measure/x9 even x ; see Giacomin et al. (1995) or H~iggstr6m (1997a) for more details. A counterpart to Theorem 8.6 can be found in Georgii and Higuchi (2000). The W i d o m  R o w l i n s o n lattice model: Consider the setup of Section 3.5, with equal activities )~+ = )~_ = ~. > 0 for the plus and minus particles. For ~. > 1 we have two distinct periodic ground states ~/+ ~ + 1 and r/_ ~  1 . From Sect/+ tion 4.4 we know that the associated limiting Gibbs measures/z + = limAl,s #Z,A
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and/z~ exist. Moreover, Theorem 4.17 asserts that a phase transition occurs for some activity ~. ifand only i f # + ( X ( x )  1) > # + ( X ( x ) =  1 ) for somex 6 s Now, it turns out that in this model not only hypothesis (8.1.2) holds, but that the nonuniqueness of the Gibbs measure is equivalent to agreement percolation, not only for the square lattice but for any graph. This comes from the nature of the randomcluster representation of Section 6.7, which is related to the sites rather than the bonds of the lattice, and is a curious exception from the fact that, on the whole, the WidomRowlinson model is less amenable to sharp results than the Ising model. However, by the reasons discussed in Section 6.7, this result does not carry over to the multitype WidomRowlinson lattice model with q > 3 types of particles. Theorem 8.13 Consider the WidomRowlinson lattice model on an arbitrary graph (s "~) for any activity ~ > O. Then the following statements are equivalent. (i) The Gibbs measure for the parameter k is nonunique. /7+
(ii) /z + (x < ~ ~ ) > Ofor some, and thus all x ~ s Sketch of Proof: Consider #z,Ao+ for some finite A. In the same way as the randomcluster representation of Section 6.1 was modified in Section 6.2 to deal with boundary conditions, we can modify the siterandomcluster representation r/+ of Section 6.7 to obtain a coupling of/zz, A and a wired siterandom cluster distribution ~pl,Z,A, so that analogues of Propositions 6.22 and 6.23 hold. As a counterpart to (6.3.3) and by the specific nature of the siterandomcluster representation, we then find that ~+ ( X ( x )
/~.,A

1)
o+ ( X ( x ) =  l )   ~ l
 /~).,A
p,2,A
/)+
(x+~0A)
0+ (x < ~ 0A)
  JZ~.,A
for all x c s In the limit A 1' Z; we obtain by an analogue to (6.3.1)
~(X(x)
/7+
 1 )  ~*~ ( X ( x ) =  1 ) = ~*~ (x ~
~ oo),
and the theorem follows immediately. To conclude this subsection, we note that hypothesis (8.1.2) also holds in other models. We mention here only the AshkinTeller model (Ashkin and Teller, 1943), a fourstate model which interpolates in an interesting way between the fourstate Potts and the socalled fourstate clock model, which is also accessible to randomcluster methods; we refer to Salas and Sokal (1996); Chayes and Machta (1997); Pfister and Velenik (1997), and Chayes et al. (1998).
1
8.5
Random geometry of equilibrium phases
105
Percolation of groundenergy bonds
So far in this section we considered a number of models which are known to show a phase transition, and asked whether this phase transition goes hand in hand with agreement percolation. These results run under the heading "phase transition implies percolation", even though for the square lattice we established results of converse type coming from the impossibility of simultaneous occupied and vacant percolation on Z 2. We now take the opposite point of view and ask whether "percolation implies phase transition". More precisely, we want to deduce the existence of a phase transition (at low temperatures or high densities) from a percolation result. In fact, such an idea is already implicit in Peierls' (1936) and Dobrushin's (1965) proof of phase transition in the Ising model, and is an integral part of the PirogovSinai theory. For models with neighbor interaction as in the Hamiltonian (2.5.1), the underlying principle can be sketched as follows. At low temperatures (or high densities), each pair of adjacent spins (or particles) tries to minimize its pair interaction energy. Note that this minimization involves the bonds rather than the sites of the lattice. So, one expects that bonds of minimal energy  the groundenergy bonds  prevail, forming regions separated by boundaries that consist of bonds of higher energy. Such boundaries, which are known as contours, cost an energy proportional to their size, and are therefore typically small when fl is large. This implies that the groundenergy bonds should percolate. Now, the point is that if the spins along a bond can choose between different states of minimal energy then this ambiguity can be transmitted to the macroscopic level by an infinite groundenergy cluster, and this gives rise to phase transition. In other words, the classical contour argument for the existence of phase transition can be summarized in the phrase: groundenergy bond percolation together with a (clearcut) nonuniqueness of the local ground state implies nonuniqueness of Gibbs measures. We will now describe this picture in detail. We consider the cubic lattice/2 = Z d of dimension d > 2 with its usual graph structure. For definiteness we consider the Hamiltonian (2.5.1) for some pair potential U : S x S + R. We can and will assume that the selfpotential V vanishes; this is because otherwise we can replace U by m
1 U ' ( a , b)  U(a, b) + ~~[V(a) + V(b)],
a, b ~ S,
(8.5.1)
which, together with the selfpotential V t  0, leads to the same Hamiltonian. Let m = min U (a, b) a,bES
be the minimal value of U.
(8.5.2)
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Given an arbitrary configuration o 6 f2, we will say that an edge e = {x, y} 13 is a groundenergy bond for tr if U(cr(x), ~r(y)) = m. The subgraph of Z d consisting of all vertices of Z d and only the groundenergy bonds for cr splits then off into connected components which will be called groundenergy clusters for or. We are interested in the existence of infinite groundenergy clusters, and we also need to identify specific such clusters. Unfortunately, the BurtonKeane uniqueness theorem 5.17 does not apply here because, for any Gibbs measure, the distribution of the set of groundenergy bonds fails to have the finiteenergy property. We therefore resort to considering groundenergy clusters in any fixed twodimensional layer of Zd; the uniqueness of planar infinite clusters can be shown in our case. (An alternative argument avoiding the use of planar layers but requiring stronger conditions on the temperature has been suggested by Fukuyama (2000).) In fact, we have the following result. Theorem 8.14 Consider the Hamiltonian (2.5.1) on the lattice s = Z d, d > 2, with neighbor interaction U and no selfpotential, and let 79 be any planar layer in s (So 79 = s for d = 2.) If ~ is large enough, there exists a Gibbs measure lz ~ G(~H) which is invariant under all 79preserving automorphisms of s and all symmetries of U such that #(3 a unique infinite groundenergy cluster in 79)  1. In the above, a symmetry of U is a transformation r of S such that U(ra, rb) = U (a, b ) f o r all a, b E S; such a r acts coordinatewise on configurations.
Theorem 8.14 is a particular case of a result first derived by Georgii (1981 a) and presented in detail in Chapter 18 of Georgii (1988). We will sketch its proof below. The remarkable fact is that this type of percolation often implies that # has a nontrivial extremal decomposition, so that there must be a phase transition. This happens whenever the set Gu = {(a, b) 6 S x S : U(a, b) m}
(8.5.3)
of bond ground states splits into sufficiently disjoint parts. To explain the underlying mechanism (which may be viewed as the core of the classical Peierls argument, and a rudimentary version of PirogovSinai theory) we consider first the standard Ising model. Example 8.15 The Ising ferromagnet at zero external field. In this model, we have as usual S = {1, 1}, U ( a , b ) =  a b for a , b c S, m =  1 , and Gu = { (  1,  1), (1, 1) }. Hence, either all spins of a groundenergy cluster are negative, or else all these spins are positive. In other words, each groundenergy cluster is either a minuscluster or a pluscluster. This implies that {3 a unique infinite groundenergy cluster in 79} C A_ U A+,
1
Random geometry of equilibrium phases
107
where A_ and A+ are the events that there exists an infinite cluster of negative, resp. positive, spins in 79. For the Gibbs m e a s u r e / z of Theorem 8.14 we thus have/Z (A_ U A+) = 1 and, by the spinflip symmetry of U and thus/Z,/Z (A_) = /Z(A+). H e n c e / z ( A _ ) > 0 and/Z(A+) > 0, so that the measures/Z = / z ( . I A  ) and/Z+  /Z(.IA+) are well defined. Since A_, A+ are tail events, it follows immediately that/Z,/Z+ are Gibbs measures for/~H. Also, A _ N A + is contained in the event that there are two distinct groundenergy clusters in 79 , and therefore has/zmeasure 0. H e n c e / Z  and/Z+ are mutually singular, whence IG(/~H)I > 1. The same argument as in the preceding example yields the following theorem on phase transition by symmetry breaking. A detailed proof (in a slightly different setting) can be found in Georgii (1988, Section 18.2). T h e o r e m 8.16 Under the conditions of Theorem 8.14, suppose that the set Gu defined by (8.5.3) admits a decomposition G u  G1 U ... U GN such that (1) the sets Gn, 1 < n < N, have disjoint projections, i.e., if ( a , b ) 6 Gn, (a', b') E Gn', and n ~= n', then a 7/= a', b ~= b', and (2) for any two distinct indices n , n f ~ {1 . . . . . N} we have r ( G n )  Gn' for some transformation f of S x S which is either the reflection, or the coordinatewise application of some symmetry of U, or a composition of both.
Then, if 13 is sufficiently large, there exist N mutually singular Gibbs measures /ZN E G(fl H), invariant under all even automorphisms of 79 and such that /z l . . . .
/z n (3 an infinite ncluster in 79)  1
[or all 1 < n < N. In particular, there exist N distinct phases for ~ H. In the above statement, an infinite ncluster for a configuration o is an infinite cluster of the subgraph of 79 obtained by keeping only those edges e 6 /3 with (~r(x), ~r(y)) c Gn, where x is the endpoint of e in the even sublattice s and Y 6 s is the other endpoint of e. Also, an even automorphism of 79 is an automorphism of Z d leaving s n "]') invariant. We illustrate this theorem by applying it to our other standard examples. E x a m p l e 8.17 The Ising antiferromagnet in an external field. We have again S  {  1 , 1}, but the interaction is now U(a, b)  ab  ~ ( a + b) for some constant h 6 R. (Here we applied the transformation (8.5.1).) If Ihl < 2d then m =  1 and G u  {(  1, 1), (1,  1) }. G u splits up into the singletons
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G 1 = {(1,  1) } and G2 = {(  1, 1) }. This decomposition meets the conditions of the theorem; in particular, G1 and G2 are related to each other by the reflection of S • S. Consequently, there exist two mutually singular Gibbs measures # 1 and lZ2 which are invariant under even automorphisms and have an infinite cluster of chessboard type, either with plus spins on the even cluster sites and minus spins at the odd cluster sites, or vice versa. Example 8.18 The Potts m o d e l . In this case, S = {1 . . . . . q} for some integer q > 2 and U ( a , b)  1  2l{a=b}. Again m =  1 , and Gu = {(n, n) 9 ! < n < q }. Theorem 8.16 is obviously applicable, and we recover the result that for sufficiently large fl there exist q mutually singular, automorphism invariant Gibbs measures, the nth of which has an infinite cluster of spins with value n. Example 8.19 The h a r d  c o r e lattice gas. This model has state space S  {0, l} and neighbor interaction U of the form U ( a , b ) = ~ i f a = b = 1, and U(a, b ) = log~.Zd(a + b) for all other (a, b) 6 S 2. Here ~ > 0 is an activity parameter, and we have again used the transformation (8.5.!). For ~ > 1 we have Gu = {(0, 1), (1, 0)}, so that Theorem 8.16 applies. Since multiplying U with a factor/3 amounts to changing ~, we obtain that for sufficiently large ~ there exist two distinct Gibbs measures with infinite clusters of chessboard type, just as for the Ising antiferromagnet at low temperatures. Example 8.20 The W i d o m  R o w l i n s o n lattice model. Here we have S = {1 ' 0, 1} and U ( a , b )  cx~ i f a b   1 ' U ( a , b ) =  log~ 2d ( [ a [  ~  [ b f ) o t h e r w i s e , a , b 6 S. I f ~ > 1 t h e n G v  { (  1 ,  1 ) , ( 1 , 1 ) } . Theorem 8.16 thus shows that for sufficiently large ~. there exist two translation invariant Gibbs measures having infinite clusters of plus resp. minusparticles. Although the results in the above examples are weaker than those obtained by the randomcluster methods of Section 6 (when these apply), the ideas presented here have the advantage of providing a general picture of the geometric mechanisms that imply a phase transition, and Theorem 8.16 can quite easily be applied. Moreover, the ideas can be extended immediately to systems with arbitrary state space and suitable interactions. In this way one obtains phase transitions in anisotropic plane rotor models, classical Heisenberg ferromagnets or antiferromagnets, and related Nvector models; see Georgii (1988, Chap. 18). One can also consider nextnearest neighbor interactions, and thus obtain various other interesting examples; for this one has to consider percolation of groundenergy plaquettes rather than groundenergy bonds, which is the setup used by Georgii (1988). Last but not least, the symmetry assumption of Theorem 8.16 can often be replaced by either some direct argument, or a Peierls condition in the spirit of the PirogovSinai theory (see Georgii (1988, Chap. 19)). One such extension will be used in our next example.
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Example 8.21 Firstorder phase transition in the Potts model. Consider again the Potts model of Example 8.18, and suppose for simplicity that d = 2. Any translate in Z 2 of the quadratic cell {0, 1}2 is called a plaquette. For a given configuration cr 6 f2, a plaquette P is called ordered if all spins in P agree, disordered if no two adjacent spins in P agree, and pure if one of these two cases occurs. If q (the number of distinct spin values) is large enough then, for arbitrary fi, there exists an automorphism invariant Gibbs measure # supported on configurations with a unique infinite cluster of pure plaquettes. This variant of Theorem 8.14 is due to Koteck3~ and Shlosman (1982), see also Georgii (1988, Section 19.3.2). Clearly, each cluster of pure plaquettes only contains plaquettes of the same type, either ordered or disordered. For some specific critical value fie(q) both possibilities must occur with positive probability; this follows from thermodynamic considerations, namely by convexity of the free energy as a function of/~ (Koteck2~ and Shlosman, 1982; Georgii, 1988). Conditioning on each of these two cases yields two mutually singular Gibbs measures with an infinite cluster of ordered resp. disordered plaquettes. Furthermore, all spins of a cluster of ordered plaquettes must have the same value, so that by symmetry the "ordered" Gibbs measure can be decomposed further into q Gibbs measures with infinite clusters of constant spin value. As a result, for large q and fi = fie(q) there exist q + 1 mutually singular Gibbs measures which behave qualitatively similar to the disordered phase for fi < fie(q) resp. the q ordered phases for fi > ~c(q). This is the firstorder phase transition in the Potts model for large q. For further discussions we refer to (Koteck3~ and Shlosman, 1982; Wu, 1982; Lannait et al., 1986, 1991) and the references therein. We now give an outline of the proof of Theorem 8.14. Sketch proof of Theorem 8.14: For simplicity we stick to the case d  2. For any inverse temperature/~ > 0 and any square box An ~ [  n , n  112 N Z 2 we 9per write ~/~,n for the Gibbs distribution relative to fi H in the box An with periodic boundary condition. The latter means that An is viewed as a toms, so that (i, n 1) ~ ( i ,  n ) and ( n  1, i) ~ (  n , i ) f o r / c [  n , n  1] N Z; the Hamiltonian Hnper in An with periodic boundary condition is then defined in the natural way. per Let I~ p e r be an arbitrary limit point of the sequence ~#~,n)n>l" Evidently, lZpe r has the symmetry properties required of tz in Theorem 8.14, and ll~per E ~ ( ~ H ) . To establish percolation of groundenergy bonds we fix some ot < 1 and consider the wedge W = {x = (Xl,X2) E /~ : Xl ~ 0, Ix21 < O~Xl}. Let A W be the event that there exists an infinite path of groundenergy bonds in W starting from the origin. We want to show that ll~per (Aw) > 3/4 when/~ is large enough. Suppose ~ r Then there exists a contour crossing W, i.e., a path V in the dual lattice s  Z 2 + (1, 89 which crosses no groundenergy bond for and connects the two halflines bordering W. For each such path y we will
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establish the contour estimate
#per (y is a contour) __ 0 is such that m + 26 is the second lowest value of U. Assuming (8.5.4) we obtain the theorem as follows. The number of paths of length k crossing ~V is at most ck 3~ for some c < c~ depending on or. Hence 1 k (3tSle~) k < k>l 4
#per ( A ~ ) < c Z
for sufficiently large /~. By the rotation invariance of ~per, it follows that lZper(AO) > 0, where A0 is the intersection of AW with its three counterparts obtained by lattice rotations. Roughly speaking, A0 is the event that the origin belongs to two doubly infinite groundenergy paths, one being quasihorizontal and the other quasivertical. Since #per is invariant under horizontal and vertical translations, the Poincar6 recurrence theorem (or the ergodic theorem) implies that the event A ~ = {~ E f2 9Ox~ ~ Ao for infinitely many x in each of the four halfaxes}
has also positive #perprobability. Each configuration in A ~ has infinitely many quasihorizontal and quasivertical groundenergy paths in each of the four directions of the compass, and by planarity all paths of different orientation must intersect. Therefore all these paths belong to the same infinite groundenergy cluster which has only finite holes, and is therefore unique. Hence A ~ is contained in the event B that there exists a groundenergy cluster surrounding each finite set of /2, and #per ( B ) > 0. As B is a tailevent and invariant under all automorphisms of s and all symmetries of U, the theorem follows by setting #   lzper( 9I B ) . It remains to establish the contour estimate (8.5.4). For this it is sufficient to show that per #/~,n (V is a contour) _< (IS e/~) I• (8.5.5) when n is so large that 9/is contained in An. This bound is based on reflection positivity and the chessboard estimate, which are treated at length in Georgii (1988, Chap. 17). Here we give only the principal ideas. The basic obserper. for any vation is the following consequence of the toroidal symmetry of #/~,n i 6 {0. . . . . n  1}, the configurations on the two parts An,+ i 
{x ~ A n "
x~ >_ i or xl Pc then m(fl, Jr) > O, and with Pprobability 1 there exist q distinct phases for the interaction flJ. In particular, this holds when p(rr) > Pc and t5 is large enough. Another way of stating this result is the following. Suppose zr = (1  P)30 + prr+ with Jr+ = zr(I]0, ~ [ ) , and let L+(fl)  f o eftrr+ (dt) be the Laplace transform ofzr+. (Note that then/5(/5, rr) = p ( 1  L + ( 2 / 3 ) ) and p(fl, re) > p ( 1 q L+(2/3)).) Then there is no phase transition for p < Pc, whereas for p > Pc the critical inverse temperature tic(P, rr+)  / 3 c ( r r ) is finite (and decreasing in p) and satisfies the bounds p  Pc
< L+(2 tic(P, Jr+)) Pc the critical inverse temperature satisfies the logarithmic bounds P
pc
 In ~
P
_< 2 / 3 c ( p , 3 1 ) ~   In
p
pc Pq
For q  2, the diluted Ising model, assertion (ii) of Theorem 9.2 gives the slightly sharper upper bound ~c(p, ,~l) < tanh 1 (Pc~p).
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Example 9.4 The case o f p o w e r law singularities. Suppose 7r+(dt) = F(a)ltaletdt is the Gamma distribution with parameter a > 0. Then L+ (/3)  (1 +/3) a, so that for p > Pc the critical inverse temperature satisfies a power law with exponent  1 / a "
( Pc ) 1/aP
Pc
1 < 2/~c(p, zr+) < 
( Pcq ) 1/a

P
Pc
Examples with other kinds of singularities can easily be produced (Georgii, 1984). Theorem 9.2 is due to Aizenman et al. (1987b). Earlier, a generalized Peierls argument was used by Georgii (198 l b) to show that for the diluted Ising model (q  2) in d = 2 dimensions a phase transition occurs almost surely when p > pc  1/2 and fl is large enough. In fact, this paper dealt mainly with the case of site dilution, in which sites rather than bonds are randomly removed from the lattice, and which in the present framework can be described by setting J(xy)  ~ ( x ) ~ ( y ) for a family (~(X))x~Zd of Bernoulli variables; the Jb are thus 1dependent. This was continued in Georgii (1984, 1985) for a class of random interaction models including the randombond Ising model as considered here, obtaining improved bounds on ~c(p, Jr+) for d = 2 as p $ 1/2. Extensions, in particular to d ___ 3, were obtained by Chayes, Chayes and Fr6hlich (1985). The diluted Ising model with a nonrandom external field h g= 0 does not exhibit a phase transition; this was shown by Georgii (1981b) for s = Z d and recently extended to quite general graphs by H~iggstr6m et al. (1999). For the diluted Ising model there is also a dynamical phase transition at the point p  Pc. For p > Pc and/~c(1, 61) < /~ < ~c(P, 61) the relaxation to equilibrium is no longer exponentially fast (Alexander et al., 1998). This illustrates that uniqueness of the Gibbs measure does not in itself imply the absence of a critical phenomenon. Beside such dynamical phenomena, there are also some static effects of the disorder in the uniqueness regime, albeit these are perhaps less remarkable. These are the subject of the next subsection.
9.2 Mixing properties in the Griffiths regime As we have seen above, the diluted Ising ferromagnet shows spontaneous magnetization when p > Pc and fl > tic(P)  tic(P, 61), and multiple Gibbs measures for flJ exist almost surely. In the uniqueness region when still p > Pc but fl < tic(P) we need to distinguish between two different regimes. At high temperatures when actually fl < tic  tic(l), the critical inverse temperature of the undiluted system, we are in the socalled paramagnetic case. This is comparable to the usual uniqueness regime for translation invariant Ising models. At intermediate temperatures, namely when/~c < fl < tic(p), we encounter different behavior
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arising from the fact that the system starts to feel the disorder. This regime is called the Griffiths regime, since it was he (Griffiths, 1969) who discovered in this parameter region the phenomenon now called Griffiths' singularities. He studied sitediluted ferromagnets, but the arguments remain valid also in the bonddiluted case. The basic fact is the following: adding a complex magnetic field h to the Hamiltonian of the diluted Ising model we find that the partition function in a box with plus boundary conditions, as a function of h, can take values arbitrarily close to zero. The reason is that typically a large part of the box is filled by a huge cluster of interacting bonds, giving a contribution corresponding to an Ising partition function in the phase transition region. The radius of analyticity of the free energy around h = 0 is thus zero. In other words, the magnetization m(fl, p, h) cannot be continued analytically from h > 0 to h < 0 through h = 0 when p > pc and/3 > tic. So, the presence of macrosopic clusters of strongly interacting spins (on which the spins show the lowtemperature behavior of the corresponding translation invariant system) gives rise to singular behavior. Related phenomena show up in a large variety of other random models, though not necessarily in the form of nonanalyticity in the uniqueness regime; in general it may be difficult to pinpoint their precise nature. Nevertheless, we will speak of the Griffiths phase or the Griffiths regime whenever such singularities are expected to occur, even when a proof is still lacking. These terms then simply indicate that the usual hightemperature techniques cannot be applied as such. As another illustration we consider a random Ising model with unbounded, say Gaussian coupling variables Jb. Then fl Jb is also unbounded, even for arbitrarily small/3, and with high probability a large box contains a positive fraction of strongly interacting spins. In particular, there is no paramagnetic regime, and the whole uniqueness region belongs to the Griffiths phase. For this reason, it is a nontrivial problem to show the uniqueness of the Gibbs measure. For example, the standard Dobrushin uniqueness condition encountered in (7.1.5) (cf. Dobrushin (1968b) and Dobrushin and Shlosman (1985b)) is useless in this case; similarly, a naive use of standard cluster expansion techniques fails. These methods are bound to fail since they also imply analyticity which is probably too much to hope for (even though we cannot disprove it). In the following we will not deal with the singular behavior in the Griffiths phase. Instead, we address the problem of showing nice behavior, which we specify here as good mixing properties of the system. We shall present two techniques: the use of randomcluster representations, and the use of disagreement percolation.
Application of randomcluster representations: Consider a random Ising model. Spins take values ~(x) = 41, and the formal Hamiltonian is H ( o )  
Jbcr (x)cr (y).
b=(xy)El3
(9.2.1)
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We setl3 boundary estimates measures
= 1. Letlzj, A be the associated Gibbs distribution in A ~ s with condition 7/ ~ ~2. For many applications it is important to have good on the variational distance ]][]/x on A C A (see (7.1.2)) of these with different boundary conditions.
Definition 9.5 The random spin system above is called exponentially weakmixing with rate m > 0 if f o r some C < oo and all A E s and A C A P
(
)
m a x []P'J,A  P],A ][A rl,r/~f2
< C[AI emd(A'Ac)
(9.2.2)
where d(A, A c) is the Euclidean distance o f A and A c.
Various other mixing conditions can also be considered. A stronger condition requires that the variational distance in (9.2.2) is exponentially small in the distance between the set A and the region where the boundary conditions 17 and q' really differ. One could also restrict A and/or A to regular boxes. See Dobrushin and Shlosman (1985b, 1987); Stroock and Zegarlinski (1992); Martinelli and Olivieri (1994); Cesi et al. (1997a, b); van den Berg (1997) and Alexander et al. (1998). Let us comment on the significance of the exponential weakmixing condition above. Remarks: (1) Suppose condition (9.2.2) holds. A straightforward application of the BorelCantelli lemma then shows that for any m ~ < m m a x [[#J,A  # ,A [[A _< Cj]A[ em d(A'AC)
with some realizationdependent Cj < cxz Palmost surely. Integrating over ~' ' for for any Gibbs measure # j we find in particular that # j  lima l.zd #J,a all ~, implying that # j is the only Gibbs measure (and depends measurably on 0 J). Moreover, noting that #j(A]B)  f #j,A(A)#j(dJTIB) for A 6 .T/x and B 6 fAC, we see that this realizationdependent Gibbs measure # j satisfies the exponential weakmixing condition sup
] # j ( A I B )  #j(A)I < CjIAIe m'd(A'A~).
(9.2.3)
A~.T'A,BEf'AC,Uj(B)>O (2) Condition (9.2.3) above also implies an exponential decay of covariances. Let f be any local observable with dependence set A ~ [ and g be any bounded observable depending only on the spins off A, where A C A. Also, let 6 ( f ) = maxc~,~, I f ( o )  f(cr~)l be the total oscillation of f and 6(g) that of g. The covariance # j ( f ; g) of f and g then satisfies Palmost surely the inequality ]#J(f; g)l < C j [ A l 6 ( f ) 6 ( g ) e  m ' d ( A ' a c ) / 2 .
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Indeed, a short computation shows that [ttJ,h(f; g)l is not larger than the lefthand side of inequality (9.2.3) times (~(f)3(g)/2, cf. inequality (8.33) in Georgii (1988). If g is local, a similar inequality holds for covariances relative to finite volume Gibbs distributions in sufficiently large regions with arbitrary boundary conditions. We will now investigate the conditions under which the random Ising system with Hamiltonian (9.2.1) is exponentially weakmixing. We start from the estimate ! ,7  /'L~,A[IA X~ there are multiple Gibbs measures. The proof of this result splits naturally into two parts: first, we need to demonstrate uniqueness of Gibbs measures for X sufficiently small, and secondly we need to show nonuniqueness for ~. sufficiently large. The first half can be done by a variety of techniques. For instance, one can partition R d into cubes of unit sidelength and apply disagreement percolation (Theorem 7.1). Two observations are crucial in order to make this work: that the conditional distribution of the configuration in such a cube given everything else only depends on the configurations in its neighboring cubes, and that the conditional probability of seeing no point at all in a cube tends to 1 as X + 0, uniformly in the neighbors' configurations. The more difficult part, the nonuniqueness for large ~., was first obtained by Ruelle (1971) using a Peierlstype argument. Here we shall sketch a modem stochasticgeometric approach using a randomcluster representation. This approach is due mainly to Chayes et al. (1995) (but see also Giacomin et al. (1995)), and works in showing both parts of Theorem 3.4. The socalled continuum randomcluster model is defined as follows. Definition 10.3 The continuum randomcluster distribution qbx,A with intensity X for the compact region A C R a is the probability measure on ~2A with density 1 2k(x) , f ( x )  ZX,A
x 6 f2A
(10.2.1)
with respect to the Poisson process rcx,A o f intensity X" here Zx,A is a normalizing constant and k(x) is the number o f connected components o f the set = Ux~xB ( x , 1/2).
In analogy to the correspondence between the lattice WidomRowlinson model and its randomcluster representation in Propositions 6.22 and 6.23, we obtain the continuum randomcluster model by simply disregarding the types of the points in the WidomRowlinson model, with the same choice of the parameter X. Conversely, the WidomRowlinson model is obtained when the connected components in the continuum randomcluster model are assigned independent types, plus or minus with probability 1/2 each. To see why this is true, note that for any x E f2A there are exactly 2 k(x) elements of f2A x ~A which do not contradict the hard sphere condition of the WidomRowlinson model and which map into x when we disregard the types of the points. Besides the randomcluster representation, we can also take advantage of stochastic monotonicity properties. Let us define a partial order _~ on f2 • f2
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127
by setting (x, y) _ (x', yt)
if
x c x' and y _ y',
(10.2.2)
so that, a configuration increases with respect to % if pluspoints are added and minuspoints are deleted. A straightforward extension of Theorem 10.4 below then implies that the Gibbs distributions for the WidomRowlinson model have positive correlations relative to this order. The methods of Sections 4.1 and 4.3 can therefore be adapted to show that the WidomRowlinson model on R d at intensity )~ admits two particular phases #+ and #~, where #+ is obtained as a weak limit of the Gibbs measures on compact sets (tending to R d) with the boundary condition consisting of a dense crowd of pluspoints, and #~ is obtained similarly. We also have the sandwiching relation #x ~_9 # •
/z+
(10.2.3)
for any Gibbs measure # for the intensity )~ WidomRowlinson model on R d, so that the uniqueness of the Gibbs measures is equivalent to having #~  / z +. The Gibbs measure for the WidomRowlinson model on a box A with "plus" (or "minus") boundary condition corresponds to the wired continuum randomcluster model 4~,A on A where all connected components within distance 1/2 from the boundary count as a single component. Arguing as in Sections 6.3 and 6.7 we find that uniqueness of the Gibbs measures for the WidomRowlinson model is equivalent to not having any infinite connected components in the continuum randomcluster model. Let ~.c be as in Theorem 10.1. Theorem 10.2 follows if we can show that the continuum randomcluster model 4~),A with sufficiently large intensity ~ stochastically dominates 7rX1,A for some ~.1 > ~.c, whereas 4~1,A __59 7r)v2,A for s o m e ~.2 < ~,c when ~. is sufficiently small To this end we need a point process analogue of Theorem 4.8, which is based on the concept of Papangelou (conditional) intensities for point processes. Suppose bt is a probability measure on f2A which is absolutely continuous with density f (x) relative to the unit intensity Poisson process 7rl,A. For x ~ A and a point configuration x ~ f2a not containing x, the Papangelou intensity of # at x given x is, if it exists, f ( x U {x})
~.(xlx) = ~
/(x)
.
(10.2.4)
Heuristically, ~.(xlx)dx can be interpreted as the probability of finding a point inside an infinitesimal region dx around x, given that the point configuration outside this region is x. Alternatively, )~(. I') can be characterized as the RadonNikodym density of the measure f #(dx)Y]x~x 8(x,x\{x}) on A x f2A, the socalled reduced Campbell measure of #, relative to the Lebesgue measure times # (Georgii and Ktineth, 1997). It is easily checked that the Poisson process rrX,A has Papangelou intensity )~(xlx) = )~.
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The following point process analogue of Theorem 4.8 was proved by Preston (1977) under an additional technical assumption, using a coupling of socalled spatial birthanddeath processes similar to the coupling used in the proof of Theorem 4.8. Later, the full result was proved by Georgii and Ktineth (1997) by a discretization argument. Theorem 10.4 Suppose lZ and ft are probability measures on f2A with Papangelou intensities )~(. I') and ~.(. I') satisfying ~(xlx) _< ~(xl~)
whenever x ~ A and x, ~ ~ f2A are such that x c ~. Then # " ~.c 2Xmax1 yields the presence of unbounded connected components in the same limit. Theorem 10.2 follows immediately. It is important to note that this approach does not allow us to show that the nonuniqueness of the Gibbs measures depends monotonically on s The reason is similar to that for the lattice WidomRowlinson model in Section 6.7" the righthand side of (10.2.5) fails to be increasing in x. It thus remains an open problem whether or not one can actually take Uc = s in Theorem 10.2. There are several interesting generalizations of the WidomRowlinson model. Let us mention one of them, in which neighboring pairs of particles of the opposite type are not forbidden, but merely discouraged. Let h 9[0, c~) + [0, c~] be an "interspecies repulsion function" which is decreasing and has bounded support. For A C R d compact and )~ > 0, the associated Gibbs distribution #h,~,A on f2A x f2A is given by its density f(x, y)
1 
Zh,~.,A
exp(
~ xex,y~y
h(lxy,)).
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relative to zrz,A x rrz,A. Infinite volume Gibbs measures on ~ x f2 are then defined in the usual way. Lebowitz and Lieb (1972) proved nonuniqueness of Gibbs measures for large ~ when h (x) is large enough in a neighborhood of the origin. Georgii and H~iggstr6m (1996) later established the same behavior without this condition, and for a larger class of systems, using the randomcluster approach. This involves a generalization of the continuum randomcluster model, which arises by taking the random connection model of Section 10.1 with connectivity function g(x) = 1  e  h ( x ) and biasing it with a factor 2 ~(z), where k(z) is the number of connected components of a configuration z of points and edges. To establish the phase transition behavior of this "softcore WidomRowlinson model" (i.e., uniqueness of Gibbs measures for small ,k and nonuniqueness for large X) one can basically use the same arguments as the ones sketched above for the standard WidomRowlinson model. However, due to the extra randomness of the edges some parts of the argument become more involved. In particular, there is no longer a deterministic bound (corresponding to K m a x ) o n how much the number of connected components can decrease when a point is added to the randomcluster configuration; thus more work is needed to obtain an analogue of the first inequality in (10.2.6). To conclude, we note that the WidomRowlinson model on R d has an obvious multitype analogue with q > 3 different types of particles. This multitype model still admits a randomcluster representation from which the existence of a phase transition can be derived (Georgii and H~iggstr6m, 1996). There is, however, no partial ordering like (10.2.2) giving rise to stochastic monotonicity or an analogue of (10.2.3).
Acknowledgments
It is a pleasure to thank J. L. Lebowitz for suggesting (and insisting) that we should write this review. We are also grateful to L. Chayes, A. C. D. van Enter and J. L6rinczi who looked at parts of the manuscript and made numerous suggestions, and to Y. Higuchi for discussions on Proposition 8.5.
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Wolff, U. (1989). Collective Monte Carlo updating for spin systems. Phys. Rev. Lett. 62, 361364. Wu, E Y. (1982). The Potts model. Rev. Mod. Phys. 54, 235 268. Zahradnfk, M. (1984). An alternate version of PirogovSinai theory. Commun. Math. Phys. 93, 559581. Zahradn~, M. (1987). Analyticity of lowtemperature phase diagrams of lattice spin models.J. Star. Phys. 47, 725755. Zahradnfk, M. (1996). A short course on the PirogovSinai theory. Rendiconti di Matematica.
2
Exact Combinatorial Algorithms: Ground States of Disordered
Systems
M. J. Alava Laboratory of Physics, Helsinki University of Technology, PO Box 1100, HUT 02015, Finland
P. M. Duxbury Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan State University, East Lansing, MI 48824, USA
C. F. Moukarzel Instituto de Ffsica, Universidade Federal Fluminense, 24210340 Niteroi, RJ, Brazil
H. Rieger Institut fiJr Theoretische Physik, Universit#t des Saarlandes, 66041 SaarbrScken, Germany
PHASE TRANSITIONS
Copyright (D 2001 Academic Press Limited
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1 Overview
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2 Basics of graphs and algorithms 2.1
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Graph notation and overview of network problems
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Algorithmic complexity . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.3
Basic Algorithms
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3 Flow algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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M a x i m u m flow/minimum cut . . . . . . . . . . . . . . . . . . . . . . . .
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M i n i m u m  c o s t  f l o w problems
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4 M a t c h i n g algorithms
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Introduction and definitions . . . . . . . . . . . . . . . . . . . . . . . . .
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A u g m e n t i n g paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Matching problems
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5 Mathematical p r o g r a m m i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Linear and convexcost p r o g r a m m i n g
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6 Percolation and minimal path . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1
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Connectivity percolation
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G r e e d y algorithms and extremal processes . . . . . . . . . . . . . . . . .
7 R a n d o m Ising magnets
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Randomfield magnets and D A F F
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Ising spin glasses and Euclidean matching . . . . . . . . . . . . . . . . .
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8 Line, vortex and elastic glasses . . . . . . . . . . . . . . . . . . . . . . . . . .
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Disordered flux arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Arrays of directed polymers
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Disordered elastic media
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9 Rigidity theory and applications
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Rigidity theory
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Rigidity percolation on triangular lattices
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Connectivity percolation
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Applications to soft materials . . . . . . . . . . . . . . . . . . . . . . . .
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10 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Overview
This review provides an introduction to combinatorial optimization algorithms and reviews their applications to groundstate problems in disordered systems. We focus on problems which are solvable in polynomial time, and for which exact largescale (e.g. a million sites) solutions are possible. This excludes some old favorites such as the travelingsalesman problem for which no exact polynomial algorithm is known. Nevertheless, we demonstrate that many difficult groundstate problems, such as: the randomfield Ising model and diluted antiferromagnets in a field; spin glasses in two dimensions; domain walls in randombond magnets; arrays of directed polymers; periodic elastic media; and rigidity percolation, are exactly solvable in polynomial time. In addition to their intrinsic interest, these problems provide a set of systems on which heuristic algorithms should be checked for accuracy and convergence. The intent of this review is to collect together and introduce the relevant combinatorial algorithms (Sections 25) and to illustrate and review their applications to physics problems (Sections 69). Our introduction and survey of the relevant optimization algorithms begins in Section 2, which introduces the basic graph terminology and some of the mathematical notation we use throughout the review. We define what is meant by a polynomial algorithm and emphasize the importance of such algorithms. Section 2 also introduces the basic search algorithms (breadthfirst search and depthfirst search) and methods for finding minimal paths (e.g. Dijkstra's method) and minimal spanning trees (e.g. Prim's algorithm). Section 3 covers flow algorithms (augmenting path and preflow methods), starting with the maximumflow problem and continuing to the minimumcostflow problem. Section 4 introduces matching algorithms, with particular emphasis on bipartite matching. Section 5 completes the algorithm survey by outlining the connection between the integer optimization algorithms described in Sections 24 and the problem of mathematical (e.g. linear)
programming. We describe in detail some of the key algorithms, which we write in "pseudocode" which is typical of the computer science literature. These algorithms, which had their origins concurrently with the development of "computer science" as an independent discipline, are being continuously refined. Nevertheless the basic ideas about search, path, tree, flow and matching methods were developed in the 1950s. However, important conceptual developments have occurred recently such as preflow methods for flow problems and interiorpoint
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methods for linear programming. There are excellent accessible textbooks on algorithms (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990) as well as more specialized books and reviews covering flow algorithms (Ford and Fulkerson, 1962; Ahuja et al., 1993a, b), matching (Lov~sz and Plummer, 1986) and mathematical programming (Papadimitriou and Steiglitz, 1982; van Leeuwen, 1990; Rockafellar, 1984; Wright, 1997). Efficient free software is available on the internet (see note after the References). Ones that we have downloaded and used include: the very broad LEDA library; the efficient flow algorithms by Goldberg; and the interiorpoint convexcost programming codes by Wright. In some cases, e.g. rigidity percolation, the algorithmic theorems were available (Hendrickson, 1992) but the first implementations were carried out by physicists (Jacobs and Thorpe, 1995; Moukarzel and Duxbury, 1995). Sections 69 cover the applications of combinatorialoptimization methods to a wide variety of problems in disordered systems. Since these sections cover many different problems, it is not possible, in one review, to do justice to all contributions to each of them. However, we do strive to include all of the papers which use exact combinatorial algorithms, which are in the polynomial class, to study these problems. In a few cases we also allude to some important advances in nonpolynomial, but exact methods, for example in the spin glass problem. There are an enormous number of useful heuristic (approximate) methods which we do not exhaustively cover, though we do refer to them when relevant. The applications discussed in Sections 69 rely upon mappings between optimization problems, discussed in Sections 24, and physics problems. Some of these mappings are quite easy to see, but others are quite sophisticated and require considerable formal development (e.g. rigidity theory). In many cases, we devote separate subsections to the formal development and to the description of the key mappings. In Section 6, we discuss the connectivitypercolation problem (Stauffer and Aharony, 1994) and the minimalpath problem (HalpinHealy and Zhang, 1995; L~issig, 1998). Precise algorithms have traditionally played a key role in the analysis of these two problems. We clarify the relationship between these traditional algorithms and optimization methods. In particular: Prim's algorithm for minimum spanning tree is essentially the same as the invasion algorithm for percolation; while Dijkstra's algorithm finds minimal paths, as does the transfermatrix method, however, Dijkstra's method is more general as it allows overhangs. In addition, both the Prim and Dijkstra algorithms work by generating a growth (Barabasi and Stanley, 1995) or invasion front which sweeps through the lattice. The invasion rules are in both cases extremal or greedy, and they generate rough growth fronts without tuning any variables. That is, they display "selforganized" critical behavior (Bak et al., 1987).
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In Section 7, we discuss random Ising magnets (Young, 1998). In particular: interfaces in randombond magnets; randomfield ferromagnets (RFIM) and diluted antiferromagnets in a field (DAFF) and; spin glasses and Euclidean matching. In each of these problems, there is a different mapping to an optimization problem. The optimization methods are restricted to ground states or excitations generated in some way from the ground states. Nevertheless groundstate properties capture much of the most interesting and difficult aspects of disordered systems. For example, metastability, lack of selfaveraging, chaos and other phenomena which are difficult to analyze precisely are most evident at zero temperature. In the jargon of the renormalization group, these problems are frequently governed by a zerotemperature fixed point. As in the directedpolymer problem, the functional renormalization group has been quite successful in predicting the exponents for randommanifold and randomsurface problems. The first precise confirmations of these results in (2 + 1) dimensions have resulted from the application of optimization methods. However, the continuum theory of the RFIM and DAFF phase transitions in three dimensions remain ambiguous, despite some high precision data generated using optimization methods (Section 7.3.3). Section 8 covers models for disordered vortex matter (Blatter et al., 1994). Several limiting cases of this very general problem can be solved exactly using optimization methods. These limiting cases are derived in Section 8.2. All of the tools developed in Sections 24 find application in various limiting cases of disordered vortex matter. One particularly surprising result is that the minimumcostflow method finds the exact groundstate configuration of a set of N directed polymers with gfunction interaction (Section 8.3). Another solvable limit is the problem of a periodic elastic medium. Until recently there was a controversy about whether the ground state of periodic elastic media was in the same class as conventional rough surfaces in (2 + 1) dimensions, or whether there is a superrough phase in the ground state. Optimization methods have convincingly confirmed the renormalizationgroup prediction of a superrough phase (Section 8.5.5). However, further extensions of the optimization methods have allowed the inclusion of dislocations in the periodic elastic medium. This destroys the superrough Bragg glass state at low temperatures (Section 8.5.6). Rigidity theory and its applications (Thorpe and Duxbury, 1999) to rigidity percolation, and soft condensed matter is the subject of Section 9. The basic model here is simple to define. Consider a lattice whose vertices are connected together with stiff Hooke springs. Actually they could be connected together by any centralforce springs. The key point is that the only restoring force is axial. Two points connected together by a spring are free to move as long as they maintain their axial separation. Now we want to know how many such springs are needed to make a set of points rigid with respect to
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each other. Of course we can solve for the forces using Newton's laws but that is tedious. Instead there are combinatorial methods which just count degrees of freedom and compare that freedom with the number of constraints. Rigidity theory (Section 9.2) makes this counting precise and turns it into a powerful tool which identifies flexible, rigid and overconstrained regions of soft materials. This theory also enables a mapping of rigidity theory to an optimization problem (bipartite matching), which in turn has led to the precise analysis of rigidity percolation on triangular lattices. There are many analogies with connectivity percolation, which can be seen as one limit of the general rigidity problem (Moukarzel and Duxbury, 1999). Further extensions to soft condensed matter, such as proteins, are also starting to develop (see Section 9.5). We make a few closing remarks in Section 10.
2
2.1
Basics of graphs and algorithms
Graph notation and overview of network problems
A graph ~(V, A)(Harary, 1969; West, 1996; van Lint and Wilson, 1992)consists of a vertex set V and an arc set A (= edge set E). Each member of the edge set (i, j ) 6 A connects two of the vertices i 6 V and j 6 V in the vertex set. We use A(i) to denote the set of arcs connected to node i. We use bond, arc and edge interchangeably. Similarly with node, site and vertex. Bipartite graphs are a very important subclass of graphs, and frequently are easier from a computational standpoint. A bipartite graph/3(X, Y, A) is a graph in which the node set V = X U Y can be partitioned into two parts X and Y, such that all of the arcs in the graph connect nodes in X with nodes in Y. Nearestneighbor hypercubic lattices are bipartite, while triangular and facecenteredcubic lattices are not. Most physics applications involve sparse graphs which have a small number of arcs incident on each node. Many of the upper bounds on algorithmic performance occur on dense or complete graphs where a very large number of arcs are incident on each node. Complete graphs correspond to infiniterange models where every node is connected to every other node. We emphasize the algorithmic methods and bounds for sparse graphs. We assign a cost, r and/or a capacity, Uij , t o each edge in the graph. The capacities, Uij, a r e restricted to be nonnegative integers, while the costs may be arbitrary integers. In many of the applications discussed in this review U ij and r a r e integers which are randomly drawn from uniform, Gaussian or bimodal distributions. Once the graph nodes or edges have been assigned flow capacities or costs, they are frequently called networks, and they are studied using network
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optimization methods (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984; Cormen et al., 1990; van Leeuwen, 1990). We consider directed arcs, in which an arc (i, j) has cost Cij and capacity Uij for a path which goes from node i to node j. An arc in the opposite direction (j, i) may have a different cost Cji and a different capacity Uji. Undirected arcs are the special, symmetric case, in which Cij  Cji and Uij  Uji. In the physics applications, it is usual to consider either the undirected case, or the special directed case where either U ij  0 o r u ji " O. Note that, even in these special cases, the algorithmic procedures for flow problems (Ford and Fulkerson, 1962; Ahuja et al., 1993a, b) frequently generate a residual graph which has forward and reverse arcs with different costs or capacities. Thus, in the algorithms, we must keep track of a graph in which Cij ~ Cji and Uij ~ Uji. When we refer to a graph G(V, A), we mean a graph with both forward and reverse arcs which have, in general, different values for their costs, capacities and flows. In minimumpath and minimumspanningtree problems each bond is assigned a cost, Cij, and the problem is to optimize the appropriate cost function. In the maximumflow problem, each bond is assigned a capacity Uij and the problem is to find the maximum flow which the network can sustain, given those capacities. In minimumcost flow each bond is assigned a cost and a capacity. The problem is to find the lowest cost flow, given the capacity constraints and a set of sources and sinks. The sources and sinks are introduced at the vertices of the graph. Several other vertex and edge variables will be defined and used in the more detailed sections describing the algorithmic solutions to these networkoptimization problems. In graph algorithms we frequently use the concept of a path, e.g. Pst, between two specified nodes s and t. Any such path consists of a connected sequence of edges. In many problems we are interested in optimal or lowestcost paths. In flow (Ahuja et al., 1993b) and matching problems (Lov~isz and Plummer, 1986), we often iteratively find the optimal solution by selecting paths and augmenting the solution using them. We restrict all of the variables and parameters in the problems to the integers. This restriction maintains the property that an exact solution is possible in polynomial time, but also enables us to model problems like directed polymer arrays using integer flow lines. Before proceeding to the algorithms, we give a brief overview of the ideas and notation of computational complexity, in particular we give a more precise definition of what we mean by a polynomial algorithm.
2.2 Algorithmic complexity Given a graph, ~(V, A), a problem is polynomial (P) if its solution can be found in a time bounded above by a polynomial in the number of arcs and/or number
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of vertices in the graph (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990). The number of nodes in the graph is denoted by IVI, and the number of arcs is IAI. If an algorithm is in P, then it is O(IAI x IVIY), where x and y are finite. The big 0 denotes the scaling behavior of the upper bound on the computational time. The lower bound, which we shall not use, is denoted by big f2. The upper bound is not typical of most physics applications, and we use ~ IAlxlvI y to denote the scaling behavior of a particular application (this is the average computation time). The discovery of an exact polynomial algorithm for a hard combinatorial problem is usually of great theoretical and practical (in terms of computational efficiency) significance. This review is devoted to P problems and their applications. The notation for problems which do not have polynomial solutions is less intuitive. NP (nondeterministic polynomial) refers to problems for which a solution can be verified in polynomial time. Clearly the class NP contains the class P. NPcomplete refers to NP problems which: (i) can be verified in polynomial time; (ii) for which there is no known exact polynomial algorithm (on "classical" computers) and; (iii) which are related to each other by polynomial reductions. This means that if a polynomial algorithm is found for one NPcomplete problem, polynomial algorithms exist for all the others. Three examples of NPcomplete problems are: the travelingsalesman problem; satisfiability (whether a set of logical clauses contradicts itself); Hamiltonian circuits and; 3matching (a generalization of bipartite matching). The NPcomplete class is now rather large, so it appears very unlikely that the NPcomplete problem class will eventually be proven to have P solutions. Nevertheless efficient algorithms can be found for some "typical" cases of NPcomplete problems. On the other hand, certain realizations of NPcomplete problems are very difficult to solve, and there is speculation that the difficult regimes are critical in the statisticalmechanics sense. (Proximity to a phase transition does frequently, but not always, increase the computational time in the groundstate problems discussed in Sections 69). Finally, it should be noted that in practice an algorithm which is not polynomial can still be competitive with P algorithms. The classic example is the simplex method for linear programming, which is not polynomial. Simplex methods remain competitive (in practice) with interiorpoint methods (which are in P) even for large sparse problems.
2.3
Basic Algorithms
We discuss three basic classes of graph algorithm: search (Section 2.3.1); minimal path (Section 2.3.2) and; minimal spanning tree (Section 2.3.3) (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990). These problems are solved exactly by greedy algorithms (Papadimitriou and Steiglitz, 1982; Lov~isz and Plummer,
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1986; Cormen et al., 1990) which work by choosing moves which are locally optimal. This locality condition makes their implementation very efficient. We show that for sparse graphs the computational time of efficient algorithms for these problems are O(IAI). In contrast, generic implementations are O(IVI2).
2.3.1
Search algorithms
Consider a connected graph {7(V, A) containing a vertex set V and arc set A. A connected graph has sufficient arcs such that a connected path exists between any two nodes i and j. We input the graph connectivity, i.e. a list of arcs which are present (the parameters Uij and Cij do not play any role in the calculation). There are two basic search strategies on {7(V, A)" breadthfirst search and depthfirst search. These strategies are closely related, though depthfirst search is more complicated. Breadthfirst search (Cormen et al., 1990; West, 1996) labels each vertex with the minimum number o f bonds l(i) which must be crossed in order to go from a starting site s to site i. This is found by starting at s, which is assigned label l(s)  O, and by growing or burning outward from that site. In the algorithm below we define S to be the set of labeled sites, and S to be the set of unlabeled sites. Clearly S U S = V. We also use a vector called p r e d ( i ) , which identifies the site from which the breadthfirst search arrived at the current site. This is important in reconstructing breadthfirst paths and is used in the flow algorithms described in Section 3.
algorithm begin
Breadthfirst
search
S{s}, s=V\{s} l(s)  O,
pred(s) = 0
while ISI < IVI do begin choose ( i , j ) 9 l(i) : mink{l(k)lk e S}, j e S, ( i , j ) e A then l ( j ) = l(i) + l , S  S \ { j } , S : S U { j } , pred(j)  i end m
end (Here we use a U b to denote the union of two sets a and b and a \ b to denote the set remaining after the elements of set b are removed from the set a.) The above algorithm is very inefficient if the choose operation above requires a search over all i 6 S at each step. However, it is easy to keep an ordered list of sites, those at the growth front, from which the next growth site is selected. An efficient algorithm generates and adds to the list all of the next candidates for burning each time a local advance of the growth front occurs. The sites which are added to
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this list have a monotonically increasing label (1(i)). Thus sites are added to the top of the list (which has highest label) and growth is to the site at the bottom of the list (which has the lowest label). This list is often called a "firstinfirstout" or FIFO list (Cormen et al., 1990). The algorithm ends when the FIFO list is empty. Using this procedure ensures that breadthfirst search is O(IA[). In the percolation literature the label l(i) has been called the chemical distance (Stauffer and Aharony, 1994). This and other applications of breadthfirst search will be discussed in Section 6.2. Depthfirst search (Cormen et al., 1990; West, 1996) labels in the same way as breadthfirst search, except it labels only one of the next level unlabeled sites at each iteration, and growth is always from the highest available label. Clearly a strategy for choosing which of the next level sites to label is needed. This choice depends on the application. A common and useful choice, for a planar graph, is to "stay as far to the right as possible" or "as far to the left as possible". These procedures identify the perimeter of a graph. Depthfirstsearch paths may lead to a "dead end", in which case it is necessary to "backtrack" to the highest available label which is not at a dead end. Depthfirst search is the basis of algorithms for the hull in percolation. Finding the backbone in connectivity percolation is possible once a graph has been separated into its biconnected parts. An O(IAI) algorithm to find the biconnected parts (and hence the backbone) has been available since the early 1970s (Tarjan, 1972), but it was only noticed recently by the physics community (Grassberger, 1992a, 1999). There are several methods available for finding the backbone. The standard method in the physics community is the burning algorithm which does a forward and backward breadthfirst search to iteratively remove dangling ends (Herrmann et al., 1984). However, a forward and reverse depthfirst search of the whole graph finds the hull of the backbone in two sweeps of the graph, and is more efficient. Nevertheless, high precision results have been found using efficient implementations of burning (Rintoul and Nakanishi, 1992, 1994), at the percolation threshold. More recently matching (Moukarzel, 1998a) methods have also been used to find the backbone and have also lead to highprecision results. Nevertheless, all of the available backbone algorithms remain storage limited, as some information at each node of the graph remains necessary. For this reason the very large calculations carried out for the infinite cluster (see Rapaport (1992) and Lorenz and Ziff (1998)) remain out of reach for the backbone.
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2.3.2
Exact combinatorial algorithms
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Shortest path
Given a set of costs Cij o n each arc of a graph, we calculate the distance label d(i), which is the cost of a minimumcost path (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990) from a starting node s to the node i. Dijkstra's algorithm, which works for nonnegative costs, is a labelsetting algorithm to solve the shortestpath problem. It is a labelsetting algorithm because it finds the exact distance label correctly at the first attempt. In contrast, labelcorrecting algorithms iteratively approach the exact distance label, and work even if some of the costs are negative, provided there are no negativecost cycles in the graph. In order to reconstruct the set of bonds which make up the minimalcost path from site s to site i, the algorithms also store a predecessor label, pred(i), which stores the label of the previous site from which the minimal path reached i. The set of minimal paths from a starting site s to all of the other sites in the graph forms a spanning tree, Tp. An example is presented in Fig. 2.1. As will be shown in Section 6, the minimalpath problem is closely related to a polymer in a random medium and hence to growth problems. Both labelsetting and labelcorrecting algorithms use the key properties which shortest paths obey: (i) For each arc belonging to the shortest path tree from node s:
d(j)  d(i) +
(i, j)inTp.
Cij
(ii) For each arc which does not belong to a shortest path:
d(j) 0,
otherwise.
(2.3.3)
The proof of these properties relies on the spanningtree structure of the set of minimal paths, namely that each site of the tree has only one predecessor. Thus if there is a bond with c/~ < 0 which is not on the minimalpath tree, adding that bond to the tree and removing the current predecessor bond for site j (which from condition (2.3.1) has zero reduced cost), leads to a reduction in the cost of
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113t 2 ~lst 1 ~I~6~5 ~12~1 l'q
, F1
I]!to 1skis3 ~Izll~9 161191k~ Iz~9141 2
3
151
Io
k s'lsl Fig. 2.1 The tree of minimal paths from the source node (shaded) to all other nodes in a directed square lattice (The arcs of the graph only allow paths which are in the positive {01 } and positive {10} directions). All bonds between nearest neighbors are labeled with their costs, but only the tree of minimal paths is shown. Each node is labeled with the cost of a minimal path from the source to that node. (Generated using the demonstration programs from the LEDA library (see note at end of References).)
the minimalpath tree. Thus any tree for which there exists a bond with c/~ < 0, is not a minimalpath tree. For later reference we also note that a directed cycle, W, has the property,
Z cd Z Cij, (i,j)eW (i,j)~W
(2.3.4)
which follows from (2.3.1)(2.3.3). We now discuss Dijkstra's method for the minimum path, which works by growing outward from the starting node s in a manner very similar to the breadthfirst search. At each step Dijkstra's algorithm chooses to advance its growth front to the next unlabeled site which has the smallest distance from the starting node.
algorithm Dij kstra begin S={s}, s : v\{s}
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d(s) = O, pred(s) = 0 while [SI < IVI do begin choose (i, j ) 9 d ( j )  mink,m{d(k) + Ckmlk E S, m E S, (k, m) E A} then S   S \ { j } , SSU{j}, pred(j)=i end end
The algorithm maintains the minimal distance growth front by adding the node j E S with minimal distance label d (j). The proof that d (j) generated in this way is actually a minimalcost label proceeds as follows: (i) Assume that we have a growth front consisting of sites which are labeled with their minimal path lengths to the source s. (ii) The next candidate for growth is chosen to be a site which is not already labeled, and which is connected to the growth front by an arc (i, j) E A. (iii) We choose the site j for which d ( j ) is minimal d ( j ) = min~,m {d (k) + Clcm,k E S , m E S, (k,m) EA}. m
(iv) Because of (iii) and because all of the costs are nonnegative there can be no path from the current growth front to site j which has smaller distance than d ( j ) . This is because any such path must originate at the current growth front and hence must use a nonoptimal path to generate any alternative path to j (negative costs can compensate for locally nonoptimal paths from the growth front and hence Dijkstra's method is restricted to nonnegative costs.) Minimal path with positive costs is an example of a globaloptimization problem which is exactly solved by local growth (greedy) dynamics. The name greedy was invented for the problem of a maximal weight forest (MWF), which is related to minimal spanning tree by a simple transformation. MWF works by choosing the highestweight edges at each step, hence the term greedy. The term extremal has been used in the physics community to refer to algorithms that choose either largestcost or smallestcost possibilities at each step. We discuss this in a more general context in Section 6.4. The "generic" Dijkstra's algorithm scales as o(Ig12), if the choose statement in the above algorithm requires a search over all the sites in the lattice. It is easy to do much better than this by maintaining a list of active sites at the growth front (as in breadthfirst search). However, now we must choose the lowestcost site from among this list. Thus the potential growth sites must be ordered according to their distance label. This ordering must be reshuffled every time a
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new growth site (with a new distance label) is added to the list. In the computer science community this is typically done with heaps which consist of a treelike data structure. Heap reshuffling is O(lnlAI) which reduces the algorithmic bound to O(IAIlnlAI). However, if the bond costs are integers(as we consider here), the site distance labels themselves can be used as pointers. Thus we can set up a queue with the distance label as pointers and the site labels with that distance in the queue(or, in computer science terminology, we use buckets). The number of buckets that is required, nb, is n b > 2C where C = max(i,j){Cij } is the maximum cost. For example if the costs are chosen from the set 1, 2 . . . . . 10, then C  10. As long as C is finite, and the graph is sparse, buckets are very efficient both in speed and storage (Dobrin and Duxbury, 1999; Duxbury and Dobrin, 1999). Using buckets, the next growth site is immediately given as one of the sites with smallest distance pointer. In addition, since each potential growth site has a distance label, we also know where to add any new growth sites in the queue. The queue constructed in this way contains up to z (z is the coordination number of a site) copies of each growth site. Only one of these copies is used in constructing Te. The redundant copies are deleted once a site has been added to Te. Using distance pointers, we have implemented a Dijkstra's algorithm for integer costs which scales as O(IAI). Note that breadthfirst search is the special case of identical costs (i.e. Cij : constant) in Dijkstra's method. As we said, Dijkstra's algorithm is a labelsetting algorithm. We now describe a labelcorrecting method, which solves the minimalpath problem when there are negative costs provided there are no negative cycles (i.e. closed directed paths W with E(i,j)~w Cij < 0). Equations (2.3.1) and (2.3.2) form a basis for the following labelcorrecting algorithm. The idea is to start with a nonoptimal spanning tree and to iteratively remove bonds with negative reduced costs from it. This iteration proceeds until no negative reducedcost bonds remain. algorithm L a b e l  c o r r e c t i n g begin d(s) : 0 a n d pred(s) : 0
d ( j ) : o o for each node j E V\{s} while some arc (i,j) s a t i s f i e s d(j)>
begin
d(i)}cij do
d ( j ) : d ( i ) } cij
pred(j)  i end end Initially the distance labels at each site are set to a very large number (except the reference site s which has distance label d(s) = 0). This method requires that
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the starting distance labels d ( j ) > exact values and the choice of d ( j ) = cx~ ensures that. In practice it is efficient to grow outward from the starting site s. The algorithm may sweep the lattice many times until the correct distance labels are identified. The worst case bound on running time O(min{lVl2lAlC, IAl21Vl}) with C = max{Icijl}, which is pseudopolynomial. An alternative procedure is to sweep the lattice once to establish approximate distance labels and then to iterate locally until local convergence is found. This FIFO implementation has complexity O(I VI IAI). Note that if there are negative cycles, the instruction d ( j ) = d(i) + Cij would decrease some distance labels ad (negative) infinitum. However, if there are negative cycles, one can detect them with an appropriate modification of the labelcorrecting code: One can terminate if d(k) <  n C for some node i (again C = max [cijl) and obtain these negative cycles by tracing them through the predecessor indices starting at node i. This will be useful in the negativecyclecancelling method for minimumcost flow (Section 3.2.2).
2.3.3
Minimal spanning tree
The minimal spanning tree (Papadimitriou and Steiglitz, 1982; Cormen et al., 1990) of a connected graph with arc costs Cij is a tree which: (i) visits each node of the graph and; (ii) for which Y~tree cij is a minimum. Prim's algorithm and Kruskal's algorithm are two methods for finding the minimal spanning tree (Cormen et al., 1990; West, 1996). Prim's algorithm is very similar in structure to Dijkstra's algorithm, although the physics is very different. In fact (see Section 6.2), as has been noted (Barabasi, 1996), Prim's algorithm is essentially equivalent to the invasion algorithm for percolation. In Prim's algorithm we start by choosing the lowest cost bond in the graph. The algorithm then uses the two sites at the ends of this minimal cost bond as the starting sites for growth. Growth is to the lowest cost bond which is adjacent to the growth front. The algorithm terminates when every site has been visited. The cost of the minimal spanning tree is stored in Cr, and the bonds making up the minimal spanning tree are stored in T.
algorithm Prim begin choose (s, r) 9 Csr then S  {s, r} , S while ISI < IVl do begin choose (i, j ) 9
 mink,m{Ckml(k, m ) e A} V\{s, r},
T  (s, r ) ,
C r = Csr
ci]  mink,m{Ckmlk e S, m e S, (k, m ) e A}
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then S  S \ { j } ,
S = S U {j},
CT  CT + Cij,
T  T U (i, j )
end end It is evident that Prim's algorithm is almost identical to Dijkstra's algorithm. The only difference is in the choose instruction, which gives the cost criterion for the extremal move at the growth front. In the minimalpath problem, one chooses the site at the growth front which has the minimumcost path to the source, while in the minimalspanningtree problem one simply chooses the minimumcost bond. Both of these problems lead to spanning trees, but Prim's has lower total cost (it is less constrained). In Dijkstra's algorithm one is checking the cost of the path from the tested site all the way back to the "source" or "root" of the tree. It is thus more nonlocal. As we discuss in Section 6.3, Dijkstra's algorithm leads to selfaffine growth fronts, while Prim's method leads to selfsimilar growth fronts. Using the same distance pointer strategy as described above for Dijkstra's method, Prim's algorithm is O(IAI). Heap implementations of Prim's method are O(IAllnlAI) as for Dijkstra's method. An alternative method for finding the minimal spanning tree is Kruskal's algorithm, which nucleates many trees and then allows trees to merge successively into one tree by the end of the procedure. This is achieved by allowing growth sites to nucleate off the growth front. That is, growth occurs at the lowest cost unused bond which does not lead to a cycle, regardless of whether it is on or off the growth front. Efficient Kruskal's algorithms have similar efficiency to heap implementations of Dijkstra's method (i.e. O(IAllnlAI)).
3
3.1
Flow algorithms
Introduction
In this chapter we introduce flow problems and describe the algorithms which solve them (Ford and Fulkerson, 1962; Goldberg and Tarjan, 1988; Goldberg et al., 1990; Goldberg, 1992; Ahuja et al., 1993a, b). As their name suggests the characteristic feature of flow algorithms is that a quantity of flow is injected at one or more vertices of a graph(these are the sources) and the same quantity of flow is removed at another set of vertices(the targets or sinks). We define Xij >~ 0 to be the flow passing from node i to node j, and at each node in the graph flow is conserved. The vector x = {xij} describes the flow in all of the edges in the graph. Note that the requirement that Xij >_ 0 does not exclude flow from node j to node i. Any such flow is represented by Xji >__O. In general the flow algorithms generate asymmetric networks where the costs Cij ~ Cji. Although the flows are positive, the costs may be negative, provided there are no negativecost cycles.
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Flow problems are familiar in disordered systems, in the form of resistor networks, flow in porous media etc. The flow algorithms described here solve efficiently problems which have nonlinear (but convex) characteristics, and which can also have a threshold for perfect plastic response. A third important feature of combinatorial flow algorithms is the discreteness imposed by requiring integer flows. In particular, a flow of one unit along a path can be used to model a nonintersecting polymer. More surprisingly, arrays of polymers with deltafunction interactions can be modeled using integer flows (see Section 8.3). In Section 3.2 we introduce the maximumflow problem in which each edge in a graph has a maximum flow capacity (blij). In this problem we will find the maximum flow between two vertices s (the source) and t (the target) given that each bond in the network can transport a maximum flow of blij. Note that in the maximumflow problem there is no optimization in the sense of a cost function. We are only interested in the maximum flow. The problem is defined by a set of constraints (0 < xij ~ uij), rather than by a cost. Associated with the maximum flow is a minimum cut (see Section 3.2.1) and frequently the minimum cut is relevant to the physics applications. In Section 3.3 we introduce the minimumcostflow problem. In this problem there is a cost function C(x)  Y~(i,j)hij(xij) associated with the flow, xij in each bond and a capacity constraint, blij , o n each bond. Minimumcost flow thus includes maximum flow as the special case in which all of the bond costs are constant (independent of the flow). However, maximum flow is a sufficiently important special case that it is usually treated separately. More particularly it may be solved more efficiently than the general minimumcost problem. The minimumcostflow problem has polynomial algorithms if the cost function h(x) is convex. One convexcost function is very familiar namely P = y~ x 2 Rij, which is the power in a resistor network. Minimumcostflow algorithms thus find the current flow in resistor networks. This is not very exciting. However, two important generalizations which are also solved by minimumcost algorithms are: flow in resistor networks in which each bond has a fixed maximum current Uij , and; nonlinear resistors, which correspond to a nonlinear cost function, h(x) ~ x ~, with c~ > 1 to preserve convexity, and ot integer to preserve the integer character of the flow. The constraint 0 < xij ~ blij can be interpreted as a perfectlyplastic response (i.e. a threshold after which the flow in a bond is constant) and is a hard problem if attempted by conventional means. Several other, not so obvious, physics applications are described in Sections 7 and 8. Flow problems have a wide variety of applications including: traffic, phone communications and water distribution networks. There are many excellent books and review articles devoted to this class of problem.. The classic work by Ford and Fulkerson (1962), who invented the augmenting path method, has been continuously refined over the years. The preflow methods introduced by Tarjan and Goldberg (Goldberg and Tarjan, 1988; Ahuja et al., 1993b) have provided new insights
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and have led to significantly improved algorithms. Recent stochastic algorithms which find the minimum cut by iterative contractions also look promising, but are beyond the scope of this review. Integer minimumcost flow is related to integer programming and in engineering and operationsresearch textbooks is frequently discussed in that context (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984). However, the dedicated flow algorithms are significantly faster than convexcost programming codes (see Section 5), so for largescale analysis (as is typical of physics problems) it is important to use algorithms like those discussed below.
3.2 3.2.1
Maximum flow/minimum cut Basic ideas and definitions
Let us define a capacitated network as a graph ~(V, A), each of the arcs (i, j) 6 A has a capacity, lgij > O, which we take to be nonnegative. A flow in the network ~(V, A) is a set of nonnegative integers, Xij, subject to the capacity constraint, Igij , for each bond 0 < xij < uij u j) ~ A (3.2.1) and to a massbalance constraint (i.e. flow conservation) for each node. In many applications (e.g. the physics applications in Sections 7.2 and 7.3) the special case in which there is one source, s, and one sink, t, arises. This is sometimes called the min(st)cut/maxflow problem. In this case, the massbalance constraints are,
Z Xij  ~ Xji = {jl(i,j)Ea} {jl(j,i)Ea}
v V 0
fori = s for i  t else
(3.2.2)
where v is the value of the flow from the source s to the target (or sink) t. The maximumflow problem for the capacitated network G is to find the flow x that has the maximum value v under the constraints (3.2.1) and (3.2.2). When the maximum flow is being transported from s to t, all of the flow capacity between s and t is being used. However, not all of the arcs carry flow equal to their capacity. In fact there exists a bottleneck or minimum cut on which the capacities Uij a r e saturated. In the physics applications, the relevant flow geometry is quite often like that shown in Fig. 3.1. To discuss the relation between maximum flow and minimal cut, we introduce some notation: A cut is a partition of the node set V into two subsets S and S  V\S denoted by [S, S]; We refer to a cut as an stcut if s E S and t 6 S; The forward arcs of the cut [S, S] are those arcs (i, j ) 6 A with i E S and j 6 S, the backward arcs those with j E S and i E S; The set of all forward arcs of [S, S] is denoted (S, S); The capacity of an stcut is defined to be u[S, S] 
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Fig. 3.1 A flow network. Each arrow has a capacity uij, and a flow 0 < Xij < uij. The "source" node is the leftmost node and the "target" node is the rightmost node. The saturated bonds (i.e. xij = uij ) on the "cut" have been removed. Actually only the "forward arcs" on the cut are saturated. (This figure was generated using the graphing routines provided with the LEDA package (see note at end of References).)
Y~(i,j)~(s,s) Uij. Note that the sum is only over forward arcs of the cut. The minimum cut is an stcut whose capacity u is minimal among all stcuts. Now we state the maxflow/mincut theorem of Ford and Fulkerson (Ford and Fulkerson, 1962; van Lint and Wilson, 1996), which is a key result in network flows. T h e o r e m (mincut/maxflow) In a transportation network G(V,A) the maximum value of x over all flows {Xij} is equal to the minimum value u[S, S] over all cuts [S, S].
Proof: Let x be a flow, v its value and [S, S] an stcut. balances for all nodes in S we have
Then, by adding the mass
l) ~ i~ES{ ~ XijZ Xji}  Z Xij Z Xji" 9 {jl(i,j)6A(i)} {jl(j,i)6A(i)} (i,j)E(S,S) (i,j)E(S,S) (3.2.3) Since
xij ~ uij
and
Xji >__0
the following inequality holds
13
0(a source) or b(i) < 0 (a target). The minimumcostflow problem is (Ahuja et al., 1993b): minimize
z(x)
~
hij(xij),
(3.3.1)
(i,j)EA subject to the massbalance constraints,
Z
xij 
{j[(i,j)EA}
Z Xji = b(i) {jI(j,i)EA}
u E V,
(3.3.2)
and the capacity constraints
0 ~ Xij 0 to s by arcs with capacity Usi = b(i) (ii) Connect all of the nodes with b(i) < 0 to t by arcs with capacity Uit  b ( i ) . Flow conservation requires ~~i b(i) = 0 so that the flow into the network is equal to the flow out of the network. The quantity h ij (xij) is the cost function and may be a different function on each bond. The cost functions hij (xij) can be any convex function, that is, Yx, y, and 0 ~ [0, 1]
hij(Ox Jr(1  O)y) < Ohij(x) 4 (1  O)hij(Y)
(3.3.4)
Linear cost is the special c a s e , h ( x i j )  r There are faster algorithms for linear cost than for convex cost, but for the applications considered here, the difference is not large. The general convexcost case is as simple to discuss as the linearcost case, so we discuss the general algorithm. In minimalcost algorithms, the residual network G(x), corresponding to a flow, x, again plays an essential role. The residual capacities are defined in the same way as in the maximumflow problem. However, the residual cost is not so simple. We need to know the cost of augmenting the flow on arc (i, j ) , when there is already a flow Xij in that arc. In the general convexcost problem, we always augment the flow by one flow unit. The costs incurred in adding or subtracting one unit of flow between nodes i and j are represented by residual costs. The key subtlety is that if a forward arc has positive cost Cij, the residual cost on the r may be negative. This arises because if a flow Xij > 0 exists reverse a r c , cji, on a positivecost arc (i, j), injecting a compensating flow from node j to node i r in the cancels the flow Xij which requires that there be a negative residual cost, Cji' reverse arc. The residual costs defined below reflect this possibility. Because we have defined Xij >__0 and Xji >~ 0, we must treat three cases. Although this seems cumbersome now, in the later analysis it significantly simplifies the discussion. The residual costs for the general convexcost problem are as follows. (i) If Xij >~ 1, =~ Xji = 0 
hij(xij),
1) 
hij(xij).
crj(xij) "~ h i j ( x i j Jr 1) Cji(Xi j)  h i j ( x i j (ii)
If
xji >_
l , ==>
xij   0 c j i ( x j i ) = h j i ( x j i + 1) Cij(Xji)  h j i ( x j i  l )
(iii)
If
xij  O,
and
(3.3.5)

hji(xji), hji(xji).
(3.3.6)
xji " 0 Ci~ (0) = hij (1)  hij (0), c~.i(O)  h j i ( 1 )  hji(O).
(3.3.7)
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As seen in the second of (3.3.5) and (3.3.6), negative residual costs may occur when reducing the flow in an arc. The residual network G(x) is then a graph with residual capacities rij = u i j  x i j + x j i , and residual costs found from (3.3.5)(3.3.7). An intuitively appealing way of thinking about the convexcost problem is to replicate each arc, (i, j) many times, with each replicated arc having capacity one. The kth replicated arc has cost h i j ( k )  h i j ( k  1). As flow is pushed along the arc (i, j), the first unit of flow goes into the first replicated arc, the second unit of flow in the second replicate etc. When the flow is reversed, the flow is cancelled first in the highest replicated arc p r o v i d e d the cost f u n c t i o n is convex. That is, we need h i j ( k )  h i j ( k  1) > h i j ( k  1)  h i j ( k  2) so that this replication procedure makes sense. Unfortunately no analogous procedure is possible when the convexity condition is violated. In the case of linear costs there is no need to replicate the arcs as the cost for incrementing the flow does not depend on the existing flow. As discussed in Section 8 many of the physics problems map to linear or convex minimumcost flow with nonnegative costs. A simple example of using minimumcost flow to model two directed polymers in a random pinning potential is presented in Fig. 3.2, where each arc has capacity one. This ensures that only one unit of flow can pass through any arc and models a contact repulsion between polymers. Note that this contact repulsion is on the arcs, so that more that one polymer can still be incident on a node. Node disjoint paths are more difficult to model. Nevertheless, arc disjoint paths provide a strong contact repulsion and should capture the essence of nonintersecting polymers. We now discuss two methods for solving minimumcostflow problems, namely the negativecyclecancelling method (Section 3.3.2) and the successiveshortestpath method (Section 3.3.3), both of which rely on residualgraph ideas. The latter is more efficient, but is restricted to nonnegative costs as it uses Dijkstra's method to find lowestcost paths.
3.3.2
Negativecyclecanceling
algorithm
The idea of this algorithm is to find a f e a s i b l e flow, that is, one which satisfies the massconservation rules, and then to improve its cost by cancelling negativecost cycles. A negativecost cycle in the original network is also a negativecost cycle in the residual graph, so we can work with the residual graph. Moreover, flow cycles do not change the total flow into or out of the network, and they do not alter the massbalance conditions at each node. Thus, augmenting the flow on a negativecost cycle maintains feasibility and reduces the cost, which forms the basis of the negativecyclecanceling algorithm. This is formalized as follows. Theorem (negative cycle) A feasible solution x* is an optimal solution of the
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Fig. 3.2 Finding the exact ground state of two nonintersecting directed polymers in a random pinning potential. The energy (cost) of each arc is indicated. The arcs attached to the source (top node) and target (bottom node) have zero cost. a) The flow computed using the minimumcostflow method. Note that in the central horizontal bond, flow cancels, b) I'he groundstate configuration of two directed polymers.
minimumcostflow problem, if and only if the residual network ~(x*) contains no negativecost cycle.
Proof: Suppose the flow x is feasible and ~(x) contains a negative cycle. Then a flow augmentation along this cycle improves the function value z(x), thus x is not optimal. Now suppose that x* is feasible and G(x*) contains no negative cycles and let x ~ ~= x* be an optimal solution. Now decompose x ~  x* into augmenting cycles, the sum of the costs along these cycles is c  x ~  c . x*. Since G(x*) contains no negative cycles c . x ~  c . x* _ O, and therefore c . x ~ = c . x* because optimality of x* implies c . x ~ _< c . x*. Thus x ~ is also optimal. [] A minimumcost algorithm based on the negativecostcanceling theorem, valid for graphs with convex costs and no negativecost cycles in ~7(0), is given below.
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algorithm C y c l e c a n c e l i n g
begin
establish calculate
171
(convex costs)
a f e a s i b l e flow x the r e s i d u a l costs
cri as in eqs.
while ~(x) c o n t a i n s a n e g a t i v e cost cycle do
(3.3.53.3.7)
begin
end
end
use some a l g o r i t h m to i d e n t i f y a n e g a t i v e a u g m e n t one unit of flow in the cycle u p d a t e cri(xij) and r/j.
cycle
W
To begin the algorithm, it is necessary to find a feasible flow, which is a flow which satisfies the injected flow at each of the sources, the extracted flow at each of the sinks, and which satisfies the massbalance constraints at each node. A robust procedure to find a feasible flow is to find a flow which satisfies the capacity constraints using the maximumflow algorithm (e.g. using the preflow algorithm). To detect negative cycles in the residual network, (7(x), one can use the labelcorrecting algorithm for the shortestpath problem presented in Section 2.3.2. In the linear cost case, the maximum possible improvement of the cost function is O(IAICU), where C = m a x Icij] and U = max uij. Since each augmenting cycle contains at least one arc and at least one unit of flow, the upper bound on the number of augmenting cycle iterations for convergence is also O(IAICU). Negativecycle detection is O(I vzl) generically, but for sparse graphs with integer costs it is O(IA)[). Thus for sparse graphs with integer costs, negative cycle canceling is O([AI2CU).
3.3.3
Successiveshortestpath algorithm
The successiveshortestpath algorithm iteratively sends flow along minimalcost paths from source nodes to sink nodes to finally fulfill the massbalance constraints. A pseudoflow satisfies the capacity and nonnegativity constraints, but not necessarily the massbalance constraints. Such flows are called infeasible as they do not satisfy all the constraints. The successiveshortestpath algorithm is an infeasible method which maintains an optimalcost solution at each step. In contrast, the negativecyclecanceling algorithm always satisfies the constraints (so it is a feasible method) and it iteratively produces a more optimal solution. The imbalance of node i is defined as
e(i)  b(i) +
~ Xji ~ Xij. {jI(ji)EA} {jl(ij)Ea}
(3.3.8)
If e(i) > 0 then we call e(i) the excess of node i, If e(i) < 0 then we call it the deficit. The successiveshortestpath algorithm sends flow along minimalcost
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paths from excess sites to deficit sites until no excess or deficit nodes remain in the graph. Dijkstra's algorithm (Section 2.3.2) is efficient in finding minimumcost paths, but it only works for positive costs. The successiveshortestpath algorithm uses Dijkstra's method to find augmenting paths, but to make this work, we have to develop a different sort of residual network with positive reduced costs (remember that the residual costs can be negative, see (3.3.5)(3.3.7)). Surprisingly, this is possible. To construct positive reduced costs from which the optimal flow can be calculated, we must first introduce the concept of a node potential 7r(i) and some beautiful results related to it. The reduced costs used in the successiveshortestpath problem are inspired by the reduced costs, c/~, introduced in the shortestpath problem (2.3.1)(2.3.3). c/~ has the attractive feature that, with respect to the optimal distances, every arc has a nonnegative cost. To generalize the definition (2.3.1) so that it can be used in the minimumcostflow problem, one defines the reduced cost of arc (i, j) in terms of a set of node potentials zr (i), C~j  Cij  7r(i) + 7r(j).
(3.3.9)
We impose the condition that a potential is only valid if Cij > 0 as for reduced costs in the minimal path problem. Note that the residual costs defined in (3.3.5)(3.3.7) appear here. All of the quantities in (3.3.9) depend on the flow X i j , though we don't explicitly state this dependence. From the definition (3.3.9), it is evident that the reduced costs, c~j, have the following properties. (i) For any directed path P from k to l:
(i,j)EP
(i,j)EP
(ii) For any directed cycle W:
Z (i,j)EW
Z cirj" (i,j)EW
In particular, property (ii) means that negative cycles with respect to crj are also negative cycles with respect to c~j. We define the residual network GJr (x) to be the residual graph with residual capacities defined as before, but with reduced costs as given by (3.3.9). The next step is to find a way to construct the potentials zr(i). This is carried out recursively, starting with zr(i)  0 when there is no flow in the network. The procedure for generating potentials iteratively relies on the potential lemma given below.
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Lemma (potential) (i) Given: a valid node potential zr(i)" a set of reduced costs, c~j; and a set of distance labels, d ( i ) (found using cTj > O) then the potential Jrt(i) ~t
rr(i)  d ( i ) also has positive reduced costs, cij > O. ~t
(ii) cij  0 for all arcs (i, j) on shortest paths.
Proof: Properties (i) and (ii) follow from the analogous properties for the minimal path (see (2.3.1)(2.3.3)). To prove (i), using (2.3.1)(2.3.3), we have, d ( j ) < d ( i ) + yf cij, then we have, cij~!  c r j  ( r r ( i )  d ( i ) ) + ( : r ( j )  d ( j ) ) = c ~ j  d ( j ) + d ( i ) > O. For (ii) simply repeat the discussion after replacing the inequality by an equality. [] Now that we have a method for constructing potentials, it is necessary to demonstrate that this construction produces an optimal flow.
Theorem (reduced cost optimality) A feasible solution, x*, is an optimal solution of the minimumcost flow problem if and only if there exists a set of node Jr >__0 V(i, j) in ~ r (x*). potentials, :r(i), such that Cij Proof: For the implication "r suppose that cTj > 0 V(i, j). Since G(x*) is optimal, it contains no negative cycles. Thus by property (ii) above, an arbitrary potential difference may be added to the costs on each arc on each cycle W. For the other direction "=~" suppose that G(x*) contains no negative cycles. Denote with d(.) the shortest path distances from node 1 to all other nodes. Hence d ( j ) < d ( i ) + c i j V(i, j) 6 G(x*). Now define re   d then cTj = cij + d ( i )  d ( j ) > O. Hence we have constructed a set of node potentials associated with the optimal flow. [] The reduced cost optimality theorem proves that each optimal flow has an associated set of potentials, while the potential lemma shows how to construct these potentials. The final step is to demonstrate how to augment the flow using the potentials. To demonstrate this, suppose that we have an optimal flow x and its associated potential zr(i) which produces reduced costs c~j which satisfy the reduced cost optimality condition. Suppose that we want to add one unit of flow to the system, injecting at a source at site, k, and extracting at a site I. Find a minimal path, Pkl, (using the reduced costs c~j) from a excess site k to a deficit site I. Now augment the flow by one unit for all arcs (i, j ) ~ Pkl. We call this flow augmentation 3. The following augmentation lemma ensures that this procedure maintains optimality.
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(flow a u g m e n t a t i o n ) The flow x' : x + 6 is optimal and it satisfies the reduced cost optimality conditions.
Lemma
Proof:
Take re and ~' as in the potential Lemma and let P be the shortest path from node 7T t s to node k. Part (ii) of the potential lemma implies that u (i, j ) ~ P 9cij  O. Therefore r
2gt
7/t
= Cij
 O. Thus a flow augmentation on (i, j ) 6 P might add ~t
(j, i) to the residual network, but Cji  O, which means that the reduced cost 7T t
optimality condition Cji > 0 is still fulfilled.
[]
The strategy for the successiveshortestpath algorithm is now clear. Given a set ofexcess nodes E = {ile(i) > 0} and a set of deficit nodes D = {ile(i) < 0}, we iteratively find minimal paths from a node i 6 E to a node j 6 D until no excess or deficit remains. The minimal path for flow augmentation is found using Dijkstra's method on the residual network with the reduced costs given by (3.3.9). After each flow augmentation, the node potentials are recalculated using the potential lemma.
algorithm
Successive
shortest
paths
(convex
costs)
begin
x:0,
Jr(i) = 0 is
while t h e r e begin
compute find the all
a node s with
the r e d u c e d c o s t s c~(x) (Eq. 3.3.9) s h o r t e s t p a t h s d ( i ) f r o m s to o t h e r n o d e s in ~(x) w 9 r " to the r e d u c e d
choose a n o d e t w i t h
augment unit
e ( s ) > 0 do
f l o w on
the
compute 7r(i) = 7r(i) 
costs
c.I~]
e(t)< 0 shortest
path
from
s
to
t
b y one
d(i)
end end
Since we worked hard to construct a system with positive reduced costs, the "find" operation above can be carried out using Dijkstra's algorithm. If we denote the sum of all the sources to be v  Z i l b ( i ) > o b(i), then the number of flow augmentations needed to find the optimal flow is simply v. Each flow augmentation requires a search for a minimumcost path from a node k c E to a node 1 E D which for sparse graphs and integer flows can be efficiently accomplished with Dijkstra's method, which is O(IAI). Thus for integer flows on sparse graphs with positive costs (as is typical of the physics applications) the successiveshortestpath algorithm is O(vlAI).
2 Exactcombinatorial algorithms 4
4.1
175
Matching algorithms
Introduction and definitions
Given a graph G(V, E) with node set V and edge set E, a matching M C E is a subset of edges, such that no two are incident to the same node (Lov~isz and Plummer, 1986; Graham et al., 1995). An edge contained in a given matching is called matched, other edges are free. A node incident to an edge e 6 M is covered (or matched) others are exposed (or free). A matching is perfect if it leaves no exposed nodes. If e = (u, v) is matched, then u and v are called mates. An alternating path is a path along which the edges are alternately matched and unmatched (for example the highlighted path in Fig. 4.1a). A bipartite graph is a graph which can be subdivided into two sets of vertices, say X and Y, such that the arcs in the graph (i, j) only connect vertices in X to vertices in Y, with no arcs internal to X or Y. Nearestneighbor hypercubic lattices are bipartite, while the triangular and facecenteredcubic lattices are not bipartite. The more general weightedmatching problems assign a nonnegative weight (= cost), w, to each edge e = (i, j). M is a maximumweight matching if the total weight of the edges in M is maximal with respect to all possible matchings. There is a simple mapping between maximumweight matchings and minimumweight matchings, namely: let Cij = W  wij, where Wij is the weight of arc (i, j) and W > m a x ( i , j ) ( t o i j ). A maximum matching o n toij is then a minimum matching o n cij. Later in this section we show that minimum (and hence maximum) matchings on bipartite graphs can be mapped to a minimumcostflow problem. An historical introduction to matching problems, whose origins may be traced to the beginnings of combinatorics, may be found in Lov~isz and Plummer (1986). Matching is also related to thermal statistical mechanics because the partition function for the twodimensional Ising model on the square lattice can be found by counting dimer coverings (= perfect matchings) (Lov~isz and Plummer, 1986). This is a graph enumeration problem rather than the optimization problems we consider here. As a general rule, graph enumeration problems are harder than graphoptimization problems. Due to the fact that all cycles on bipartite graphs have an even number of edges, matching on bipartite graphs is considerably easier than matching on general graphs. In addition, maximumcardinality matching and maximumweight matching on bipartite graphs can be easily related to the maximumflow and minimumcostflow (respectively) problems discussed in Section 3. Further, rigidity percolation (even on nonbipartite graphs like the triangular lattice) is related to maximumcardinality matching on bipartite graphs (see Section 9). We thus treat bipartite matching separately (Section 4.3.1). The more complex problem of matching on general graphs, which are used to solve spinglass models, is discussed in Section 4.3.2.
176 (
M . J . Alava e t al. ) ..........................
)
Fig. 4.1 a) A matching (thick) is a subset of edges with no common end. An augmenting path (shaded) starts and ends at exposed nodes, and alternates between unmatched (thin) and matched (thick) edges, b) Interchanging matched and unmatched edges along an augmenting path increases the cardinality of the matching by one. This is called augmentation and is the basic tool for maximummatching problems.
4.2 Augmenting paths The algorithms for maximum matchings are based on the idea of successive augmentation, which is analogous to the augmentingpath methods for flow problems (see Section 3.2.2). An augmenting path A p with respect to M is an alternating path between two exposed nodes. An augmenting path is necessarily of odd length, and, if G is bipartite, connects a node in one sublattice, U, with a node in the other sublattice, V. Clearly, if matched and free edges are interchanged along Ap, the number of matched edges increases by one. Therefore if M admits an augmenting path it cannot be of maximum cardinality. Similarly a matching M cannot be of maximum weight if it has an alternating path of positive weight, since interchanging matched and free edges would produce a "heavier" matching. The nonexistence of augmenting paths is a necessary condition for maximality of a matching. It is also a sufficient condition. A central result in matching theory states that repeated augmentation must result in a maximum matching (Berge, 1957; Norman and Rabin, 1959). Theorem (augmenting path) (i) A matching M has maximum cardinality if and only if it admits no augmenting path. (ii) A matching M has maximum weight if and only if it has no alternating path or cycle of positive weight. Proof: (i) =~ is trivial. To prove r assume M is not maximum. Then some matching M' must exist with IM~I > IMI. Consider now the graph G' whose edge set is E ~ = M A M ~ (the symmetric difference of M and M~). Clearly each node of G~ is
2
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incident to at most one edge of M and at most one edge of M t. Therefore nodes in Gt have at most two incident edges and the connected components must be either simple paths or cycles of even length, and all paths are alternating paths. In all cycles we have the same number of edges from M as from M ~ so we can forget them. But since IM'I > IMI there must be at least one path in G~ with more edges from M ~than from M. This path must necessarily be an augmenting path. (ii) =, is again trivial. Assume M is not maximum. Some matching M t must therefore exist with w(M ~) > w(M). Consider again ~ = (V, MAMa). By the same reasoning as before, we conclude that ~ must contain at least one alternating path or cycle of positive weight. [3
4.3 Matching problems 4.3.1
Matching on bipartite graphs
Consider a bipartite graph, B(U, V, E), where U is the set of nodes on one sublattice and V is the set of nodes on the other sublattice. It is conventional to draw bipartite graphs as shown in Fig. 4.2, with the two sublattices joined by arcs, which can only go from one sublattice to the other. Now assume that an initial matching M is given (which can be the empty set), as in Fig. 4.2a. It is natural to look for alternating paths starting from exposed nodes (If there are no exposed nodes, M is maximum. Stop). An efficient way to do this is to consider all alternating paths from a given exposed node simultaneously in the following way: build a breadthfirstsearch (BFS) (see Section 2.3.1) starting from an exposed node, for example node vs, as described in Fig. 4.3a. In the BFS tree, node v5 corresponds to level 0. All its adjacent edges are free. They lead to nodes u3 and u5 at level 1, which are covered. Now since we must build alternating paths, it doesn't make sense to continue the search along free edges. Therefore we proceed along matched edges, respectively, to nodes Vl and v2. From these we follow free edges to u l and u2, and then matched edges to nodes v3 and v4. In the last step, node u4 is found exposed. Therefore (u4, v4, u2, vl, u3, v5) is an augmenting path. After inverting it, the augmented matching shown in Fig. 4.2b is obtained. If no exposed node were found when the B FS ends, then node v5 will never be matched and can be forgotten because of the following result (van Leeuwen, 1990), which is valid for general graphs: Theorem If there is no augmenting path from node uo at some stage, then there never will be an augmenting path from Uo. In practice, there is a difference between the search technique described here and the usual BFS since the searches from oddnumbered levels are trivial. They always lead to the mate of that node, if the node itself is not exposed. Therefore
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\
a)
b) ~
'
\
Fig. 4.2 a) An initial matching is given for a bipartite graph, b) The enlarged matching obtained after inverting the augmenting path discovered from node v5 (see Fig. 4.3).
v+3
Fig. 4.3 a) The BFS tree built from exposed node v5 in Fig. 4.2a. Dashed lines represent nontree edges to already visited nodes. The search finishes when an exposed node u4 (double circle) is found, b) The auxiliary tree obtained after removing oddlevel nodes and identifying them with their mates. After inverting the augmenting path {v5, u3, Vl, u2, v4, u4}, the enlarged matching in Fig. 4.2b is obtained.
the search can, in practice, be simplified by ignoring oddnumbered nodes and going directly to their mates, as shown in Fig. 4.3b. The search for alternating paths can be seen as a usual BFS on an auxiliary graph from which oddlevel nodes
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Exact combinatorial algorithms
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have been removed by identifying them with their mates. The basic augmenting path algorithm is then as follows (Papadimitriou and Steiglitz, 1982). algorithm ( M a x i m u m  c a r d i n a l i t y begin
bipartite
matching)
e s t a b l i s h a i n i t i a l f e a s i b l e m a t c h M on B c o n t a i n s an e x p o s e d n o d e u E U do
B(U, V,E)
while begin
Find an a l t e r n a t i n g
path,
to an e x p o s e d n o d e invert the a l t e r n a t i n g
update M
P,
v E V path,
from u E U P
end end The best known implementation for this algorithm is due to Hopcroft and Karp (1973). It runs in time O(IEI 9[ , / ~ ) and is based on doing more than one augmentation in one step. There is also a simple way in which to map the maximumcardinalitymatching problem to the maximumflow problem. Let /3  (V, U, E), and define B' by adding a source node s and a target node t, and connecting all nodes in V to s, and all nodes in U to t, by edges of capacity 1. Now let all edges e E E have capacity 1. Because of the integer flow theorem, maximum flows in/3' are integral. Every flow of size f thus identifies f matched edges in/3, and viceversa. Since maximum flows in 0  1 networks are computable in O(IEI 9~/IV[) time, so are maximumcardinality matchings on /3. The mapping of matching problems to flow problems also applies to the maximumweightmatching problem on bipartite graphs. This is also known as the assignment problem, because it can be identified with optimal assignment, e.g. of workers to machines, if worker i E U produces a value tOij working at machine j E V. In a similar manner to that described in the previous paragraph, this problem can easily be formulated as a flow problem. Again add a source node s, a sink node t; connecting s to all nodes in U by unitcapacity, zerocost arcs; all nodes in V to t by unitcapacity, zerocost arcs. Also interpret edge eij as a directed arc with unit capacity and c o s t w ( e i j ) = W  e i j , where W > max(i,j)(tOij ). The solution to the minimumcostflow problem from s to t is then equivalent to the maximumweight matching which we seek.
4.3.2 Matchingon general graphs Maximum matching on general graphs is considerably more difficult because of the presence of oddlength cycles, which are absent on bipartite graphs. Consider starting a BFS for alternating paths from an exposed node a at level 0 (Fig. 4.4).
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h 9
d/////
Y
e A
a
b
9
v
I
w
..
a)
9
i
j
h 9
a 9
b)
v
k
x
I p n
b _
'~
B
Fig. 4.4 a) A blossom is an oddcycle which is as heavy as possible in matched edges, i.e. contains 2k + 1 edges among which k are matched, b) The reduced graph obtained by shrinking a blossom to a single node contains the same augmenting paths as the original graph.
And consider what happens when searching an evenlevel node x (necessarily covered). Let (x, y) be an unexplored edge incident to x. If y is exposed, we have found an augmenting path, and the augmentation proceeds as usual. If y is covered there are two possibilities: if y is oddlevel, nothing special happens. If y is marked as even, there is a special situation: there are two evenlength alternating paths, one from a to x and one from a to y, and therefore (x, y) closes a cycle of oddlength. Let c be the last node common to both alternating paths (necessarily at an even level). The odd cycle including c is called a blossom B, and c its base. A blossom is essentially an oddlength alternating path from c to itself, as depicted in Fig. 4.4a. Its presence may conceal an existing augmenting path, like for example {a, b, c, i, j, k, x, y, f, e, d, h, n, p} in Fig. 4.4, which would not be discovered by the BFS since edge (d, h) would never be explored. A blossom might also make us "find" an augmenting path where none actually exists, like for example {a, b, c, d, e, f, y, x, k, j, i, c, b, a} in Fig. 4.4a. The first polynomialtime algorithm to handle blossoms is due to Edmonds (1965). Edmonds' idea was to shrink blossoms, that is, replace them by a single
2 Exactcombinatorial algorithms
181
node/3 obtaining a modified graph ~ as shown in Fig 4.4b. The possibility of shrinking is justified by the following theorem due to Edmonds.
Theorem (Edmonds) There is an augmenting path in G if and only if there is an augmenting path in ~. The existence of a blossom is discovered when edge (x, y) between two evenlevel nodes is first found, and its nodes and edges are identified by backtracking from x and y until the first common node is found (c in our example), which is the blossom's base. Once identified, the blossom is shrunk, replacing all its nodes (among which there might be previously shrunk blossoms) by a single node, and reconnecting all edges incident to nodes in the blossom (necessarily uncovered edges) to this one. The search proceeds as usual, until an augmenting path is found, or none, in which case a is abandoned and no search will ever be started from it again. If an augmenting path is found, which does not involve any shrunk blossom, it is inverted as usual. If it contains blossomnodes, they must be expanded first and one has to identify which way around the blossom the augmenting path goes. This may need to be repeated several times if blossoms are nested. After inverting the resulting augmenting path, a new search is started from a different node. A simple implementation of these ideas runs in O(IVI 4) time (Papadimitriou and Steiglitz, 1982). The fastest known algorithm for nonbipartite matching is also O(IE[ 9~/Igl) time (Micali and Vazirani, 1980).
5
5.1
Mathematical programming
Introduction
There are many books devoted to linear programming (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984; Wright, 1997) and its applications. The problems described in Sections 24 can be cast as linear or, more generally, convexprogramming problems. In this chapter we give a brief introduction to mathematical programming and its relation to minimumcostflow problems and to matching on nonbipartite graphs. Linear programming is the problem of optimizing (minimizing or maximizing) a linear cost function while satisfying a set of linear equality and/or inequality constraints. Linear programming was discussed by George B. Dantzig around 1947, although L.V. Kantorovich had introduced and solved a related problem in 1939. The simplex method for linear programming was published by Dantzig in 1949. From the outset it was realized that flow problems are a special case of mathematical programming with flow conservation at each node being a set of
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linear constraints. Special implementations of the simplex method for network flow can be much more efficient than the general simplex method and are called networksimplex methods. Since 1949, generations of workers in the fields of operations research, economics, finance and engineering have been trained to formulate and solve linear and nonlinearprogramming models, particularly in the framework of simplex methods. Mathematicalprogramming methods have had a much smaller impact on modeling in the physical sciences. If the variables over which we optimize can have continuous variation, then linear programming (LP) is known to be in P, that is, it is solvable in polynomial time. Polynomial bounds on continuous linear programming were first demonstrated for ellipsoid algorithms. However, ellipsoid algorithms are inefficient in practice, in contrast to the N P simplex method which is efficient in practice. In 1984, Karmarkar introduced a second class of P algorithms for linear programming and from this work the interiorpoint methods have developed. Interior point methods are efficient, especially for large sparse LP and convexcost problems (Wright, 1997). In many problems, the variables over which we wish to optimize are restricted to be integers. This class of linearprogramming problem is called integer linear programming or ILE In contrast to continuous LP, ILP does not, in general, have polynomial bounds. In addition the simplex method is not designed for this general class of problem. However, all of the P problems which we have discussed in Sections 24 can be mapped to an ILP problem, and it is only these network ILP problems which are known to have polynomial bounds. The general ILP class includes many of the outstanding "hard" combinatorial problems such as the travelingsalesman problem.
5.2 5.2.1
Linear and convexcost programming Problem definition
The standard form (Papadimitriou and Steiglitz, 1982; Rockafellar, 1984; Wright, 1997) of the primal LP problem is: minimize the cost function,
cost  ~
Ci X i ,
(5.2.1)
and xi >_0.
(5.2.2)
i
subject to the linear constraint equations, AYb,
If we seek an integer linearprogramming solution, we have the additional requirement that xi be an integer. Convexcost programming is the generalization
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to the case where the cost is a convex function ie. Cost ~ hi (xi), where hi is a convex function. For the problem defined by (5.2.1) and (5.2.2) to have a solution, we need the length (n) of the vector of variables s to be larger than the number of linear constraints (m), otherwise the problem is overconstrained or infeasible. The matrix A is an m • n array which may couple the variables xi to each other. Linearprogramming problems may be written in a variety of different forms, and various tricks are needed to reduce them to the standard form. An important alternative form of linear programming is the duallinearprogramming form. The dual problem is stated as follows: maximize the cost function, cost = Z bi 1Bi , (5.2.3) i
subject to the linearconstraint equations, t ~ A  Y, with  C o _< 1/3 i < CO
(5.2.4)
There are a variety of relations between the primal and dual problems, with one of the most important being,
Lemma (strong duality) If either the primal or dual problem has an optimal solution, then both possess optimal solutions and the two optimal costs are equal.
5.2.2
Minimumcostflow as mathematical programming
Define the flow in each bond Xij > O, to be the variable over which we are to optimize; the sources at each site to be co > bi >_  c o ; the capacities on each bond to be Uij > 0 and; a linear cost function, Z i j CijXij then the minimumcostflow problem can be written as: (Ahuja et al., 1993b)" Minimize the cost function
cost  Z
CijXij,
(5.2.5)
ij
subject to the flow constraints, MY  b, with 0 _< xij < blij.
(5.2.6)
Other than the capacity restriction on the flow Xij, this is in primal linearprogramming form (compare with (5.2.1) and (5.2.2). It is simple to remove the upper constraint o n xij. Just introduce a set of slack variables, ~ij, ~ij   U i j   X i j .
(5.2.7)
This is a set of linear equations, which is added to the set Ms  b which impose conservation of flow at each vertex. This minimumcost problem is now in primal linearprogramming form, with the variables ~ij >~ 0 and Xij > O.
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184
5.2.3
Weighted matching as linear programming
All weightedmatching problems can be thought of as matching problems on complete graphs (Lov~isz and Plummer, 1986), by just adding the missing edges with zero weight. Furthermore, the number of nodes can always be assumed to be even, since an extra node can always be added which has zeroweight connections to all others. Therefore, optimal solutions will always be perfect matchings. Thus we can formulate the maximumweight matching problem (over weights 113ij ) a s a minimization problem (over costs C i j ) over the set of perfect matchings on a complete graph, after defining the costs Cij = W  l13ij, where W is greater than or equal to the largest weight. In order to see that maximumweight matching can be formulated as a minimization problem on a complete graph with n nodes, we associate an integer variable Xij t o each edge eij, w i t h xij = 1 meaning that edge (i, j) is in the matching and Xij   0 meaning that it is not. By convention X i j " X j i and Xii  O. We are then left with the following linear minimization problem. minimize Z
CijXij
(5.2.8)
i,j
subject to n
Z
X i j "
1
i = 1,, n
(5.2.9)
j=l
xij > 0
1 < i p, mark site j as tested and do not test it again. For p > Pc, the Leath algorithm grows a cluster of infinite size, while for p < Pc, the growth of the cluster ceases at a finite distance from the source. Very high precision results for percolation have been found using this method, for example, lattices of size 20483 have been studied (Lorenz and Ziff, 1998). The HoshenKopelman algorithm (Hoshen and Kopelman, 1976) grows many clusters simultaneously by assigning cluster labels to each new cluster that is nucleated during growth. This is done "rowbyrow" (or "layerbylayer" in three dimensions) and must take into account the merging of different growing clusters. This merging is accounted for by defining "equivalence classes" which are sets of cluster labels that are set to be equivalent due to merging. Simulations on lattices of up to 4 • 1011 sites have been carried out using this method (Rapaport, 1992). (iii) Invasion or "greedy" algorithms When calculations are carried out at fixed concentration p as above, it is necessary to "tune" the value of p to find the critical concentration pc. The threshold value, Pc, must be known very accurately in order to find the critical exponents to high precision. However, it is possible to define a dynamics which automatically
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stays at the critical threshold. Though this behavior has been known for some time (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984) it has recently become a prototype for the more general class of dynamical phenomena now known as "selforganized criticality" (SOC) (Bak et al., 1987; Furuberg et al., 1988; Cieplak et al., 1996). The invasion algorithm for percolation (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984) is very similar to the Leath algorithm (Leath, 1976a, b). A growth front evolves outward from a seed site s. Each site adjacent to the growth front is again assigned a uniform random number rj. However, the new feature is that growth is always to the site which has smallest rj. That is, from all the uninvaded sites adjacent to the growth front, choose the one which has the minimum value of its random number, rj. A similar algorithm can be defined for bond percolation. It has been shown that at long times the value of rj adjacent to the growth front converges to pc (Wilkinson and Barsony, 1984). By construction, the greedy algorithm grows outward indefinitely. It has also been shown that the cluster grown in this way leads to the same scaling behavior as the infinite cluster at the percolation threshold (Wilkinson and Barsony, 1984). Since this algorithm "selforganizes" itself to be always at the percolation threshold, it is a pedagogical example of a SOC produced by extremal dynamics (Zaitsev, 1992; Miller et al., 1993; Paczuski et al., 1995; Sneppen, 1995). The invasion algorithm just described is equivalent to Prim's algorithm (see Section 2.3.1) (Barabasi, 1996) for minimal spanning tree, provided we start Prim's algorithm at the source site s (rather than at the cheapest bond r in the graph). Prim's algorithms is a classic "greedy" algorithm in computer science, and it is likely that many other things that are known in computer science about greedy algorithms will prove useful in the study of extremal dynamics (Papadimitriou and Steiglitz, 1982; Ahuja et al., 1993b). From the point of view of numerical calculations, the algorithms for connectivity percolation are quite satisfactory. For example, in three dimensions, the Fisher exponent r = 2.189(2), the finitesize correction exponent is S2 = 0.64(2), and the scalingfunction exponent is cr = 0.445(10) (Lorenz and Ziff, 1998). Even the backbone exponents are quite well known (see Section 9.4).
6.3 6.3.1
Minimal path Introduction and scaling theory
The minimal path problem on a graph, G(V,A), where each edge has a cost Cij, was discussed in Section 2.3.2. As demonstrated there, Dijkstra's algorithm is a invasion (growth) algorithm which solves this problem exactly. The physics community has focused on the special case of a directed path
2
Exact combinatorial algorithms
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or a DPRM (Huse and Henley, 1985; Kardar et al., 1986; Kardar, 1987; Kardar and Zhang, 1987; Derrida and Spohn, 1988; Nattermann and Lipowski, 1988; M6zard, 1990; M6zard and Parisi, 1990; Parisi, 1990; Fisher and Huse, 1991; HalpinHealy and Zhang, 1995; L~issig, 1998). In the continuum limit the DPRM is described by,
S l u } 
dz
~
+ 6pin(U(Z), Z), ].
Oz
(6.3.1)
The random pinning potential 6pin, which is usually chosen to have mean zero, is characterized by a correlation function,
(F,pin(U,Z)~pin(U', Zt))  K ( z
 z', u  u').
(6.3.2)
Frequently, the random potential is taken to be deltacorrelated K (z  z', u(z) u'(z')) = y 8 (z  z')~i(u(z)  u'(z')), with y describing the strength of the pining potential. More complicated situations, like extended defects or columnar defects, can easily be modeled by correlating the disorder in some or all directions appropriately. A generalized lattice version of the Hamiltonian (6.3.1) is,
d1 H  Z Z (uz,j  Uzl,J)P [ ,~pin(Uz), z j=l
(6.3.3)
where p = 1 if often taken for convenience. The Hamiltonian (6.3.1) is also a standard model of a fluxline (Blatter et al., 1994) in a disordered environment and of a polyelectrolyte in a frozen gel matrix (Kardar et al., 1986; HalpinHealy and Zhang, 1995). The DPRM, by its construction of being stretched in the longitudinal (z) direction, has no selfinteraction (i.e. is selfavoiding), but it can fluctuate in the transverse (u(z)) direction. One of the attractive features of the DPRM model (6.3.1) is that some beautiful analytic results may be derived for it (Forster et al., 1977; Kardar et al., 1986; Derrida and Lebowitz, 1998). Firstly, a DPRM configuration u(z) can also be regarded as the world line of a particle in d  1 (transverse dimensions) moving in a time dependent potential 7. To see this, consider the restricted partition function Z(u, z) 
Du I exp

T
dz
2
+epin(Ut(Zt),Zt)l } (6.3.4)
which is proportional to the probability for the DPRM's end point at internal coordinate z being located at u. The righthand side of (6.3.4) is the pathintegral formulation of the imaginarytime Green's function of a particle in a time dependent potential g pin (U(Z), Z), where u(z) is the coordinate of the particle, z is
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the imaginary time, and T plays the role of h. Thus it satisfies the Schr6dinger equation
Z
OZ = {
} ~ffV2 + Epin(U(Z),Z) Z.
(6.3.5)
The free energy (of the fluxline/DPRM) f (u, z) =  T In Z (u, z) then obeys the KPZ nonlinear diffusion equation
Of 0z = DV2 f + 2 IVu(z)fl 2 + r/(u(z), z),
(6.3.6)
where D = T/2cr can be interpreted as a diffusion constant, ~. =  T / c r describes Epin(U(Z),z)/T is the strength of the coupling to the nonlinearity and 17 a Gaussian noise. The correspondence f + h, demonstrates that the height fluctuations during KPZ growth are analogous to the freeenergy fluctuations in the minimal path (Kardar et al., 1986). The roughness w(L) and energy fluctuations AE2(L) of a DPRM of length L, obey the scaling relations =
w2(L) = [ll2]av  [1112 ~ ' L 2~',
(6.3.7)
and its energy fluctuations
AE2(L) [E2]av [Ej2v ~, L 20.
(6.3.8)
The exponents ~" and 0 obey the exact scaling relation 0 = 2~"  1. For the DPRM in 1 + 1 dimensions, ~" = 2/3, and 0 = 1/3 are known exactly (Kardar et al., 1986). During growth, surfaces can become "kinetically" rough and their roughness is often described by the KPZ equation (Kardar et al., 1986), which, as shown above, is related to the DPRM problem. The roughness of the KPZ growth front scales as, w(t, l) .~ tt3 ffJ(1/tr (6.3.9) where l is the sample size parallel to the growth front, t is the growth time, and /5 and ~ are critical exponents. At long times, the surface fluctuations saturate to a value w ~ + U. For KPZ growth c~ = 1/2. It is then evident that ~ =/~/ot. The scaling function may be found by a mapping to the asymmetric exclusion process (Derrida and Lebowitz, 1998), and generalizations to broad distributions of "waiting times" lead to continuously varying exponents (Tang et al., 1991).
6.3.2
Algorithms for minimal path
As stated above, Dijkstra's method solves the minimumpath problem. For the detailed description below, we specialize to the case p = 1 in the discrete solidonsolid energy (6.3.3). The DPRM is a directed optimal path on the links of a
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lattice (see Fig. 6.1). For a path along the {10} orientation, the path is allowed to step forward, to the left or to the right, with an increased energy cost for motion to the left or right, which models the elasticity of the DPRM (Kardar, 1985; Kardar et al., 1986). An even simpler model is a path in the {11 } orientation (see Fig. 6.1). In this case, there is no explicit elasticity to the DPRM, but the motion is restricted to the transverse direction, and it is believed that this constraint is sufficient to maintain the DPRM universality class. The random potential 6pin(U(Z), Z) is included as a random energy on the bonds of the lattice (see Fig. 6.1). Thus the lattice Hamiltonian is simply
H  ~
ei
(6.3.10)
9x i
i where the sum is over all bonds i = (u(z), z) of the lattice and xi represents the DPRM configuration starting at 0 (at u = 0 and z = 0, see Fig. 6.1): It is xi = 1 if the DPRM passes through the bond i and xi  0 otherwise. The ground state of (6.3.10) is the minimum energy path from 0 to some point (u(z), z) at the lower boundary plane of the lattice (either fixed or free). Interpreting the energies as distances (after making them all positive by adding a sufficiently large constant to all energies), and the lattice as a directed graph, this becomes a shortest path problem that can be solved by using, for instance, Dijkstra's algorithm (see Section 2.3.2). In addition, due to the directed structure of the lattice one can compute the minimum energies of DP configurations ending at (or shortest paths leading to) coordinates (u, z) recursively (this is the way in which Dijkstra's algorithm would proceed for this particular case) (Huse and Henley, 1985). This is the same as the transfermatrix algorithm, which follows from the group property of the partition function (6.3.4): Z(u, z + 1)  ~
Z(u, u', 1) Z(u', z)
(6.3.11)
Ut
which, in the zero temperature limit, reduces to d1
E(u(z + 1)) = Minu,(z){E(u'(z)) + e(u(z + 1)) + J ~
luj(z + 1)  u)(z)[}
j=l
(6.3.12) In Fig. 6.2 we show a collection of such optimal paths in the 1 + 1 dimensional case. The transfermatrix method has been successful in studies of the DPRM in arbitrary dimensions. Simulations in 1 + 1 dimensions confirm the exact exponents (e.g. ( = 2/3) (Huse and Henley, 1985) and have yielded accurate results in 2 + 1 dimensions and in 3 + 1 dimensions (0 = 0.248(4) and 0  0.20(1) respectively) (Kim et al., 1991), and have clearly disproved the early conjecture that ( is
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M.J. Alava et al. 0
~. _ , ~ m , ,
i~__:
I
e, i
a)
b)
Fig. 6.1 Models for a DPRM. a) In the {10} orientation, b) In the {11} orientation. (From Rieger, 1998a)
Fig. 6.2 A collection of polymers of lowest energy directed along the diagonals of a square lattice with random bonds. Each polymer (crossing 500 bonds) has one end fixed to the apex of the triangle, the other to various points on its base, and finds the optimal path in between. (From Kardar and Zhang, 1987)
superuniversal (Kardar and Zhang, 1987). Questions remain about whether the KPZ equation has a finite upper critical dimension, with recent work suggesting no upper critical dimension (Castellano et al., 1998) or a critical dimension which is less than or equal to four (L~ssig and Kinzelbach, 1997), though the latter result is inconsistent with the majority of the numerical evidence (Kim et al., 1991; Tang et al., 1992; AlaNissila et al., 1993; Castellano et al., 1998). Optimization methods have recently been applied to the minimalpathproblem in (1 + 1) and have confirmed that overhangs are not important at weak disorder (Cieplak et al., 1995; Marsili and Zhang, 1998; Schwartz et al., 1998).
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However, there is an important difference between the minimalpath problem with overhangs as compared to the DPRM, namely that the growth front in the minimalpath problem with overhangs is rough (Duxbury and Dobrin, 1999). By construction, the growth surface in the transfermatrix method is fiat, because the energy (distance) labels are incremented row by row. The Dijkstra growth front generated by invading a random network obeys the same scaling behavior as (6.3.9), with time replaced by L. Invasion starts at L  0 (which is analogous to t = 0 in the growth analogy) and occurs on a strip of transverse size I d1 . Preliminary work indicates that the Dijkstra invasion front is in the KPZ class, so that the unrestricted Dijkstra algorithm finds minimal paths and generates KPZ growth fronts simultaneously (Dobrin and Duxbury, 1999).
6.4 Greedy algorithms and extremal processes In "greedy" algorithms one chooses and updates locally optimal sites (Papadimitriou and Steiglitz, 1982; Lov~isz and Plummer, 1986; Cormen et al., 1990; West, 1996). As discussed in Section 2.3, the greedy strategy is useful in solving the minimumspanningtree problem and the minimalpath problem. There are a variety of other g r a p h i c  m a t r o i d problems that are solved by greedy methods. In computerscience applications, these methods are interesting because they solve certain optimization problems efficiently and exactly. However in physics they are also interesting because they generate physically interesting and relevant growth morphologies. The greedy strategy is analogous to extremal dynamics (Duxbury and Dobrin, 1999). Extremal dynamics occurs in systems which are metastable and are driven weakly (well below the activation barriers) and so are evolving slowly (de Arcangelis et al., 1985; Duxbury et al., 1987; Zaitsev, 1992; Bak and Sneppen, 1993; Miller et al., 1993; Paczuski et al., 1995; Sneppen, 1995). A wellknown example is the BakSneppen evolution model which optimizes the "fitness" of a population by the extremal selection of the weakest species (Bak and Sneppen, 1993; Paczuski et al., 1996; Bak, 1996). The hottestbond algorithm for fuse networks (de Arcangelis et al., 1985; Duxbury et al., 1987) and the invasionpercolation algorithms (Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984) are two other important examples of extremal dynamics. Notice that in the fracture of fuse networks load enhancement plays a crucial role so that, in spite of the extremal dynamics, there is no "selforganizing" critical steadystate. The connection between an extremal dynamics and critical behavior is related to the question of how to map annealed disorder to quenched and vice versa. In the case of the KPZ equation with thermal noise, as discussed in Section 6.3, the arrivaltime mapping connects the quenched disorder of the directed polymer to the noise in the interface equation. And in both limits the system shows non
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trivial scaling. The mapping can sometimes work in the opposite direction, as in the socalled "runtime statistics", where it was used to find the fractal dimension and dynamics of invasion percolation (see Marsili, 1994). Each greedy algorithm described in Section 2 solves exactly an optimization problem, that is, it minimizes a cost function. This begs the question as to whether other extremal processes have an associated cost function which is minimized by the dynamics. Such a cost function would act as a nonequilibrium free energy which would be useful in the theoretical development of the problem.
7
Random Ising magnets
7.1
Introduction
Random Ising magnets (Young, 1998) provide a key pedagogical basis for understanding the interplay between ordering (e.g. ferromagnetism) and disorder induced by quenched impurities. Many studies indicate that the ordered phase in random magnets and random manifolds is dominated by a zerotemperature fixed point which means that the zerotemperature behavior is characteristic of the entire ordered phase. In this chapter we demonstrate that the ground state of many random Ising magnets can be found exactly using optimization algorithms. Use of these methods is starting to provide detailed tests of functionalrenormalizationgroup and replicasymmetrybreaking theories of random magnets. This is especially important at low temperatures, where MonteCarlo methods are inefficient due to strong metastability effects. We consider the "spinhalf" Ising magnets described by 7[   ~ Jij~ricrj  ~](hi + n)~ri (i,j) i
(7.1.1)
The first sum is over nearestneighbor spins and cri = 41. We identify the following special cases: (i) Interfaces in randombond magnets. Jij >__ 0, hi  0. In this case the bulk phase is ferromagnetic, but an interface is enforced by fixing the spins on two opposite faces of a lattice to have opposite orientations. The roughness and energy fluctuations of such domain walls have interesting scaling behavior. This minimalenergyinterface problem maps directly to the maximumflow problem (with the domain wall being the minimum cut) (Section 7.2.2). (ii) Randomfield magnets. Jij >__O. Randomfield magnets are ferromagnetic at low values of the random field for dimensions d > 2 (Imry and Ma, 1975;
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Imbrie, 1984), but as the randomfield strength increases, they become
frozen paramagnets. Even at zero temperature, there is then the possibility of a bulk phase transition, at which the order parameter (the magnetization) is singular. This problem also maps to maximum flow (the minimum cut defines the domain structure) (Section 7.3.2). (iii) Diluted antiferromagnets in a field (DAFF). Jij    ] J I E i E j , where Ei  0, 1. This models experimental systems such as FecZnlcF2 (Belanger, 1997), which are superexchange antiferromagnets doped with nonmagnetic impurities (Zn). A key observation is that (nearestneighbor) DAFFs on bipartite graphs map to diluted ferromagnets in an alternating field. The latter problem is solved by maximum flow (Section 7.3.2). Many of the experimental systems have bipartite structure, and are dominated by nearestneighbor superexchange, so the DAFF model is an excellent first approximation. (iv) Frustrated magnets and spin glasses. Frustrated magnets are those in which the exchange interactions themselves cannot all be satisfied. The most famous example of a frustrated magnet is a spin glass (Binder and Young, 1986). A second famous example is the axial nextnearestneighbor Ising model (ANNNI) (Selke, 1988). However spin glasses combine disorder and frustration (Toulouse, 1977) while there are a large number of magnets that are only frustrated (e.g. ANNNI). Although the twodimensional Ising spin glass with nearestneighbor interactions is solvable in polynomial time, the threedimensional problem has been proven to be N Pcomplete. Nevertheless NP optimization methods are starting to make some progress even in that problem (Section 7.4). In this section we also present some interesting recent results for the Euclideanmatching problem (M6zard and Parisi, 1988; Houdayer and Martin, 1998), which is solvable in polynomial time in arbitrary dimensions and which shows some of the features of a spin glass.
7.2
Interfaces in randombond magnets
7.2.1 Introduction and scaling theory We have already discussed an interface in a randombond magnet in two dimensions (Section 6.3) as it is equivalent to the minimalpath problem. Nevertheless we discuss some further aspects of interfaces in twodimensional randombond magnets as well as discussing results in three and four dimensions. After describing the continuum and lattice models used to study randombond interfaces, we summarize some of the scaling predictions. In Section 7.2.2, we show
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how randombond interfaces may be found, in arbitrary dimensions, by mapping to maximum flow. This is an important development as the transfer matrix is very inefficient for this problem in three and higher dimensions 9 Maximumflow methods have provided the first precise tests (see Section 7 9149 of the functionalrenormalizationgroup predictions for this problem, as well as providing detailed information about the dependence of interface roughness on the strength of the disorder and the lattice orientation. The section on randombond interfaces closes with a discussion of groundstate degeneracy in diluted networks, and some observations on the sensitivity of the ground state to a small drive field (Section 7.2.4). The continuum solidonsolid(SOS) model for randombond interfaces is, 7/({u})
f
dY
~[Vu(?)
+V(u(Y))+Hu(~)
)
,
(7.2.1)
where the single valued function u(?) measures the height (displacement) of the elastic manifold above a flat reference surface. For a ddimensional interface in a (d + 1)dimensional Ising magnet the vector ~ is ddimensional. V (u (7)) models the random fluctuations in the exchange constants, cy is the interface stiffness and is related to J for Ising domain walls. H is an applied drive field. For many types of disorder, interfaces in random magnets are selfaffine, that is they are fractals with different scaling dimensions perpendicular and transverse to the average orientation of the manifold. The equilibrium "width" and "energy fluctuations" of the manifold are defined as usual, i.e w2(L l)  [uZ]av  [u] 2 and AE2(L, 1) = [E2]av  [E]2v . As a function of the system size parallel to the interface (L) and perpendicular to the interface (1), we have, 9
'
av,
w(L,1) ~ Lr fo(1/L ~)
(7.2.2)
A E ( L , l) ~ L ~ A ~ ( 1 / / ~ ) .
(7.2.3)
The energy fluctuation exponent 0 is related to the roughness exponent ~', (Huse and Henley, 1985; Barabasi and Stanley, 1995; HalpinHealy and Zhang, 1995), 0  2~" + d  2.
(7.2.4)
From the minimalpath problem, we know that in (1 + 1) dimensions ~" = 2/3 (Huse and Henley, 1985; Kardar et al., 1986). In (d + 1) dimensions, a functionalrenormalizationgroup calculation yields ~" = 0.2083(4  d) (Fisher, 1986). Lattice models of randombond interfaces begin with the Ising magnet with Hamiltonian,
7[ =  Z JijcricrJ  Z (i,j)
Hcyi
(7.2.5)
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where Jij >__O. An interface is enforced by fixing the spins on two opposite faces of the lattice in opposite orientations (i.e. on one side fix o'i  1 and on the other O"i " ~   1 ) . As we now demonstrate, the energy and configuration of interfaces in this randombond magnet can be found exactly in an arbitrary dimension using a mapping to the maximumflow problem.
7.2.2
Mapping to maximum flow
Although the mapping between randombond interfaces and the maximumflow problem is simpler than either the spinglass (Bieche et al., 1980) or randomfield mappings (Picard and Ratliff, 1975; Angl6s d'Auriac et al., 1985; Barahona, 1985) to optimization problems, it was noticed and used much later. Early connections between minimalcost interfaces (minimum cuts) were in the context of limiting paths in granular superconductors (Rhyner and Blatter, 1989; Riedinger, 1990, 1992). Rhyner and Blatter (1989) noted that the problem was a linearprogramming problem and solved small samples using the simplex method, while Riedinger (1990) made the connection with the maximumflow problem and applied the augmentingpathalgorithm to its solution. More recently, Middleton (1995) used the mapping to maximum flow, and the preflow method, to find highprecision values for randombond exponents in (2 + 1) and (3 + 1) dimensions (see Section 7.2.3). The mapping of randombond interfaces to maximum flow/minimum cut is quite simple, it only requires using the correct variables. First, for any spin configuration {cri}, we define
S

{i E Vl~ri  +1}
S

{i E W l c r i   1 } = V \ S
(7.2.6)
Given a spin configuration described by S, its energy (taking (7.2.5) with H  0) is given by,
~(S)
=

~ iES,jES


Jij
~
Jij+
iES,jES
~ Jij + 2 Z Jij iEV, jEV iES,jES
~
Jij
(7.2.7)
iES,jES  Ebulk ] Edw
The first term in the second of these equations is the bulk ferromagnetic energy (Ebulk), while the second term is the interface energy (Edw). The interface (or domainwall) energy, Edw, is positive and we seek to minimize it. Now we show that the domainwall energy, Eaw, is equivalent to the maximum flow in a related capacitated network, and that the domainwall configuration is exactly the same as the minimum cut in that network.
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To define the analogous maximumflow problem, first connect the sites on one face of the lattice to a new source node s, and the sites on the opposite face to a second new node, the target t (these two extra nodes are sometimes called "ghost" sites). Assign to the bonds making these connections infinite capacity. For each existing bond in the lattice make the association Uij   U j i   J i j , so that the exchange constants map to flow capacities. This construction leads to a network which is identical to that shown in Fig. 3.1. Note that we have forward and backward arcs for each pair of interacting sites in the lattice. Now we seek the maximum flow in this network, given the capacities U i j . From the minimum cut/maximum flow theorem, the maximum flow is equal to the sum of the capacities on the minimum cut (Section 3.2). The sites on one side of the cut are S, while the sites on the other are S. The capacity of a cut (S, S) is, w

u[S, S] =
Z
uij =
Edw
2 '
(7.2.8)
i6S,j6S
which is equal to the domain wall energy, as seen by comparing (7.2.8) with (7.2.7) and recalling that Uij  J i j .
7.2.3
Roughness exponent and orientation dependence
Early attempts at numerically testing the prediction for the roughness exponent in higher dimensions (~ = 0.2083(4  d) (Fisher, 1986)) used a transfermatrix algorithm (Kardar and Zhang, 1989). However, although the transfer matrix is efficient for the DPRM in arbitrary dimensions, it is very inefficient for higherdimensional manifolds. Hence the analysis based on the transfer matrix was restricted to small sizes and the exponents found were not indicative of the infinitelattice behavior. Middleton (1995) used the maximumflow mapping to obtain highprecision results. Middleton used {111 }oriented Ising systems with a domain wall pinned (at one point) in the center of the system. His data for the (2+ 1)dimensional problem is reproduced in Fig. 7.1. The best data collapse (that shown in the figures) using systems with varying lateral size (L) and height (l) yielded exponents: ~ = 0.41 (1), 0 = 0.82(2) in (2 + 1); which are consistent with the predictions of the functionalrenormalizationgroup calculations (~  0.416 in (2 + 1)), and with the scaling relation (7.2.4). In (3 + 1) a similar analysis yielded ~  0.22(1) and 0  1.45(5) (1 up to 20 and L up to 30), which again confirm the RG prediction and the scaling relation (7.2.4). Using the mapping to maximum flow, Alava and Duxbury (1996) and R~iis/inen et al. (1998) have extended the analysis of random manifolds to include the dependence of roughness on: the type and strength of the disorder and; on lattice orientation. The dependence of the interface roughness on dilution, 1  p,
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.
.
.
.
201
.
i
d=2
a)
F
'A__0 ifhi 0.
(7.3.2)
This construction is illustrated in Fig. 7.7. This construction is general as it applies for arbitrary range exchange c o n s t a n t s Jij >__0, including infinite range. The field terms may be regular and/or random so it also includes an alternating field which
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arises from the mapping to the DAFF (see below). With this construction, we can now write the energy (7.3.1), or "cost function", as
E
Z
Jij~
(7.3.3)
(i,j)EA where the spin orientation of the source and target nodes are fixed: as = 1 and at =  1 . This is now a randombond Ising model on the enlarged graph which includes s and t. We proceed in a manner very similar to that given for randombond interfaces (see Section 7.2.2). The new feature is that the spins of the ghost sites now impose the "antiperiodic" boundary conditions, and enforce an interface. We define, as in Section 7.2.2, S to be the set of up spins and S to be the set of down spins. The energy of the RFIM ground state can then be written as (this is the same as in Section 7.2.2),
E(S)  
~ (i,j)Ea
Jij + 2
~
(7.3.4)
Jij.
(i,j)E(S,S)
Now the first term (the negative term) is the energy one obtains by presuming that all of the exchange terms and all of the field terms are satisfied. The second term is the degree to which the field and exchange terms are unable to be simultaneously satisfied. Clearly the groundstate energy is minimized when the positive term in (7.3.4) is smallest. As in Section 7.2.2, making the association Uij  Jij makes it clear that the positive term in (7.3.4) is equivalent to the cut capacity of a flow problem, where flow is injected at s and removed at t. Hence, from the mincut/maxflow theorem (Section 3.2) the cut energy is equal to the maximum flow. Once the minimum cut has been identified, it defines a partition (S, S) and hence it also defines the domain structure of the RFIM ground state. It is worth stating again that this mapping applies for arbitrary range ferromagnetic interactions and arbitrary fields (for spin 1/2 magnets). However, a generalization of this mapping to Potts models, leads to a kcut or a kterminal problem, which has been proven to be in the NPclass (Dahlhaus et al., 1994). For planar graphs, including the kterminal vertices, there are complicated but polynomial algorithms. Diluted antiferromagnets in a field (DAFF) are expected, in the weak field limit, to be in the same class as the RFIM (Fishman and Aharony, 1979; Cardy, 1984). Examples of diluted antiferromagnets are materials such as, FecZnlcF2 (Belanger, 1997). c is the concentration of magnetic atoms (Fe) and can be varied. The DAFF Hamiltonian reads m
~~  Z JijEiEj~iO'j  Z OEi~ (i,j) i
(7.3.5)
where O'i  '11, Jij >_ O, (i, j) are nearestneighbor arcs on a lattice, and ei E {0, 1} with ei  1 with probability c, representing the site concentration
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UP
__db,
IF
I I
IF
I~'
M i n i m u m Cut h 4 DOWN Source
Fig. 7.7 An example of how to map the onedimensional RFIM chain into a augmented graph with two extra sites. The sites of the original chain are connected to each other with bonds of capacity J, and depending on the sign of the random field to either of the "ghost sites" with bonds of capacity Ih i I. We need to find the maximum flow from s(source) to t(target or "sink"), and its associated minimum cut.
of spins. Usually one takes Jij " J, as is relevant to experiment. The antiferromagnetic exchange competes with the external field which tries to produce a finite magnetization. For bipartite networks, with nearestneighbour interactions, it is trivial to map this problem to a ferromagnetic exchange problem in an alternating field (Mattis, 1976). To see this, divide the lattice into two bipartite sublattices A and B. Every site in A has all its nearest neighbors in B, and vice versa. Now use the "gauge" transformation (here we allow for random fields hi at each site, while for the "usual" DAFF case hi = H.),
t
{ +~ri
for for
i 6 A i EB '
t { +sihi hi = 8ihi
for for
i 6 A
i ~ B ' (7.3.6) which flips the spins on one sublattice and leaves the other sublattice unaltered. The Hamiltonian which results is, O'i  
O" i
"   Z (i,j)
Jgisjo'/o'J  Z "8i0"/ + Z "giO'/" i~A iEB
(7.3.7)
(7.3.7) is the Hamiltonian for a diluted ferromagnet in a staggered field, with one sublattice being subjected to an up field, while the other sublattice experiences a
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down field. This model is solvable by mapping to the maximumflow problem, as are more general problems in which the nearestneighbour exchange is random (though still positive), and there is a local random field. However, it is restricted to nearest neighbours as further neighbour antiferromagnet interactions lead to couplings within the sublattices.
7.3.3
The phase transition in three dimensions
The analysis of the phase transition in the randomfield Ising model has a history filled with controversy, and in some cases, acrimonious differences of opinion. An early perturbation analysis (Parisi and Sourlas, 1979) suggested that "dimensional reduction" should be valid, so that the behavior in a pure system in dimension d should correspond to randomfield behavior in d + 2. This implies that there should be no phase transition in the d  3 RFIM. However, it was then proven that the d = 3 RFIM is ordered at weak fields (Imbrie, 1984), demonstrating that dimensional reduction is incorrect. There is still no reliable ~expansion for the RFIM (Br6zin and de Dominicis, 1998). Concurrently there was a bitter conflict between experimentalists who claimed that DAFFs in three dimensions exhibited a phase transition and those who claimed that they did not. This was also resolved in favor of a phase transition (Belanger, 1997). Two dimensions is the lower critical dimension for the RFIM and there is a rigorous proof that a unique Gibb's state exists for arbitrary weak fields in two dimensions (Aizenman and Wehr, 1989), implying that there is no finite magnetization for any finite random field in two dimensions. Using the mapping to maximum flow (see Section 7.3.2), Ogielski (1986) studied the groundstate phase transition in the RFIM in three dimensions for Gaussian and bimodal disorder and clearly confirmed that there is indeed a phase transition. He concluded that the phase transition is second order (at zero temperature) for the Gaussianrandomfield case and for the bimodal case the order parameter had a tendency toward a firstorder behavior. For the Gaussiandisorder case, he found the magnetization exponent r = 0.05 and the correlationlength exponent v  1.0. The meanfield exponents for the Gaussian case (which is second order) are v = 1/2, /3 = 1/2. Several recent calculations of a similar type to Ogielski's (Swift et al., 1997; Nowak et al., 1998; Hartmann and Nowak, 1999), as well as MonteCarlo simulations (Rieger and Young, 1993; Rieger, 1995a) have confirmed many of his general conclusions. In all cases, the magnetization exponent is very small indicating, perhaps, a firstorder behavior. However, there is no indication of a latent heat. The correlationlength exponent also seems to be nonuniversal, with values near v  1.1 reported for Gaussian RFIM and DAFF, while values closer to v = 1.6 are typical of the (+h) RFIM. Compare these with the latest experiments on a highquality DAFF sample, that indicate a correlation
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length exponent of v ~ 0.87 (Slanic et al., 1999). A typical scaling analysis for the DAFF is presented in Fig. 7.8, where the staggered magnetization is M [Iml]av, and the corresponding "disconnected" susceptibility is Xdis • L3[m2]av. Here m is the sublattice magnetization for a given realization of the disorder, and the average is over disorder. The scaling behaviors M  L  M v l f I ( ( A  A c ) L l/v) and Xais = L  y / v X ( ( A  A c ) L l/v) are assumed. As remarked above the values of v and fl found in these simulations are quite close to those reported by Ogielski (1986). However, the small size of fl suggests a firstorder behavior in that quantity, and which motivates a detailed look at the behavior of the magnetization as a function of field. An example of the field dependence of the magnetization, for a single realization of a DAFF, is presented in Fig. 7.9. This figure suggests that the magnetization decays via a sequence of jumps, rather than in a continuous manner. These jumps are "averaged out" in the data of Fig. 7.8. There is then a question about how the size of the magnetization jumps change as the system size is increased. Angl6s d'Auriac and Sourlas (1997) and Sourlas (1999) have done an interesting analysis in which they show that the size of the magnetization jumps ji (e.g. the five largest ones) appearing in data such as Fig. 7.9 should be plotted according to Ci
ft"
ji = j ~ + Li~(1 +  ~ ) 6 j i .
(7.3.8)
From this analysis it is concluded that the magnetization jumps do not disappear in the infinitelattice limit. Instead, there is a finite, and large, jump in the magnetization in the infinitelattice limit, for example for the DAFF, they find that the magnetization jump j ~ ~ 0.55. In addition, the correlationlength exponents v found from the finitesizescaling analysis of (7.3.8) are nonuniversal and are significantly larger than the results found using conventional analysis (Ogielski, 1986; Swift et al., 1997; Nowak et al., 1998; Hartmann and Nowak, 1999). This clearly raises questions about the nature of the RFIM phase transition in three dimensions, and it is likely that optimization methods will greatly assist in their resolution. Analysis of the transition in four dimensions agrees with both the meanfield limit, and recent hightemperature series expansions. For binary disorder the transition is discontinuous, and for Gaussian disorder it is continuous (Swift et al., 1997).
7.3.4
Domains and domain walls in the R F I M
At sufficiently high random fields in three dimensions, and at all finite fields in one and two dimensions, random fields destroy ferromagnetic longrange order in the ground state. We define the "breakup lengthscale", Lb, to be the size scale at
2
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Exact combinatorial algorithms
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~,
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F i g . 7.8 S c a l i n g plots o f the s t a g g e r e d m a g n e t i z a t i o n M = [[m
[]av a n d
15
the c o r r e s p o n d i n g
s u s c e p t i b i l i t y gdis  L3[m2]av for the D A F F on a c u b i c lattice w i t h site c o n c e n t r a t i o n c = 0.50. T h e f o l l o w i n g v a l u e s w e r e f o u n d Ac = ( H / J ) c
 0.62, v = 1.14,/3 = 0.02,
 3.4. ( F r o m H a r t m a n n and N o w a k , 1999)
I
I
0.8
0.6 0.4 t
0.2
rl
I _
o
~~.,'
I _I   
I _
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_
_
_
_ i, . . . . .
I
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1
2
3
4
5
6
7
A
Fig. 7.9 T h e g r o u n d  s t a t e s t a g g e r e d m a g n e t i z a t i o n , m, o f a D A F F w i t h c = 0 . 5 0 and L = 15 on a c u b i c lattice, as a f u n c t i o n o f the strength o f the a p p l i e d field A. T h e g r o u n d state is d e g e n e r a t e , the m i n i m u m (solid) and m a x i m u m ( d a s h e d ) c u r v e s i n d i c a t e the variation o f m o v e r the set o f d e g e n e r a t e states. ( F r o m H a r t m a n n and N o w a k , 1999)
214
M . J . Alava et aL
which ferromagnetic domains cease to exist. In one dimension, Lb diverges algebraically on approach to A  0. In three dimensions, L b diverges algebraically on appoach to Ac, if the transition is continuous. If the transition is first order, the domain size should remain finite up to the critical point. In two dimensions, which is the lower critical dimension for the RFIM, the behavior is different. There, as suggested by Binder (1983) and Grinstein and Ma (1983), the data is consistent with, Lb ~ exp (A[1/A]2),
(7.3.9)
where A is a disorderdependent constant. This form has been confirmed, with A = 1.9 q 0.2 and 2.1 + 0.1 for bimodal and Gaussian disorder, respectively. This result is different from that obtained with finitetemperature MonteCarlo simulations for small L (Fernandez and Pytte, 1985), which demonstrates the advantages of groundstate calculations in pushing to large enough system sizes. The surfaces of the magnetic domains scale with nontrivial exponents (Esser et al., 1997). We note that for the Gaussian disorder case, Vives et al. claim that there is a phase transition to a ferromagnetic state (Frontera and Vives, 1999), which is inconsistent with the Binder prediction and is also inconsistent with the AizenmanWehr theorem (Aizenman and Wehr, 1989). Nevertheless an interesting conflict between the conventional analysis of the RFIM and the results of groundstate simulations is beginning to emerge (Sepp~il~i et al., 1998). Specifically the role of the percolation of a macroscopic domain. Preliminary work indicates that percolation of ferromagnetic order occurs in the RFIM at a fairly large value of A ~ 2. This is not too surprising since even in the strongfield limit, the 50/50 concentration of up and down fields is close to the site percolation threshold, pc  0.59. As the field strength is reduced domains grow so percolation of a macroscopic domain is expected. If this percolating domain remains fractal all the way down to A  0, there is no violation of the AizenmanWehr theorem. Nevertheless, there is a "phase transition" at the percolation point to a quasilongrangeordered phase. This appealing picture warrants an intensive numerical analysis. The renormalization calculations (Fisher, 1986; HalpinHealy, 1989) and an ImryMa argument indicate that the roughness of domain walls in (d § 1)dimensional RFIMs should have the exponent ( = (4  d ) / 3 . Early studies of RFIM domain walls in (1 § 1) used the transfermatrix method (Fernandez et al., 1983). The domainwall energy is calculated, as usual, from Edw  Eap  Ep
(7.3.10)
where Eap denotes the energy of the system with antiperiodic boundary conditions and E p the energy with periodic boundary conditions. Fig. 7.10 shows an example of a (1 § 1) RFIM domain wall.
2
Exact combinatorial algorithms
215
Fig. 7.10 An RFIM interface. The calculation is for an L = 100 square lattice with weak bimodal disorder, A = 10/17. Note the large jumps on the interface and the lack of overhangs. (From Sepp~l~ et al., 1998)
The idea that domain walls in the RFIM are selfaffine has been supported by the work by Fernandez et al. (1983) and by HalpinHealy and Herbert (1993), both using the transfermatrix technique. Moreover, later studies with combinatorial optimization further supported this claim (Moore et al., 1996; Jost et al., 1997). If RFIM interfaces are selfaffine, the variations in the domainwall energy would scale as, A E ( L ) ~ L ~ as in the randombond case discussed in Section 7.2, and the energy should be linear in L. Notice that this is in direct contrast to the argument as to why the twodimensional ground state should break up into domains: the domainwall energy has a logarithmic correction which can be measured in ground state calculations (Seppfilfi et al., 1998). The data in Fig. 7.11 shows that (1 + 1) RFIM interfaces scale, at weak disorder, with a roughness exponent close to 6/5. Weak disorder implies that the system size is smaller than the "breakup" length, L < < Lb, so that the bulk state is ferromagnetic. Once beyond the breakup length scale, the roughness exponent returns to 1 (see inset to Fig. 7.11). The origin of the superrough behavior seen in Fig. 7.11, is a broad distribution of jump sizes which is visually evident in Fig. 7.10. A quantitative analysis indicates that the distribution of jump sizes P ( 3 h ) is a stretched exponential, but also that large jumps occur quite frequently at large sample sizes as in some models of kinetic growth (Krug, 1994). For the weakdisorder regime one obtains an effective energyfluctuation exponent 0 ~ 0.9 (Sepp~il~ et al., 1998). One should finally note that the roughness exponent as determined for groundstate domain walls has nothing to do with the similar one for driven interfaces with randomfield disorder (quenched EdwardsWilkinson equation) (Kessler et al., 1991; Leschhorn, 1993).
216
M . J . Alava et aL
102 m m
101
9
/ oo~
9
10 ~
A
10~ 10~
.
10 ~ .
.
.
.
.
.
.
i
102 L
10 2 .
.
.
.
.
i0 a .
.
.
103
Fig. 7.11 Scaling of the interface width for an RFIM with bimodal disorder. A = 2/3 (empty triangles) and 3/2 (filled squares). The line indicates a leastsquares fit with a roughness exponent ~ = 1.20 4 0.05. The inset shows the crossover in interface properties with increasing system size (A = 10/9). (From Sepp~il~iet al., 1998) 7.3.5
Sensitivity and degeneracy
In this section we describe studies of the properties of the paramagnetic state of the RFIM and the DAFE In this regime, the minimum cut gives the complete domain structure of these complex ground states. We study the stability of these ground states to small perturbations (sensitivity) (Alava and Rieger, 1998), and in the case of the DAFF and the (+h) RFIM, the groundstate degeneracy (Bastea and Duxbury, 1998; Hartmann, 1998c). In spin glasses (Binder and Young, 1986; Rieger, 1995c; Young, 1998) it is well known that small changes of parameters like temperature or external field cause a complete rearrangement of the equilibrium configuration (Fisher and Huse, 1986; Bray and Moore, 1987). This is sometimes called "chaos" in analogy with the sensitivity to initial conditions exhibited by chaotic dynamical systems. This chaos has experimentally observable consequences like reinitialization of aging in temperature cycling experiments (Refregier et al., 1987; Lefloch et al., 1992; Mattsson et al., 1993; Rieger, 1994) and has also been investigated in numerous theoretical works (Ritort, 1994; Kisker et al., 1996; Kondor, 1989; Kondor and Vdgs6, 1993; Franz and NeyNifle, 1995). A slight random variation of the quenched disorder has the same effect on the groundstate configurations. In theoretical calculations, chaos with respect to temperature changes is harder to observe than chaos with respect to disorder changes (NeyNifle and Young, 1997), and the latter phenomena has been used to quantify spinglass chaos in numerical investigations (Bray and Moore, 1987; Rieger et al., 1996).
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217
Fig. 7.12 A twodimensional RFIM ground state plus the perturbationinduced changes. The original spinorientations are indicated in grey for oi = + 1 and white for cri  1, the flipped spins are indicated in black. L = 320, A  2 and 6  0.1. (From Alava and Rieger, 1998)
This type of chaos has also been demonstrated in the directed polymer in a random (bond) medium (Zhang, 1987; Feigel'man and Vinokur, 1988; Nattermann and Lyksutov, 1992), where it is quantified using a "displacement exponent". The displacement exponent is related to the roughness exponent ~" (Feigel'man and Vinokur, 1988; Nattermann and Lyksutov, 1992) and is also related to the susceptibility of such interfaces (Shapir, 1991). The extent to which the ground state ({cri}) and the perturbed state ({a/(3)}) of a magnetic system differ, is quantified by the overlap,
q  ~
1
!
O'i~ri ( 5 ) .
Z
(7.3.11)
i
In glassy systems, thermodynamic states are typically chaotic, which means that on length scales L > L* the two spin configurations {or} and {or~} lose all correlations. In contrast, the RFIM ground state remains correlated (see Fig. 7.12) at long length scales, although there are considerable domain excitations at small length scales. The "sensitive" regions in the RFIM model concentrate on the cluster boundaries of the ground state. Fig. 7.13 presents the value of the overlap q as a function of the randomfield amplitude A in two dimensions. This figure has been obtained with a random field distribution and a perturbation distribution that have a constant probability density between A and A and  5 / 2 and 5/2, respectively, and Jij = 1. In any
218
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Alava e t al.
1.00
0.95
0.90
0.85
[] A V O 9 . . . .
~ 1
.
.
.
.
.
oL=40 &L=80 V L = 160 9L = 240 o L = 320 .
.
.
10
Fig. 7.13 Scaling of the chaos or overlap, q, of the twodimensional RFIM model with Gaussian disorder, for system sizes L = 40, 80, 160, 240 and 320 and for 6 = 0.1. (From Alava and Rieger, 1998)
dimension, the limit A ~ cx~ goes over to a sitepercolation problem, i.e. the local RForientation determines the spin state at a site. In that limit the overlap q is determined by the probability that the applied perturbation 6 will change the local field orientation. At small enough fields, the ground state (on finite lattices) is ferromagnetic and there are no "sensitive" interface regions in the ground state. The overlap then approaches one. As seen in Fig. 7.14, the groundstate degeneracy of (• RFIM also occurs at the interfaces between up and down domains in the RFIM ground state. This sort of degeneracy also exists in the DAFF (see below) and so may be of experimental relevance. These degenerate configurations are found from the degenerate cut structure using the same method as that used for degenerate randombond interfaces (see Section 7.2.4) (Bastea, 1998) (see also (Hartmann and Usadel, 1995)). The degenerate domains evident in Fig. 7.14 produce a finite groundstate entropy for a wide range of A = h / J , as demonstrated in Fig. 7.15. Prior work in one dimension (Derrida et al., 1978; Bruinsma and Aeppli, 1989; Igloi, 1994), on Cayley trees (Bruinsma, 1984), and in two dimensions (Morgenstern et al., 1981), had already indicated the singular nature of the degeneracy as a function of the field. In particular, the large degeneracies at fields (h/J) which are small rationals. However, the presence of groundstate entropy even for irrational h / J was a surprise (Bastea and Duxbury, 1998). The mechanism for this is easy to construct. The degenerate clusters at irrational A have zero field energy and the same domainwall energy in both the up and down states of the cluster. For the
2
Exact combinatorial algorithms
219
Fig. 7.14 Groundstate structure of the bimodal RFIM for A = h / J = 3/2. The spins frozen "up" are the dominant grey color, the spins frozen "down" are white, while the other shades of grey represent the spins making up the degenerate clusters. A dot indicates a field "up" and the absence of it indicates a field "down". (From Bastea, 1998)
RFIM on the square lattice, the lowest order degenerate clusters of this sort are indicated in the inset to Fig. 7.15. The number of these clusters (or twolevel systems (TLS)), nTLS, can be estimated from the expression, L2 nTLS O~ P T L S ~ Lb
(7.3.12)
where L is the linear size of the system, and Lb is the domain size. From (7.3.9), we have Lb o~ e x p [ ( 1 / A ) 2 ] . L Z / L b is the total length of interface between up and down spin domains in the system. PTLS is the probability of occurrence of a TLS at a given interface site. PTLS  pn/4, where 1/4 is the probability of occurrence of the updown pair of fields and Pn is the probability that this pair is surrounded in the ground state by frozen spins with the appropriate configurations. The entropy density is then s o~ p n e x p [  ( 1 / A 2 ) ] for A < 4 and 0 f o r A > 4. Pn is discontinuous at A  2, because the dominant TLSs for A < 2 are different than those for A > 2 (for example there are twice as many spin configurations that lead to a TLS below than above A  2). If we take the observed jump ~ 3.3, then the above argument leads to the curve given in the inset to Fig. 7.15, which is very close in form to the continuous part of the entropy presented in Fig. 7.15. In the regime Ac < A _< A., we can consider the ground state to be composed of a frozen background in which is embedded a set of largely noninteracting free superspins (for each independent cluster). The ground state thus contains a large number of magnetic twolevel systems (Coppersmith, 1991). This gives rise to a
220
M . J . Alava et aL
q)
0.06
5
@ i
2
0.03
0
o
I
2
hlJ
3
hlJ
4
Fig. 7.15 The groundstate entropy of the twodimensional RFIM as a function of h / J = A. The inset shows the smallest twolevel systems (TLSs) at irrational A and an estimate of the entropy (from (7.3.12)) produced by them. A dot indicates where the local random field favors the "up" spin direction. The system sizes used were from 10 x 10 to 130 x 130 and the entropy was found as the slope of the line (ln D).vs.N, where D is the degeneracy, N is the system size (total number of spins) and the average is over the disorder (1000 samples were used). (From Bastea and Duxbury, 1998)
paramagnetic response at low temperatures for both the RFIM and DAFF in this regime. The natural groundstate order parameter for the paramagnetic response in the regime Ac < A < A , is the magnetization mpferro for the RFIM and the staggered magnetization mpstaggered for the DAFE These quantities can be calculated by applying an appropriate infinitesimal field or by polarizing all of the degenerate domains in a given orientation. The basic features of the groundstate degeneracy are then reflected in the groundstate paramagnetic magnetization. In three dimensions, the entropy remains zero at low A, reflecting the ferromagnetic state for A < Ac. While for large fields A > A, = 6 in three dimensions, the field polarizes all of the spins and there is no degeneracy. Calculations of the paramagnetic order parameter for the DAFF, mpstaggered, for square and cubic lattices are shown in Fig. 7.16. (The situation is similar for the RFIM.) There is a strong sublattice paramagnetic response for all Ac < A < A,, with spikes at certain rational values. These figures are for a DAFF with dilution p = 0.9, but only the details change as p is varied, at least when p > Pc,site. In the regime A < Ac there is a spontaneous staggered magnetization, and low temperature measurements (such as neutron
2
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221
0
0.2
d E ffl
E
0
0.1
2
4
6
H/J
i
0
,
J
i,,
1
i
2
H/J
I
3
i
,
4
5
Fig. 7.16 The order parameter for the groundstate paramagnetism (mpstaggered) of the DAFF on square and cubic (inset) lattices. (From Bastea and Duxbury, 1998)
scattering and NMR) should be influenced by both the "staggered paramagnetic" response and the spontaneous staggered magnetization. The existence of the additional order parameter mpferro in the case of the +h RFIM, and rnpstaggered in the case of the DAFF may complicate the analysis of the phase transition in three dimensions, particularly because it has been shown that these degenerate clusters are powerlaw distributed close to the phase transition (Bastea and Duxbury, 1999).
7.4 7.4.1
Ising spin glasses and Euclidean matching Introduction and overview
The spinglass problem has been a central problem in theoretical physics since the introduction of the EdwardsAnderson model (Edwards and Anderson, 1975).
222
M. J, Alava et aL
The EdwardsAnderson model is "][   Z Jij~ (i,j)
(7.4.1)
where (i, j) runs over nearestneighbor pairs on a ddimensional lattice, and ai are Ising spins. The exchange couplings Jij a r e independent random variables. In particular the + J model is defined by P ( J i j ) = p6(Jij  J) + (1  p)3(Jij + J)
(7.4.2)
where p is the density of ferromagnetic bonds. Continuum distributions of exchange constants, for example Gaussian or uniform distributions, are also of interest. This model captures the essential features of disorder and frustration (Toulouse, 1977; Vannimenus and Toulouse, 1977; Mrzard et al., 1987; Mrzard and Monasson, 1994), which are believed to promote spinglass behavior. Frustration is illustrated in Figure 7.17 for a plaquette on a square lattice. If the number of antiferromagnet interactions in a square loop is odd, no spin configuration can simultaneously satisfy all of the exchange interactions. Plaquettes with an odd number of negative interactions are therefore frustrated. A similar conclusion is valid for any closed loop. It is easy to see that a frustrated loop always encloses an odd number of frustrated plaquettes, i.e. there is a "topological charge" associated with isolated (not paired) frustrated plaquettes, the effects of which can be measured by performing a "loop integral" in the same way that electric charge (but not pairs of opposite sign) can be detected by measuring the electric field on a surface enclosing it. There are several similarities (Kirkpatrick, 1977; Fradkin et al., 1978) between shortrange spinglass systems and gauge theories, which justifies making these analogies. The solution of the infiniterange version of the EdwardsAnderson model (Kirkpatrick and Sherrington, 1978; Parisi, 1980) has provided a compelling vision of the nature of the spinglass ordered state. In this model there are an infinite set of ultrametrically related ground states. It has been argued that this structure is destroyed in finite dimensions and instead there are a finite number of degenerate ground states (in spin glasses with continuous distributions of exchange constants) and that there are powerlaw distributions of droplet excitations which dominate the low temperature behavior of spin glasses (Fisher and Huse, 1986). Despite an enormous amount of computational work (mostly Monte Carlo) and analytical analyses (Binder and Young, 1986; Young, 1998), the spinglass ordered phase is not well understood. The twodimensional spinglass groundstate problem was the first magnetism problem to be mapped to a polynomial optimization (matching) problem (Bieche et al., 1980), however, it was soon realized that threedimensional spin glasses
2
4
,,,,,
Exact combinatorial algorithms
o3
4
223
+
o3
+
,.I,.I1
o
+ 2
a)
1
2
b)
Fig. 7.17 a) If the number of negative interactions around a plaquette is even, it is always possible to satisfy all four interactions, b) If the number of negative couplings is odd, this is not possible and the plaquette is said to be frustrated.
and twodimensional spin glasses in a field are in the N Pcomplete class (Barahona, 1982). In Section 7.4.2 we discuss the mapping of spinglass problems to matching (planar graphs) and to maximum cut (general case). In Section 7.4.3 we review calculations of the exact ground states of spin glasses in two dimensions. In Section 7.4.4, we describe the Eulcidean matching problem, which is a toy model for spinglass behavior, and which is solvable in polynomial time using matching algorithms.
7. 4.2
Mapping to optimization problems
(i) Mapping to a matching problem We first show how a nearestneighbour spinglass problem on a square lattice, with free boundaries, is mapped to a matching problem on a general graph (Bieche et al., 1980; Barahona, 1982; Bendisch et al., 1994; Palmer and Adler, 1999). Associate with each unsatisfied bond an "energy string" joining the centers of neighboring plaquettes sharing this bond, and assign to each energy string a "length" equal to I Jij I. Clearly the energy of the system equals the total length of these energy lines, up to a constant. Let us for example put all spins pointing upwards, so that each negative bond will be unsatisfied and thus cut by an energy string (Fig. 7.18a). Unfrustrated plaquettes have by definition an even number of strings crossing their boundary, therefore energy lines always enter and leave unfrustrated plaquettes. Frustrated plaquettes on the other hand have an odd number of strings, and therefore one string must begin or end in each frustrated plaquette. These observations hold for any spin configuration. If boundary conditions are open or fixed, some of the energy strings can end at the boundary, whereas for periodic boundary conditions frustrated plaquettes occur in pairs and are paired by energy lines.
224
M.J.
Alava
e t al.
e:
:. . . . .
o
;
e [ ......
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6
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,, ,I4 , ,
......
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,
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9
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'a)
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,, 9
9
b)
c)
Fig. 7.18 Thick lines represent negative couplings. Energy strings (dotted) are drawn perpendicular to each unsatisfied coupling. Frustrated plaquettes (odd number of negative couplings) are marked by a dot. a) All spins up configuration, b) A ground state, c) Another ground state. Clearly, finding a ground state is equivalent to finding a minimum length perfect matching (Section 4) in the graph of frustrated plaquettes (Bieche et al., 1980; Barahona, 1982; Barahona et al., 1982). If IJij] = J, i.e. all interactions have the same strength, Fig. 7.18b shows one possible ground state. An equivalent ground state is obtained by flipping all spins inside the gray area in Fig. 7.18c, since the numbers of satisfied and unsatisfied bonds along its contour are equal. Degenerate ground states are related to each other by the flipping of irregularly shaped clusters, which have an equal number of satisfied and unsatisfied bonds on their boundary. If the amplitudes of the interactions are also random, this large degeneracy of the ground state will, in general, be lost, but it is easy to see that there will be a large number of lowlying excited states, i.e. spin configurations which differ from the ground state in the flipping of a cluster such that the "length" of unsatisfied bonds on its boundary almost cancels the length of satisfied ones. Similar ideas apply to threedimensional systems (Fradkin et al., 1978), although the problem is not a matching problem. In this case, the unitlength segments through the centers of frustrated plaquettes always form closed rings. One isolated negative bond produces four frustrated plaquettes. Fig. 7.19 shows some simple examples. If one associates a unit square perpendicular to each unsatisfied bond, it is easy to see that an open surface is formed whose boundary is the ring of segments through the centers of frustrated plaquettes. Thus minimizing the energy is equivalent to minimizing the extension of this spanning surface, very much like in the foam problem. Barahona (1982) has shown that finding a threedimensional groundstate is A/Phard, so it is unlikely that a polynomialtime algorithm will ever be found for this problem.
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225
Fig. 7.19 Energy surfaces in a frustrated threedimensional Ising system. Finding a minimum energy configuration is equivalent to finding a minimum spanning surface with fixed boundaries. Thick black lines represent negative interactions. Frustrated plaquettes are indicated with a dot, and it is always possible to draw a closed ring (black line) through centers of frustrated plaquettes. Energy surfaces (gray) are perpendicular to unsatisfied bonds, and span these rings, which are not planar in the general case.
(ii) Mapping to a cut problem In analogy to (7.2.7) for the interface in randombond ferromagnets and to (7.3.3) for the RFIM, the problem of finding the ground state of the spinglass Hamiltonian H defined in (7.4.1) is again equivalent to finding a minimal cut (S, S), see (7.2.6), in a network m
min~_ H'(o')  min(s,g )
Z
Jij  min(s,g)Ecut,
(7.4.3)
(i,j)E(S,S)
where H:  ( H + C ) / 2 with C  Z(ij) Jij a constant. However, now the capacities U ij  Jij of the underlying network are no longer nonnegative, so the minimal cut we seek cannot be identified with a minimumcut/maximum flow problem. Note that for the DAFF (7.3.5) it was possible to gauge away the sign of antiferromagnetic couplings Jij. This is not possible when positive and negative couplings are mixed randomly. The problem of minimizing Ecut is the same as the problem of maximizing  E c u t , which is the well known maximumcut problem in combinatorial optimization. This formulation is obviously more flexible than the matching formulation, but the general maximumcut problem is not solvable in polynomial time. If the field is zero and the graph is planar, minimumcut can be mapped into a "Chinese postman" problem (Barahona et al., 1982), which can be solved in polynomial time by a method due to Edmonds. For the case of periodic boundary conditions without field, Barahona (unpublished) has found a polynomialtime algorithm although it is reported (De Simone et al., 1996a) to be too slow to be useful.
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M.d. Alava et aL
But even in the nonpolynomial cases, it is still possible to find exact ground states in reasonable times (Barahona, 1982; Kawashima and Suzuki, 1992; De Simone et al., 1996a, b; Klotz and Kobe, 1994), and in fact it seems to be a nonpolynomial branchandcut algorithm (De Simone et al., 1996a, b) which gives efficient running times for the spinglass problem, even in two dimensions. In what follows we would like to sketch the idea of an efficient but nonpolynomial algorithm (De Simone et al., 1996a) for the maximumcut problem formulated in (7.4.3). Let us consider the vector space R A. For each cut [S, S] define X (s'~) E R A, the incidence vector of the cut, by X~es's) 
1 for each
edge e  (i, j) E (S, S) and X~es's)  0 otherwise. Thus there is a onetoone correspondence between cuts in G and their {0, 1}incidence vectors in R A. The cutpolytope Pc(G) of G is the convex hull of all incidence vectors of cuts in G" Pc(G)  conv{x (S'S) E R A IS c_ A}. Then the maxcut problem can be written as a linear program max {u~x Ix E Pc(G)} (7.4.4) since the vertices of Pc(G) are cuts of G and vice versa. Linear programs (see Section 5.1) usually consist of a linear cost function u r x that has to be maximized(or minimized) under the constraint of various inequalities defining a polytope in R n (i.e. the convex hull of finite subsets of R n) and can be solved for example by the simplex method, which proceeds from comer to comer of that polytope to find the maximum (see (Lawler, 1976" Chv~ital, 1983; Derigs, 1988)). The crucial problem in the present case is that it is A/'79hard to write down all inequalities that represent the cut polytope Pc(G). However, it turns out that partial systems are also useful, and this is the essential idea for an efficient algorithm to solve the general spinglass problem as well as the travelingsalesman problem and other "mixedinteger" problems (i.e. linear programs where some of the variables x are only allowed to take on some integer values, like 0 and 1 as in our case) (Lawler et al., 1985; Thienel, 1995). One chooses a system of linear inequalities L whose solution set P (L) contains Pc(G) and for which Pc(G)  convex hull {x E P(L)Ix integer}. In the present case these are 0 < x < 1, which is trivial, and the socalled cycle inequalities, which are based on the observation that all cycles in G have to intersect a cut an even number of times (have a look at the cut in Fig. 7.18 and choose as cycles for instance the paths around elementary plaquettes). The most remarkable feature of this set L of inequalities is that the separation problem for them can be solved in polynomial time. The separation problem for a set of inequalities L consists in either proving that a vector x satisfies all inequlaities of this class or to find an inequality that is violated by x. A linear program can be solved in polynomial time if and only if the separation problem is solvable in polynomial time (Gr/3tschel et al., 1988). For those that can be solved in polynomial time" the cuttingplane algorithm, starting from some small initial set of inequalities,
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iteratively generates new inequalities until the optimal solution for the full subset of inequalities is feasible. Note that one does not solve this linear program by the simplex method since the cycle inequalities are still too numerous for this to work efficiently. Due to insufficient knowledge of the inequalities that are necessary to describe P c ( G ) completely, one may end up with a nonintegral solution x*. In this case one branches on some fractional variable Xe (i.e. a variable with x e ~_ {0, 1}), creating two subproblems: one having Xe = 0 and the other having Xe = 1. Then one applies the cutting plane algorithm to both subproblems, hence the name branchandcut. Note that in principle this algorithm works for any dimension, boundary conditions, and in a finite field. However, there are realizations of it that run fast (e.g. in two dimensions) and others that run slow (e.g. in three dimensions) and there is ongoing research which attempts to improve on the latter. A detailed description of this rather complex algorithm can be found in Thienel (1995).
7. 4.3
Groundstate calculations in two dimensions
In their pioneering work, B ieche et al. (1980) studied the groundstate behavior of the ( + J ) spin glass as a function of the fraction x of antiferromagnetic bonds. From simulations of systems of 22 x 22 spins they deduced that ferromagnetic was destroyed at x*  0.145. This zerotemperature transition is detected by the appearance of fracture lines which span the system, i.e. paths along which the number of satisfied and dissatisfied bonds is equal, and which can thus be inverted without any cost in energy. The authors also looked at the fraction of spins in connected components, defined as sets of plaquettes which are matched together in any groundstate. Later investigations by Barahona et al. (1982) located the loss of ferromagnetism at a somewhat lower density of antiferromagnetic bonds, x* ~ 0.10, suggesting that in the regime 0.10 _< x _< 0.15 a random antiphase state exists which has zero magnetization but longrange order. This state is, according to the authors, characterized by the existence of magnetic walls (which are different from fracture lines) across which the magnetization changes sign. Thus the system is composed in this regime of "chunks" of opposite magnetization, so < M > 0 although the spinspin correlation does not go to zero with distance. At x = 0.15 a second transition occurs, this time due to the proliferation of fracture lines, and rigidity (longrange order) is lost since the system is now broken into finite pieces which can be flipped without energy cost. Their conclusions were supported by later work using zerotemperature transfermatrix methods (Ozeki, 1990). Freund and Grassberger (1989), using an approximate algorithm to find lowenergy states on systems up to size 210 • 210, located the ferromagnetic transition at x* = 0.105 but found no evidence of the random antiphase state. The
228
M . J . Alava e t al.
largest twodimensional systems studied to date using exact matching algorithms appear to be 1800 • 1800 square arrays (Palmer and Adler, 1999). Bendisch et al. (1994) examined the ground state magnetization as a function of the density p of negative bonds on square lattices, and concluded that 0.096 < x* < 0.108, but their finitesize scaling analysis is not the best one could think of. They did not analyze the morphology of the states found, so no conclusion can be reached regarding the existence of the random antiphase state. More recently Kawashima and Rieger (1997), compared previous analyses of the ground state of the twodimensional ( i J) spin glass, in addition to performing new simulations. They summarized the results in this area in the phase diagram given in Fig. 7.20. Kawashima and Rieger have found that the "spinglass phase" is absent and that there is only one value of Pc. They thus argue for a direct transition from the ferromagnetic state to a paramagnetic state, for both site and bondrandom spinglass models. Their analysis is based on the energy difference A E = E p  E a , where E p is the groundstate energy with periodic boundary conditions and Ea is the groundstate energy with antiperiodic boundary conditions. The scaling behavior (Bray and Moore, 1985a), [AE]av ~ L p ,
[ A E 2 ] a v ~ L 20
(7.4.5)
was assumed. In a ferromagnetic state p = 1 and 0 = 2, while in a paramagnetic state p < 0 and 0 < 0. However, in an ordered spinglass state we have p < 0, and 0 > 0. Although the conclusion of this analysis was the absence of a spinglass phase, the exponent they found, 0  0.056(6) for p < Pc, is small. Although, in our view, the numerical evidence that there is no finitetemperature spinglass transition in the twodimensional EdwardsAnderson model with a binary bond distribution is compelling, one should note that a different view has been advocated (Shirakura and Matsubara, 1995). A defectenergy calculation similar to the one described above has been presented (Matsubara et al., 1998) to support this view. It was shown that the probability distribution of IAEI does not shrink to a deltafunction centered at A E for L + ~ , instead it maintains a finite width. However, even if l i m L _ ~ [AEI = A E ~ > 0 a f i n i t e value of this limit indicates that the spinglass state will be unstable with respect to thermal fluctuations since arbitrarily large clusters will be flipped via activated processes with probability e x p (  A E ~ / T ) at temperature T. The correct conclusion is then that there is no finiteT spinglass transition in the twodimensional EdwardsAnderson model with binary couplings. The twodimensional Ising spin glass with a binary ( + J ) bond distribution is in a different universality class than the model with a continuous bond distribution. The degeneracies, which are typical for a discrete bond distribution, are absent for the continuous case for which the ground state is unique (up to a global spin flip). Even in the continuous case, the ground state is found using a
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P T
...................
SG(?)/
0 5
..,(2) (1) Pc Pc
"
1
P Fig. 7.20 Phase diagram of a twodimensional (+J) Ising model, with fraction p  1  x of ferromagnetic bonds. (From Kawashima and Rieger, 1997)
minimal weighted matching algorithm (with the modification that now not only the length of a path between two matched plaquettes counts for the weight, but also the strength of the bonds laying on this path). The latest estimate for the stiffness exponent of the twodimensional Isingspinglass model with a uniform bond distribution between 0 and 1 obtained via exact groundstate calculations (Rieger et al., 1996) is [AE2]av cx L 20
with
0 = 0.281 + 0.002,
(7.4.6)
which implies that in the infinite system arbitrarily large clusters can be flipped with vanishingly small excitation energy. Therefore the spinglass order is unstable with respect to thermal fluctuations and one does not have a spinglass transition at finite temperature. Nevertheless, the spinglass correlation length ~ (defining the length scale over which spatial correlations like [(Si Si+r)Z]av decay) will diverge at zero temperature as ~ ~ T 1/u, where v is the thermal exponent. A scaling theory (for a zerotemperature fixedpoint scenario as is given here) predicts that v = 1/101 which, using the most accurate MonteCarlo (Liang, 1992) and transfermatrix (Kawashima et al., 1992) calculations gives v = 2.0 4 0.2, is inconsistent with (7.4.6). This is certainly an important unsolved puzzle, which might be rooted in some conceptional problems concerning the use of periodic/antiperiodic boundary conditions to calculate largescale lowenergy excitations in a spin glass (these problems have first been discussed in the context of the XY spin glass
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M . J . Alava et al.
(Kosterlitz and Simkin, 1997) and the gauge glass (Kosterlitz and Akino, 1998)). Further work in this direction will be rewarding. Next we would like to focus our attention on the concept of chaos in spin glasses (Bray and Moore, 1987). This notion implies an extreme sensitivity of the spin glass state with respect to small parameter changes like temperature or field variations. There is a length scale ~.the socalled overlap lengthbeyond which the spin configurations within the same sample become completely decorrelated if compared for instance at two different temperatures
CAT [((riCri+r)T(Cricri+r)T+AT]av ~ e x p (  r / ~ . ( A T ) ) .
(7.4.7)
This should also hold for the ground states if one slightly varies the interaction strengths Jij in a random manner with amplitude 6. Let {or} be the ground state of a sample with couplings Jij and let {ort} be the ground state of a sample with couplings Jij 4 SKi j, where the Kij a r e random (with zero mean and variance one) and 3 is a small amplitude. Now define the overlap correlation function as !
!
C~(r) = [o'io'i_Fr (Tio'i+r]av ~ ?(r31/ff),
(7.4.8)
where the last relation indicates the scaling behavior we would expect (the overlap length being ~. ,~ 61/C) and ~" is the chaos exponent. Rieger et al. (1996) confirmed this scaling prediction with 1/~"  1.2 4 0.1 by exact groundstate calculations of the twodimensional Ising spin glass with a uniform coupling distribution, and the corresponding scaling plot for C~(r) is shown in Fig. 7.21. The twodimensional Ising spin glass in a homogeneous field (i.e. with an additional term h Z i Si in the Hamiltonian) is already a much harder problem that cannot be solved in polynomial time. Nevertheless efficient methods exist for this case (see Section 7.4.2) so that large systems can still be solved exactly in a reasonable computational time (Barahona, 1994; De Simone et al., 1996a, b; Rieger et al., 1996). A nonzero external field h induces a nonvanishing magnetization m = N 1 y~N=I SO in a system with ground state {sO}. The relation between magnetization and field strength is highly nontrivial in general and motivates the introduction of a new exponent 3h characterizing this relation in the infinite system (L + oo) for small fields (h Z
Iei[O(x)O(x+e~)A(x'x+e~)] 1 12.
(8.2.4)
X,O/
On the righthand side we have put the discrete version, the coordinates x now denoting discrete lattice points, ot  1, 2, 3 the space directions and A(x, x + e~) 
f x+e~ ds A
(8.2.5)
dX
are the phase shifts induced by the vector potential A. The latter can be split into an external (quenched part) A ext, induced by the applied magnetic field, and a fluctuating part a, that is responsible for screening effects. Changing the notation from coordinates x to lattice indices i for a simple cubic lattice one thus obtains from (8.2.4) the wellknown XYHamiltonian nxy
  J
L~ (i,j)
cos(0/ 
Oj  A i j  ~ .  l a i j )
1
+ ~L ~ [ V x a] 2
(8.2.6)
[]
where J is the interaction strength and )~ the screening length or penetration depth and the sum is over nearestneighbour sites. The last term describes the magnetic energy (which is f d3xB 2 in (8.2.1) with B  V x A the microscopic magnetic field), and is the sum over all plaquettes of the lattice, where the curl is given as the directed sum of the aij around one plaquette.
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In the limit of infinite screening length (X ~ oo) the fluctuating vector potential is irrelevant, which is a situation that is typically assumed in the context of Josephsonjunction arrays and granular superconductors, for which the Hamiltonian (8.2.6) with ~. + oo has originally been proposed (Ebner and Stroud, 1985; John and Lubensky, 1986) HX~.~ Y oo
  Z Jij (i,j)
cos(0/

Oj  A i j ) ,
(8.2.7)
Oi n o w representing the superconducting phase in a particular grain with index i of the two or threedimensional array and Jij (x 1/Rij describes the strength of the Josephson coupling between grain i and j, with Rij being the corresponding resistance. In the nonhomogeneous situation one has two sources of randomness, 1) the phase shifts Aij c a n be random due to the random location and distance of the superconducting grains, and 2) the couplings Jij a r e random due to the random resistance. Two different models arise: (i) The XY spin glass, in which Jij is random and Aij = 0 or constant. The lowercritical dimension for the appearance of an XY spinglass state is believed to be four (Banavar and Cieplak, 1982; Morris et al., 1986; Jain and Young, 1986). (ii) The gauge glass, in which Jij = J constant and Aij E [0, o'] is distributed randomly over a continuous range (0 < o < 2Jr) measures the strength of the disorder, here we discuss only the case of strong disorder, i.e. o"  27r. It is believed to be in a different universality class from the XY spin glass, since it does not have the global reflection symmetry Oi + Oi, and its lower critical dimension is believed to be three, leading to the presence of a vortexglass state in superconductors in three dimensions (Fisher, 1989). In the strongscreening limit the vortexglass state is destroyed as is demonstrated in Section 8.4.
8.2.5
Perturbing the flux lattice
When a homogeneous superconducting material is placed in a magnetic field of strength H > Hcj, a triangular vortex lattice, with lattice constant a = (2/~/~)1/2(~o/B)1/2 , is formed. In the presence of weak disorder, the flux lattice is distorted, which we describe by the d  1component displacement field ui(z). (In the continuum description we use R and u(R, z)). Note that when setting u(Ri, z) = u i (Z) one assumes that there are no dislocations present, otherwise such a relabeling is impossible to perform unambiguously. For weak disorder one expects u(Ri, z) to be slowly varying on the scale of the lattice and one can use a continuum elastic energy (Blatter et al., 1994; Giamarchi and Le Doussal, 1995), as a function of the continuous variable u(r), with r  (r, z):
1 Hel  ~ ~,
f.
ddq z (2n) d u ~ ( q ) ~ u ~ (  q ) '
(8.2.8)
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M.J. Alava et al.
where or,/3 = 1. . . . . d  1 label the lattice coordinates, BZ denotes the Brilloun zone and ~t~ is the elastic matrix. An appropriate (dispersionfree) approximation for the highTc superconductors that can be modeled by stacks of coupled planes and therefore described by layers of twodimensional triangular lattices would then contain the three elastic moduli: the compression modulus cll, the shear modulus C66 and the tilt modulus C44: nel  1/2 f d2Rdz((Cll C 6 6 ) ( Z o t O~U~)2 ~ C66 Y~,fl(O~Ufl) 2 F C44 Z~(OzU~)2), where or, fl run over the transverse coordinates of the flux line. Here, for simplicity, we confine ourselves to a fully isotropic situation in which
c f d d r[vu (r) ]2 Hel  ~
(8.2.9)
corresponding to O~,t~(q)  cq23a~ Next we include disorder via a Gaussian pinning potential e(u, z) acting on the flux lines. The total Hamiltonian then reads
H  Hel +
f
d clr ~pin(r) p (r),
(8.2.10)
where the fluxline density at a given point r  (R, z) is given by p(r)  Z
6(R  Ri  u ( R i , z))
(8.2.11)
i
Some care has to be taken when going to the continuum description since the discrete translational symmetry u > u + Ri of the density has to be preserved in any approximation. Since the coordinates Ri only serve as an internal label of the fluxlines it is more convenient to use a label that is a function of the actual position of the fluxlines. One can introduce the slowly varying field via the implicit equation 4~(R, z) = R  u(4~(R, z), z) that has a unique solution in the absence of dislocations. Using this relabeling field q~(r) the density (8.2.11) can be rewritten (Giamarchi and Le Doussal, 1995) as p(r)  P0 det[0~p/~] ~
e iKj(p(r),
(8.2.12)
J
where Kj are the vectors of the reciprocal lattice and P0 = 1/[R] 2 = B/Cbo is the fluxline density. When one retains only the smallest reciprocal lattice vector K1 and neglects the term VRU (since fluctuations in the density are slowly varying) one obtains the standard continuum representation for the disorder energy of the fluxline lattice
ndi s ,~
f
d3repin(r) 2 p ~ 1 7 6
u(r)])"
(8.2.13)
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For concreteness, let us consider the twodimensional case. In this case the elastic energy given in (8.2.8) can be made isotropic by a rescaling of the coordinates and the full Hamiltonian reads,
f
H2d ~" 21
d2 r {K (Vu) 2 + W (r, u) }
(8.2.14)
with Gaussian distributed periodic disorder with zero mean and correlation function (W(r, u)W(r', u')) = 2g cos(K](u  u'))32(r  r'). Note that this formulation is indeed equivalent to (8.2.13). The elastic constant is proportional to ~/C11/C66and g is the strength of the disorder. In the literature the Hamiltonian (8.2.14) is often represented as the vortexfree twodimensional XY model in a random symmetrybreaking field
'fd2
HXy  ~
r {K(Vu) 2 + 2~/~cos[u(r)  0(r)]}
(8.2.15)
where the random phases 0(r) are uniformly distributed at each point and uncorrelated at different points: (0(r)0(ff)) cx 32(r  if). In the large g limit this model reduces to the randomsurface model which is solvable by optimization methods. This is what we study in Section 8.5, where the threedimensional version of this model and the effect of dislocations on it are also considered. Concluding this section we remark that at low temperatures the twodimensional model (8.2.15) and the threedimensional version of it are expected to be in a glassy phase with quasilongrange order (the Bragg glass) (Giamarchi and Le Doussal, 1994, 1995). Many of the properties of this phase, including its stability or instability with respect to dislocations, can be investigated via ground state (T = 0) calculations.
8.3 Arrays of directed polymers 8.3.1
Mapping to minimumcost flow
In Section 8.2.3 we pointed out that in the lowfield limit of highTc superconductors a directedpolymer description is appropriate. In the literature usually only the properties of a single line, or at most two lines (Tang, 1994) are studied numerically, generally with the transfermatrix method, whose computational demands increases exponentially with the number of lines. Here we demonstrate how the complete Nline problem (with local interactions) can be solved numerically in polynomial time using optimization methods. Here we consider N directed polymers. Actually the model also applies to undirected polymers as well, just as Dijkstra's method solves for minimum path
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M . J . Alava e t al.
including the possibility of overhangs (see Section 6.3). The lattice version of this model is given by the Hamiltonian H(x) = Z
eij 9Xij,
(8.3.1)
(i,j) where
Z(i,j) is a sum over all bonds (i, j) joining sites i and j of a ddimensional
lattice, e.g. a rectangular (L d1 x 1) lattice, with periodic boundary conditions (b.c.) in d  1 directions and free b.c. in one direction. The bond energies eij >>_0 are quenched random variables that indicate how much energy it costs to put a segment of polymer on a specific bond (i, j). The polymer configuration x (xij > 0 ) , is modeled by associating it with an integer flow. Each bond is allowed an integer flow which is either Xij   1 when arc (i, j ) is occupied by the polymer and Xij = 0 otherwise. This is imposed in a flow network by setting the arc capacities to be U ij = l. This restriction also ensures that there is a hardcore repulsion between polymers, on the bonds of the lattice. Note that two polymers can visit the same site, though this is improbable due to the bond repulsion. The connectivity of the polymers is ensured by the fact that flow is conserved, that is, on each site of the lattice the incoming flow balances the outgoing flow, V . x = 0,
(8.3.2)
where V. denotes the lattice divergence. Obviously the polymers have to enter, and to leave, the system somewhere. To control the number of polymers in the sample, we attach all sites of one free boundary to an extra "ghost" site (via energetically neutral arcs, e = 0), which we call the source s, and the other side to another extra "ghost" site, the target (as in Fig. 3.2). Now if we want to place N polymers in the sample, we inject a flow of N units at the source s and extract a flow of N units at the sink. (7.X)s = +N
and
(7.x)t = N,
(8.3.3)
Note that we could consider more general situations in which we put flow in and take flow out of the system at any desired location in the lattice provided the flow entering the graph is equal to that leaving. With the above construction, the problem of N nonintersecting polymers in a random media is solved exactly by solving the minimumcostflow problem with: linear cost (8.3.1); with N sources and N targets at locations which can be chosen to suit the physical configuration and, with capacity constraint Uij = 1 to ensure nonintersection. An example for two polymers is presented in Fig. 3.2. As demonstrated in Section 3.3.3, the successiveshortestpath algorithm solves this problem in a time bounded by O(IAIN). Examples are presented in Fig. 8.1.
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243
a) 30 25 20 15
25 20 15
25 20 15
10
1
1 302
' ~ 1 0 ~ 00 ~ 5
10 15 .
.
.
30 25 20 15 10
b)
i
.
25 20
o
10 ~
1
0
]~ 
Y '~10 ~
25 20 15 10
o
1
0
l
o ~o~
~
J
r
~
o
Fig. 8.1 a) Optimal polymer configurations for twodimensional samples of linear size L = l  100. The number of polymers, N, is 1,2,4,8,16,32. b) Optimal polymer configurations for a series of threedimensional samples (linear size L = 32).
8.3.2
Polymer/flux line fluctuations in d = (1 + 1) and d = (2 + 1)
The onepolymer problem (DPRM) has been discussed in Section 6.3. Whereas two lines (N = 2) are still tractable (Tang, 1994), the transfermatrix method fails to work efficiently for an increasing number of lines since its complexity grows exponentially with N. Since it is the dense limit N  p L d1 with p of order one
244
M.J. Alava et aL
which is expected to contain new physics the minimumcostflow method, which solves this problem in polynomial time, is highly desirable. We now describe some calculations which test the effect of polymer density on the roughness of directed polymers. A glance at Figure 8.1 indicates that in (1 + 1)dimensions, nonintersecting polymers are confined by their inability to pass through each. In contrast, in (2 + 1)dimensions directed polymers are far less confined by their neighbours and hence their fluctuations may grow more rapidly with increasing system size. Recall that for a single directed polymer, the roughness w 2 = [tt2]av  [tt]2v ~ 12~, where ~"  2/3 in d = (1 + 1) (Huse and Henley, 1985; Kardar et al., 1986) and ~" ~ 0.625 in d  2 + 1 (Kim et al., 1991). Consider the situation in (1 + 1) dimensions, where N polymers or flux lines are present in a sample of length 1 in the direction of the polymers and L transverse to them. Fig. 8.1 shows a number of optimal line configurations with varying density obtained with the minimumcostflow algorithm. As long as the transverse fluctuations are smaller than the mean distance between the lines, i.e. l~ < L / N  1/p the line roughness grows simply like l~, as for a single line confined to a strip of width 1/p. However, for 1~ > 1/p the lines start to feel each other (via the hardcore repulsion), and the lines begin to compete collectively for the deepest minima. When one confines the transverse fluctuations by choosing free boundary conditions (at x=l and x = L ) the line roughness saturates for 1 ~ c~ (Knetter et al., 1999) (see Fig. 8.2). Wsat(L) ~ p1 In L,
(8.3.4)
where p1 is the average distance between two neighboring lines and plays the role of the microscopic length scale. This is equivalent to saying that the meansquare displacement [uZ]av ( o f the nth line from its average position [Un]av n 9p1) is proportional to In 2 L, or that for the displacementdisplacement correlation function (of line n and n + r) one has [(Un Un+r)Z]av (X In 2 r. We will encounter this superrough behavior again further below in the context of flux line arrays and twodimensional elastic periodic media in Section 8.5. From this we learn that a hardcore repulsion between individual polymers or flux lines is sufficient to put the Nline system defined above into the universality class of the XYmodel with random phase shifts (8.2.15) defined in Section 8.2.5. We remark that similar conclusions have been reached for a model of interacting flux lines with an elastic part in the Hamiltonian (Nattermann et al., 1991). The crossover from the singleline roughness w ~ 1f to the collective behavior (8.3.4) occurs at some characteristic longitudinal length ~tl (L, p), which can be extracted by scaling the data obtained according to 
w(1, L, p) ~.. //)sat" g ( I / ~ l l )
(8.3.5)
as shown in Fig. 8.2. For system sizes L < 256 it turns out (Knetter et al., 1999) that ~11(L, p) ~ c(p) 9L 2/3, and c(p) ~, 1/p2/3, a result that is not completely
2 8.0
Exact combinatorial algorithms
o
245
o L=128
.............. L=64 o o L=32 , L=16
6.0
4.0
/
2.0
a)
0.0
10
/
/
100
1000
10000
100000
1000
10000
100000
I
L=256 1.0
[] o
[] L = 1 2 8 o L=64
................. L=32 0.8
0.6
/
//
0.4
0.2
0.0
b)
10
100
Fig. 8.2 Scaling plot of the line roughness in two dimensions for constant density: a) p = 1/32 and b) p  1/16. Each data point is averaged over 500 independent samples, the saturation roughness Wsat has been extrapolated for l + cc and obeys (8.3.4), and the characteristic crossover length scale ~11(L, p) has been chosen in such a way to achieve the best data collapse for the collective region 1/~11 >> 1. (From Knetter et al., 1999).
understood regarding the fact that the XYmodel with random phase shifts, to which the present line system should be asymptotically equivalent, is isotropic in both space directions (implying ~ll cx L). Let us finally mention that in three dimensions the superrough twodimensional result (8.3.4) will no longer hold, and one expects the n o r m a l roughness / / ) s a t ( 3 d ) "~ ~/ln L, see Section 8.5.5 and Knetter et al. (1999). As an outlook for further applications we note that besides point disorder (the energies eij all independent), columnar disorder (eij identical along vertical or
246
M.J. Alava et aL
tilted columns) can be treated (Hwa and Nattermann, 1995; Krug and HalpinHealy, 1993; Nelson and Vinokur, 1993; Tang and Lyksutov, 1993; Arsenin et al., 1994). Moreover, if there is no disorder present (e.g. eij  const.) the Nline ground state is simply given by N straight vertical lines at arbitrary positions. We can lift this degeneracy and force the polymers to arrange themselves in a periodic pattern by a columnar modulation of the local energies eij. Adding a random perturbation to these energies leads to a competition between disorder and the periodic potential, which eventually, for strong enough perturbation, leads to the destruction of the periodic structure. So for instance the disorder induced melting of the triangular (Abrikosov) lattice can be studied. It is also possible to model a driven depinning transition of the polymers (fluxlines) by adding a force term to the local energies eij which increases linearly with the space coordinate in one direction, in order to model a transport current. A less trivial extension is the introduction of a softcore repulsion, which can be modeled by allowing a multiple occupancy of a bond (Xij = 0, 1, 2 . . . . ) but punish high polymer densities with an energy eij (Xij) increasing faster than linear with the number of flux units Xij o n the bond (i, j). Thus the Nline problem with soft repulsion consists in minimizing (x)  ~ eij (xij), (i,j)
(8.3.6)
under the constraints (8.3.2) and (8.3.3). The local energy functions eij c a n be chosen arbitrarily for each bond (i, j), however, they have to be convex as for instance eij(xij) = kij . x i j with n > 1 arbitrary. The energies eij have now to be replaced by the quantity eij(Xij + 1)  eij (xij), which is the energy needed to increase the flow Xij on arc (i, j) by one unit. Since it depends on the current flow x the convexity of eij is needed to ensure that the reduced costs fulfill the inequality c u > 0 after the flow modification. Whereas with hardcore repulsion it was only possible to put N  L d1 polymers into the system, the polymer density can now be arbitrarily high and an interplay between the repulsion and the disorder effects lead to a much richer phenomenology (Rieger, 1998b). This problem is also solved by the successiveshortestpath algorithm (see Section 3.3.3).
8.4 8.4.1
Gauge (vortex) glass The strongscreening limit
The model (8.2.7) is believed to describe various aspects of bulk superconductors correctly. When we set all Aij in (8.2.7) to zero, the model is just the XY ferromagnet, which is in the correct universality class to describe the transition to the Meissner phase neglecting screening, since it has the same value for the
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order parameter everywhere. If Aij a r e chosen to correspond to a uniform field in some direction, the model exhibits a transition to a vortexlattice state, which corresponds to an antiferromagnet in the magnetic analogue. As is well known from the classical XYmodel (Jos6 et al., 1977; Kleinert, 1989) the spinwave degrees of freedoms of the Hamiltonians (8.2.7) and (8.2.6) can be integrated out and one is left with an effective Hamiltonian for the topological defects, the vortices, which are the singularities of the phase field 0 interacting with one another like currents in the BiotSavat law from classical electrodynamics, i.e. like 1/r, where r is the distance (see also Li et al. (1996)). An additional integration over the fluctuating vector potential sets a cutoff for this longrange interaction beyond which the interaction decays exponentially, and can thus be neglected. To be specific one obtains, in three dimensions, the vortex Hamiltonian (Bokil and Young, 1995) H3d_
1 Z(Ji 2 (i,j)
 b i ) G ( i  j ) ( J j  bj),
(8.4.1)
which is defined on the dual lattice, which again is a simple cubic lattice. The Ji, represent the vortex density of the phase field 0 on bond i, and there are threecomponent integer variables running from  o c to oe living on the links of the dual lattice and satisfying the divergence constraint ( V . J)i = 0 on every site i. The bi are magnetic fields which are constructed from the quenched vector potentials Aij by a lattice curl, i.e. one obtains bi as 1/(27r) times the directed sum of the vector potentials on the plaquette surrounding the link on the dual lattice which bi lives on. By definition, the magnetic fields satisfy the divergencefree condition (V 9b)i = 0 on every site, since they stem from a lattice curl. The vortex interaction is given by the lattice Green's function G ( i , j ) = J (2at)2 ~
..... N
1  exp(ik 9(ri  rj))
(8.4.2)
2 }~nd__iil COS'~n)) 7 7,.O2,
which behaves asymptotically as G ( r ) ~ r 1 exp(r/~.) in three dimensions, i.e. it is a ( I / r ) interaction screened for distances r > ~., as described above. In two dimensions the singularities of the phase field (0) are points and the corresponding vortex Hamiltonian is actually a twodimensional Coulomb gas (Fisher et al., 1991) H2 d _~ _ l
Z(ni
_ bi)G2d(i _ j ) ( n j  b j ) ,
(8.4.3)
2 (i,j)
where now ni are integers (the vortex strengths), subject to the charge neutrality c o n s t r a i n t ~"~i rti   0 and G 2d (i  j) is the given by (8.4.2) with d = 2 describing
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a logarithmic interaction G(r) ~ In(r) for distances r < ~.. The real numbers bi are given by 1/2zr times the directed sum of the quenched vector potentials on the links of the original lattice which surround site i of the dual lattice. Most of the theoretical work so far has concentrated on establishing numerically the lower critical dimension of the gaugeglass model, both with and without screening of the interactions between vortices. Without screening, there is no finite temperature transition to a vortexglass phase in two dimensions (Fisher et al., 1991; Gingras, 1992; Bokil and Young, 1995; Kosterlitz and Simkin, 1997), whereas in three dimensions there is evidence for a finite Tc, as has been found by domainwall renormalizationgroup analyses (DWRG) (Gingras, 1992; Bokil and Young, 1995; Reger et al., 1991; Kosterlitz and Simkin, 1997; Kosterlitz and Akino, 1998; Maucourt and Grempel, 1998) and finite temperature MonteCarlo simulations (Huse and Seung, 1990; Wengel and Young, 1997), though due to limited system sizes and insufficient statistics the earlier DWRG studies (Gingras, 1992; Bokil and Young, 1995; Reger et al., 1991) could not fully rule out the possibility that the lower critical dimension is exactly d = 3. Sufficiently close to the critical point (i.e. at the superconductor to normalstate phase transition), screening effects become important, since the correlation length ~ diverges more strongly than the screening length ~. and the two length scales eventually become comparable (Bokil and Young, 1995). The effect of screening was investigated (Bokil and Young, 1995) by a DWRG study and more recently (Wengel and Young, 1996) by means of a finitetemperature MonteCarlo simulation, and the results indicate that screening is a relevant perturbation, destroying the finite temperature transition in three dimensions, though the DWRG analysis could only be performed for rather small system sizes (L < 4).
8.4.2
Mapping to minimumcost flow
For any nonvanishing value of the screening length ~. (in particular for the nonscreened (;~ ~ cx~) case (8.2.7) the vortex Hamiltonians are nonlocal in the vortex variables Ji or ni. The twodimensional case is then equivalent to the Coulombglass problem, for which it is NPhard to find the ground state, and the threedimensional case is obviously still harder. These problems can in principle be solved with modem integerlinearprogramming techniques (branch and bound). Since the underlying graph is complete due to the longrange interactions, the system sizes that have been feasible up to now are very modest. However, one can argue (with a coarsegraining picture in mind) that for all finite values of the screening length ~. the vortex Hamiltonians fall into the same universality class, in particular the same as the the strongscreening limit, k  0. In this case the Green's function G ( i  j ) reduces to G(0) = 0 f o r / = j and G(i, j) = J(2rr,k0) 2 for i ~ j with exponentially small corrections (Wengel and Young, 1996). Thus
2 Exact combinatorial algorithms
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if we subtract J (2rr ~.0) 2 f r o m the interaction and measure the energy in units of J(2zrZ0) 2, one obtains the simpler Hamiltonian t4z__>O __ _1 Z ( j i ~v 2
 bi) 2 .
(8.4.4)
i
We remark that Hv is still highly nontrivial due to the divergence condition ( V . J)i : 0. Finding the ground state of the Hamiltonian in (8.4.4) subject to the constraint (V 9J)i = 0 is a minimum(convex)costflow problem and can be restated as (8.4.5) Minimize z(J)  Z Ci (Ji) i
subject to the constraint (V 9J)i  0, where the cost functions ci(Ji)  (Ji bi)2/2 have been defined. Applying the successiveshortestpath algorithm presented in Section 3.3.3, the determination of exact ground states can be performed in polynomial time.
8.4.3
Domainwall energy and chaos
As in spin glasses (see Section 7.4) the issue of the existence of a finitetemperature phase transition can be scrutinized via groundstate calculations of the Hamiltonians (8.4.1) and (8.4.3) by studying the scaling behavior of the largescale lowenergy excitations, typically done via a DWRG analysis. For the latter one usually calculates the groundstate energy once with periodic and once with antiperiodic boundary conditions in the original phase variables 0, the energy difference being the defect energy A E (L) measuring the energy cost of a domain wall of length L (see Kosterlitz and Simkin (1997) for a more refined recipe). To this end one has to carefully treat the boundary conditions of the vortex Hamiltonian (8.4.1) and (8.4.3) leading to an additional term that is nonlocal in the vortex variables Ji or ni. Note that the vortex Hamiltonian in (8.4.4) with periodic b.c. and without the additional boundary term corresponds to fluctuating boundary conditions in the gaugeglass model (Olsson, 1995; Gupta et al., 1998). Again such a nonlocal term makes the determination of exact ground states computationally hard, in particular it is not a pure minimumcostflow problem any more. To overcome this difficulty a different procedure (see below) to induce lowenergy excitations in this model has been devised (Kisker and Rieger, 1998). This procedure also avoids some conceptual ambiguities that are present in the DWRG using periodic/antiperiodic boundary conditions for the phase variables (Kosterlitz and Simkin, 1997; Kosterlitz and Akino, 1998). In the model under consideration a lowenergy excitation of length scale L is certainly a global vortex loop encircling the threedimensional torus (i.e. the
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L x L • L lattice with periodic b.c.) once (or several times) with minimum energy cost. For the pure case the global minimumenergy loop is simply a straight line that costs energy A E (L) = J L, which is exactly what one expects for a domain wall of length L in a threedimensional XYmodel, and which is also obtained from the energy difference between ground states with periodic and antiperiodic b.c. To induce a global vortex loop, without manipulating the boundary conditions, we instead manipulate the costs for flow in one particular space direction. Suppose we have found an exact ground state, which also specifies the cost required to increase the flow variables jx,y,z with respect to j0 by one unit, e.g." A c x   c i ( J Ox Jr1) ci(J? x)  j?x b x + 1/2. If we smoothly decrease the variables Ac x and apply our mincostflow algorithm to this modified problem, at some point a configuration J] that is the original ground state plus a global loop in the xdirection will appear as the new optimal flow configuration for the modified problem. This extra loop, which can be easily identified by comparing the new optimum with the original ground state, is the low energy excitation we are looking for. Its energy A E ( L ) is simply the difference H ( J 1)  H(J~ found by evaluating the vortex Hamiltonian (8.4.4). Note that energy is always positive, since it is definitely an excitation (in contrast to the usual DWRG procedure where the b.c. is modified). Four remarks are in order: (1) small, simply connected loops are not generated by this procedure, since all that can be gained in energy is lost again on the return. (2) In the pure case this procedure would not work, since at some point spontaneously all links in the zdirection would increase their flow value by one. It is only for the disordered case with a continuous distribution for the random variables bi that a unique loop can be expected. (3) Sometimes (in about 5% of the samples) the global flux changes discontinuously by more than one unit, however, we still define these to be elementary excitations of length scale L. (4) In the presence of a homogeneous external field one has to discriminate between different excitation loops. Those parallel and those perpendicular to the external field need not have the same energy (however, it turns out that the disorder averaged defect energy is identical in all directions, see below). Schematically the numerical procedure is the following: (1) Calculate the exact groundstate configuration {j0} of the vortex Hamiltonian (8.4.4). (2) Determine the resulting global flux along, say, the xaxis fx = ~ Z i
j?x.
(3) Find the cost for increasing the flow corresponding to introducing a global vortex loop in the xdirection Ac x  ci(J~ + 1)  ci(J ~  b x + 1/2.
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(4) Reduce the Ac x until the optimal flow configuration {j1 } for this minimumcostflow problem has the global flux ( f x + 1), corresponding to the socalled e l e m e n t a r y low energy excitation on the length scale L. (5) Finally, the defect energy is AE  H({j1})  H({J~ Kisker and Rieger (1998) have computed (using the procedure above) the disorderaveraged domainwall or defect energy A E as a function of the system size L, with L up to L  50. It was found that AE ~ L ~
with
00.95(3).
(8.4.6)
In this way one reestablishes that Tc  0 for the gaugeglass model in the strongscreening limit, as has been found by Bokil and Young (1995) and Wengel and Young (1996). From the stiffness exponent 0 the thermal exponent v, which describes the divergence of the correlation length, can be calculated. For Tc  O, the correlation length behaves as ~ ~ T v. By equating the thermal energy with the energy of a lowlying excitation on the length scale of the correlation length, it follows that 1
v  ~.
(8.4.7)
101
From this relation one obtains v  1.05(3), which agrees well with a result from a finitetemperature MonteCarlo simulation for the same model (Wengel and Young, 1996), where system sizes L _< 12 have been studied and a zerotemperature phase transition with v  1.05(1) has been found. Another important feature of a glassy systems is their sensitivity with respect to changes of various parameters ("chaos"). In Kisker and Rieger (1998) the change in the groundstate configuration when the random vector potentials Aij are perturbed by a small amount has been studied. To be specific, one defines ! new vector potentials by Aij  Aij + ~ij, where 6ij is randomly drawn from the interval [  8 , 6] and then one calculates the ground state for both realizations of the disorder {Aij } and {A~j}. The distance D between the resulting groundstate configurations {J} and {J'} by D ( 8 )  Y ~ i ( J i Rieger, 1998) that DL(8)  d(L61/()
j,i)2. It turns out (Kisker and
with (  3.9(2),
(8.4.8)
where d ( x ) is a rapidly increasing function of x. This implies that groundstate configurations are decorrelated on length scales L > L*, where the overlap length behaves like L* ~ 8 1/~ as a function of the perturbation strength 8. It is straightforward to consider for model (8.4.4) a homogeneous external field B ext via bi
1 7[V x Arand]i + Bext AT/"
(8.4.9)
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M.J. Alava et al.
and specifying the boundary conditions to be periodic in all space directions (corresponding to fluctuating boundary conditions in the phase variables of the original gauge glass Hamiltonian) and choosing B ext  B ez. This models a situation in which the external field points in the zdirection. It turns out (Pfeiffer and Rieger, 1999a) that the disorder averaged domain wall energy A E is independent of the value for B (for a single sample it, however, depends on B), which means that also the stiffness exponent 0 is independent of B. Moreover, one observes that the ground state rearranges chaotically with varying B (which is the manifestation of field chaos) and the overlap length L* diverges like L* ",~ B 1/r with the same chaos exponent ~" as in (8.4.8) for the disorder chaos. Finally let us remark that it would be highly desirable to extend such investigations to the case of a nonvanishing screening length ~.. However, this is not feasible with existing mappings and algorithms.
8.5 8.5.1
Disordered elastic media Elastic glasses and the randomsurface problem
As demonstrated in Section 8.2.5, a simple model to describe the effect of disorder on the periodic flux lattice is the randomphase sineGordon model (also called the elasticglass model). This, and the models derived from it are generally refered to as disordered elastic media. The lattice version of the model (8.2.15) is given by H
~_,(Ui (i,j)
 U j ) 2  ~. ~
COS(2Yr(Ui  Oi))
(8.5.1)
i
where i denote the individual sites of a square lattice, (i, j) denote arcs between neighbor sites, Oi are independent random variables each distributed uniformly over the interval [0, 1], and ~. is the coupling strength to the disorder. In the limit of infinite coupling strength ~ ~ cx~ the cosineterm in the Hamiltonian (8.5.1) forces the displacements to be ui = Oi + ni, where ni are arbitrary integers. Thus, in this limit the above model maps onto a solidonsolid (SOS) model on a disordered substrate: H  ~ _ ~ ( n i } Oi  n j  Oj) 2,
(8.5.2)
(i,j)
where hi  ni + O i denotes the total surface height at lattice site i, ni is the integer height above the random surface which is described by heights Oi (see Fig. 8.3). The models (8.2.15), (8.5.1) and (8.5.2) have been investigated intensively. The model (8.5.2) possesses a phase transition at a critical temperature Tc to a low temperature, glassy phase, in which the displacementdisplacement correlation
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Fig. 8.3 The SOS model on a disordered substrate. The substrate heights are denoted by di E [0, 1] (d i = 0 i is the substrate roughness), the number of particle on site i by ni E Z, which means that they could also be negative, and the total height on site i by h i = d i + n i .
function G ( r ) = [((u(0)  u(r))2)lav
(8.5.3)
shows an anomalous ("superrough") behavior. Two different predictions have been made for T < Tc, one proposing a loglinear behavior G ( r ) = B ( T ) ln(r), with a coefficient B ( T ) that approaches a nonvanishing constant for T ~ 0 (in contrast to thermal roughness without disorder, which would simply vanish at T = 0) and another one proposing a stronger, logsquare increase: G ( r ) = C ( T ) lnZ(r). In the language of superconductivity the former prediction leads to an algebraic decay of the order parameter correlation function [(exp{i (u(0) u(r))})]av ~ r  B ( T ) / 2 , however with an enhanced exponent, whereas the latter leads to a faster than algebraic decay. Whereas the existence of this transition is established now, the qualitative and quantitative features of the glassy lowtemperature phase are still debated. Predictions of earlier renormalizationgroup (RG) calculations (Cardy and Ostlund, 1982; Toner and Di Vincenzo, 1990; Tsai and Shapir, 1992; Hwa and Fisher, 1994b) turned out to be incompatible with results of extensive numerical simulations (Batrouni and Hwa, 1994; Cule and Shapir, 1995; Rieger, 1995b). The subsequent discovery of the relevance of replica symmetry breaking (RSB) effects in variational (Bouchaud et al., 1991; Korshunov, 1993; Giamarchi and Le Doussal, 1994) and RG (Kierfeld, 1995; Le Doussal and Giamarchi, 1995) calculations lead to a variety of new results, which are again in disagreement with the most recent numerical studies (Lancaster and RuizLorenzo, 1995; Marinari et al., 1995). All the works, analytical as well as numerical, cited so far are confined to a region around the critical temperature, though it is generally believed that the behavior at low temperatures is typical. This is the reason why an investigation of the ground states is most useful.
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8.5.2
M.J. Alava et al.
M a p p i n g to m i n i m u m  c o s t f l o w
To calculate the ground states of the SOS model on a disordered substrate with general interaction function f (x) we can map it onto a minimumcostflow model (Blasum et al., 1996; Rieger and Blasum, 1997). We define a network G by the set of nodes N being the sites of the dual lattice of our original problem and the set of directed arcs A connecting nearest neighbor sites (in the dual lattice) (i, j ) and (j, i). If we have a set of height variables ni we define a flow x in the following way: suppose two neighboring sites i and j have a positive height difference ni  n j > 0. Then we assign the flow value Xij = ni  n j to the directed arc (i, j) in the dual lattice, for which the site i with the larger height value is on the fighthand side, and assign zero to the opposite arc (j, i), i.e. Xji = O. And also x i j = 0 whenever site i and j are of the same height. See Fig. 8.4 for a visualization of this scheme. Obviously then we have: u E N "
~ Xij ~" ~ Xji. {j I (i,j)EA} {j I (j,i)EA}
(8.5.4)
On the other hand, for an arbitrary set of values for xij the constraint (8.5.4) has to be fulfilled in order to be a flow, i.e. in order to allow a reconstruction of height variables out from the height differences. This observation becomes immediately clear by looking at Fig. 8.4. We can rewrite the energy function as H (x)  Z cij (Xij), (i,j)
with
cij (x) = f (x  dij),
(8.5.5)
with dij  di  dj and f ( y ) = y2 (actually f ( y ) can be any convex function, for instance f ( y ) = [y in, with integer n, and it does not even need to be identical for all bonds). Thus our task is to minimize H(x) under the constraint (8.5.4), which is (see Section 3.3.3) a m i n i m u m  ( c o n v e x )  c o s t  f l o w problem with the massbalance constraints (8.5.4) and convex cost functions Cij (Xij). The most straightforward way to solve this problem is to start with all height variables set to zero (i.e. x = 0) and then to look for regions (or clusters) that can be increased collectively by one unit with a gain in energy. This is essentially what the negativecyclecanceling algorithm discussed in Section 3.3.2 does: the negative cycles in the dual lattice surround the regions in which the height variables should be increased or decreased by one. However, the successiveshortestpath algorithm is more efficient and solves this problem in polynomial time (see Section 3.3.3). This algorithm starts with an optimal solution for H (x), which is given by Xij = Jr 1 for dij > 1/2, xij "    1 for dij <  1/2 and Xij = 0 f o r dij E [  1 / 2 , + 1/2]. Since this set of flow variables violates the massbalance constraints (8.5.4) (in general there is some imbalance at the nodes) the algorithm iteratively removes the excess/deficit at the nodes by augmenting flow.
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Fig. 8.4 The flow representation of a surface (here a "mountain" of height n i 3). The broken lines represent the original lattice, the open dots are the nodes of the dual lattice. The arrows indicate a flow on the dual lattice, that results from the height differences of the variables n i on the original lattice. Thin arrows indicate a height difference of xij  1, medium xij 2 and thick xij 3. According to our convention the larger height values are always on the right of an arrow. Observe that on each node the massbalance constraint (8.5.4) is fulfilled. =
a r r o w s

a r r o w s

We would like to mention that this mapping of the original SOS model (8.5.2) onto the flow problem works only for a planar graph (i.e. free or fixed boundary conditions), otherwise it is not always possible to reconstruct the height variables ni from the height differences Xij. AS a counterexample consider flow in a toroidal topology (i.e. periodic boundary conditions) where the flow is zero except on a circle looping the torus, where it is one. Although this flow fulfills the mass balance constraints (which are local) it is, for the height variables, globally inadmissible. To the right of this circle the heights should be one unit larger than on left, but left and right become interchanged by looping the torus in the perpendicular direction, which causes a contradiction.
8.5.3
A m a p p i n g to m a x i m u m f l o w
The randomsurface problem maps to the minimumcost flow problem for general convexcost functions f ( x ) . In the special case f ( x ) = Ix I, there is a simple mapping to the maximumflow problem (Zeng et al., 1996). This mapping is appealing physically as well as computationally, as we now discuss. The Hamiltonian is, H  y ~ Ihi  h j l (i,j)
(8.5.6)
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with hi : ni q di, di E [0, 1]. The groundstate configuration is degenerate with modulation one lattice unit in the vertical(height) direction. A different way to represent the problem is to define the Hamiltonian on a fiat substrate but with bond energies in the vertical direction alternating between di and 1  di. That is, the disorder is "periodic" in the vertical direction. Note that the exact mapping is actually to a "brickwork" lattice which has these altemating bond energies. However, the universality class is expected to be the same for a cubic lattice with couplings having modulation two. It is obviously possible to define a bond energy having modulation of an arbitrary wavelength ~, in which case = 2 is the randomsurface problem, while ~. = oo is the randombond interface problem discussed in Section 7.2. This construction can be used in any lattice orientation. The randomsurface problem is equivalent to finding the groundstate configuration of an interface in the presence of this periodic potential in the height direction (Zeng et al., 1996). It can be treated with the maximumflow algorithm in the same way as for the randommanifold problem (Section 7.2). Note that the problem is now threedimensional, though it may be implemented in a "slab" which must be wide enough to avoid the fluctuating interface.
8.5.4
A special case solvable by matching
We now introduce a restricted SOS model which is believed to exhibit some of the important behaviors of the randomsurface model. The starting point (Zeng et al., 1996) is the triangular Ising solidonsolid (TISOS) model (Nienhuis et al., 1984) HTISOS Z J i j ( l n i  n j l  1), (8.5.7)
(i,j) where the sites i are on a triangular lattice, the height variable ni can take on only integer values, the summation is over nearest neighbors, and the coupling c o n s t a n t s Jij are chosen uniformly (and independently for each bond) in the interval [0, 1]. In contrast to the model (8.5.6), further constraints are imposed on the height variables: (1) the height difference Ini  n j[ for every bond must be either 1 or 2; and (2) the total height increment (clockwise or counterclockwise) along any elementary triangle is zero (see Fig. 8.5). One can easily check that the SOS surface so defined describes an interface along the {111 }direction of a simplecubic lattice. The TISOS model can now be related to matching or "dimercovering" on a hexagonal lattice. One identifies each bond which has Ini  njl  2 on the triangular lattice with a "dimer" on the bond of the dual lattice. The dual lattice is hexagonal (see Fig. 8.5) and any allowed surface configuration on the triangular lattice maps to a complete dimer covering of the hexagonal lattice. The constraint that the sum of heights around a loop is zero, ensures that there is one bond with
2
Exact combinatorial algorithms
I
o
0
257
I
i
i .......) 0
o f
L3 Fig. 8.5 Various representations of the TISOS model. The height variables defined on each vertex of the triangular lattice (dashed lines) are shown in the figure. The equivalent dimer coveting is indicated by the thick bonds on the dual hexagonal lattice. The corresponding two polymers are displayed as contiguous triple lines. The linear size of the system is denoted by L(= 3). Periodic boundary conditions are imposed (twisted from top to bottom). (From Zeng et al., 1996). Ini  njl  2 in each triangle. The energy of a perfect matching is simply given by E  ~ tOe (8.5.8) eeM
where M is a perfect matching and tOe E [0, l] is a random weight for the edge e. One can verify (Zeng et al., 1996) that the two different representations (interface along {111 }direction dimers) are energetically equivalent, with a transformation of the random bonds that maintains its uniform and independent distribution. To solve the minimumcost perfectmatching problem on the hexagonal lattice, we divide the lattice into its two sublattices U and V. Any edge (i, j ) in the arcset A of the graph has one end site in U and other end site in V. A source b(i) = I is connected to each of the sites in one sublattice, i 9 U, while a sink of size b ( j )   1 is connected to each of the sites in the other sublattice, j e V. The task is to find the minimumcost flow through a hexagonal lattice with costs Cij  tOgj , given the set of sources and sinks stated above, and with the constraint that each arc has capacity one, Uij l. This problem is efficiently 

258
M . J . Alava et al. 5.0
i
i
,
9 D i s o r d e r e d substr at 9T I S O S
r.~ 4.o
~'
3.0
r.~ 2.o
~
1.0
0.0
0.0
,
1
1.0
,
i
2.0
log L Fig. 8.6 Scaling of roughness in the randomsurface problem. (From Zeng et al., 1996). solved using the successiveshortestpath algorithm. The randomsurface problem on large sample sizes (up to L  420 with high statistics) have been studied using this matching method (Zeng et al., 1996).
8.5.5
The superrough phase and loop statistics
All of the optimization methods described in Sections 8.38.5 have been used to study the ground state of the the randomsurface problem. In (1 + 1), the randomsurface problem is trivial, the surface is an uncorrelated random walk and hence its mean roughness behaves as w ".~ L ]/2. In (2 + 1) dimensions, the functional renormalization group predicted that the roughness should scale as w 2 ~ (lnL) 2, instead of w 2 ~ In L as applies to thermally rough surfaces. High precision data supporting the functionalrenormalizationgroup prediction are presented in Fig. 8.6. A brief summary of the other results obtained recently is as follows: (Blasum et al., 1996; Zeng et al., 1996; Rieger and Blasum, 1997): 9 The heightheight correlation function G(r) as defined in (8.5.3) diverges like G(r) c~ log2(r) with the distance r for all nonlinearities of the local cost functions f (x)  I x l n. 9 XL

L4
Y~(i,j)[(hi

hj)2]av can be fitted to Xc = a + b log(L) +
c log2(L) (see Fig. 8.6), again indicating a log 2 dependence of the heightheight correlation function. Moreover, the coefficients a, b and c depend
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on the power n in f (x)  Ix In: c increases systematically with increasing g/.
9 By considering a boundary induced step in the groundstate configuration one sees that the step energy increases logarithmically with the system size: AE = a + b In L with b = 0.56(2). This is analogous to a domainwall or defect energy, where the defect is a dislocation line between two point defects at fixed position at a distance L (see Section 8.5.6 for a more detailed study of dislocations, in particular those in optimized positions). Furthermore the step is fractal, it's length increases like/~step (X Ldl with the fractal dimension d f ~ 1.35(2). 9 Upon a small, random variation of the substrate heights di of amplitude the groundstate configuration decorrelates beyond a length scale L* 31/C with ~" ~ 1. This illustrates the chaotic nature of the glassy phase in this model, in analogy to spin glasses. Recently (McNamara et al., 1999) the threedimensional version of the model (8.5.6) has also been investigated, which is a lattice realization of the threedimensional elastic medium defined by the Hamiltonian (c.f. Section 8.2.5) H = Hel + Hdi s as given by (8.2.9) and (8.2.13). It is found that 9 The Fouriertransformed S(k) of the displacementdisplacement correlation function G(r) as defined in (8.5.3) behaves like S(k) ~ Ak 3 with a universal prefactor A. This implies that the average roughness as defined for the twodimensional case, XL diverges logarithmically with system size. 9 By considering a twisted boundary condition analogous to the step inducing boundary conditions in two dimensions one finds that the excitation or domainwall energy increases algebraically with the system size: AE L ~ with 0 ~ 1. Furthermore, as in two dimensions, the domainwalls are fractal, their area increases like Swall cx Ldl with the fractal dimension d f ~ 2.60. 9 Upon a small, random variation of the disorder configuration of amplitude the groundstate configuration decorrelates beyond a length scale L* 6 1/c with ~"  0.39(2). The scaling relation ~"  d f / 2  O, as proposed originally in the context of spin glasses (Bray and Moore, 1987) is fulfilled. The results on the displacementdisplacement correlations are in agreement with renormalization group and variational calculations (Giamarchi and Le Doussal, 1995) and the results on the domainwall energy are consistent with the prediction of the marginal stability of the system with respect to the introduction of dislocations (Fisher, 1997).
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Fig. 8.7 {111 } interface of a simple cubic lattice and its two loop representations. An example of a groundstate interface confined between two flat {111} layers is shown. The level set at mean height consists of closed contour loops due to periodic boundary conditions. Also shown is the fullypackedloop (FPL) representation of the same interface (see text for details of construction). (From Zeng et al., 1998)
Another interesting representation of an interface in the {111 }direction of an simple cubic lattice, is in terms of loops. Given the exact shape of the {111 }interface its topography is completely characterized by a contour plot with the level spacing equal to a single step of the discrete height variable. The contour plot consists of contour loops which live along the bonds of the hexagonal lattice s The contours are closed due to periodic boundary conditions which we impose in both lateral directions. For example, in Fig. 8.7, we show all the contour loops (at the mean surface height) for the {111 } interface. The union of all the contour loops for different realizations of disorder is the contourloop ensemble. As well as this natural contourloop characterization, there exists yet another interesting loop representation of the {111}interface, the fully packed loops (FPLs). These loops owe their existence to the onetoone mapping between a {111 }interface and a complete dimer coveting of the hexagonal lattice /2. By removing the bonds of E that coincide with the dimers, we are left with a configuration of fully packed loops, as shown in Fig. 8.7, where every site of E belongs to one and only one loop. A physical realization of FPLs is magnetic
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domain walls in the ground state of the Ising antiferromagnet on the triangular lattice (Nienhuis et al., 1984). FPL models of general loop fugacity in the absence of disorder have been studied recently (Batchelor et al., 1994; Blrte and Nienhuis, 1994; Kondev et al., 1996) and were shown to be critical for values of the loop fugacity that does not exceed two. The interfaceFPL mapping thus allows one to consider the effect of quenched disorder on the critical FPL model on the honeycomb lattice with fugacity equal to one (Zeng et al., 1998).
8.5.6
Dislocation effects
Various kinds of topological defects can be present in a line lattice: Interstitials and vacancies, edge and screw dislocations (Nabarro and Quintanilha, 1980). The proliferation (via thermal or disorder fluctuations) of such dislocations can induce a topological phase transition that destroys the longrange or quasilongrange order, i.e. the fluxline lattice or the Braggglass phase, respectively. Recently the effect of dislocations on the glassy phase of the randomsurface model has been considered (Gingras and Huse, 1996; Middleton, 1998; Zeng et al., 1999). Within the elasticglass model, dislocations can be included if we treat the phase field q~(x) or the height field h (x) as a multivalued function which may jump by 42zr or 41, respectively, at surfaces which are bounded by dislocations lines. Including these effects, it has been argued (Gingras and Huse, 1996) that for weak enough disorder the system is stable with respect to the formation of topological defects. However, for strong disorder the vortex lines were predicted to proliferate and thus to destroy the (quasi)longrange order. An example of a dislocation, a pair of point defects, in the random surface model (8.5.2) is shown in Fig. 8.8. Here the energy in terms of the flow variables X i j " n i   n j is H   Z ( i , j ) ( x i j  dij) 2, where d i j   d i  dj. With the given configuration of substrate heights the absolute minimumenergy configuration of flux variables Xij a r e xij = 0 in every bond of the dual lattice, except for the thick straight line between the two dots, where Xij   1. This would actually be the starting configuration of the successshortestpathalgorithm as discussed in Section 3.3.3. This is not a feasible solution for the surface problem since the flow x is not divergence free, which implies that one cannot find a displacement or height field n i that fulfills X i j = n i   n j for all pairs (i j). This flow configuration corresponds to a flat surface with one dislocation line between the two point defects shown in the figure. Actually, in its search for a feasible solution the successiveshortestpathalgorithm will remove this dislocation by sending flow from the fight excess node to the left deficit node. Then the optimal surface (without dislocation) is flat, ni = 0. This already demonstrates that the ground state of the random surface (without dislocations) is unstable with respect to the introduction of dislocations. If the latter are allowed the energy is simply the
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Fig. 8.8 Sketch of a dislocation pair for the randomsurface model. The substrate heights di are indicated in grey scale (white means di = 0). The optimal surface would be flat (Vi n i = 0), however, the dislocations as indicated would (if allowed) decrease the total energy (see text).
 dij) 2 without the massbalance constraint global optimum of H  Y~(i,j)(xij for the variable xij. It is easy to see that half of all nodes have excess or deficit in the global optimum, which means that the global optimum is full of dislocations. Thus if dislocations are allowed, without penalty, Braggglass order is certainly destroyed.
One can ask more detailed questions about a single dislocation pair in the random surface model. For instance one can fix the position of this pair by defining a graph with one deficit and one excess node in a distance L, or one can let them choose an optimal position in the distance (Pfeiffer and Rieger, 1999b; Zeng et al., ! 999). In the first case the energy of the dislocation increases (Pfeiffer and Rieger, 1999b; Zeng et al., 1999) like In L, as has already been found (Rieger and Blasum, 1997) for the average step energy with fixed step position at the boundary (see Section 8.5.5), whereas in the latter case it decreases with L like (In L)~, with ~ = 3/2. In the framwork of matchings, dislocations arise for incomplete (nonperfect) matchings (see Fig. 8.9). In the case of an incomplete matching, there is no longer a uniquely defined height variable corresponding to displacements of the d = 2 elastic medium. An energy for the dimer configuration can then be assigned as follows: (1) a local pinning energy where vertices are covered by dimers in M; and (2) a defect energy for vertices that are not the endpoints of dimers in M. The
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Fig. 8.9 Left: a defect occurring in an otherwise perfect matching is shown. The unmatched vertex on the A (B) sublattice correspond to a negative (positive) vortex. The dashed line indicates a possible path for a defect string that connects two defects. If this were the defect string, then in the ground state with no defects, the unmatched edges along this path would become matched and the matched edges would be removed. Right: the symmetric difference between matchings for a ground state with no defects and a ground state for the same disorder realization, with a defect energy of Ec = 1.2 (384 • 384 unit cells.) Dimers are included if they belong to a matching in one of the ground states, but not both. Vortices are at the ends of the defect lines. The defect lines themselves show where the phase change due to the introduction of defects is localized. (From Middleton, 1998)
energy of a matching is then,
E
NcEc ~
We,
(8.5.9)
eEM
where Nc is the number of noncovered vertices and Ec is the core cost of a defect. Minimizing E in (8.5.9) gives the T = 0 configuration for the elastic medium with defects that have an associated core energy. Over local regions where the height is well defined, the energy is still periodic in the height (displacement) variable. It turns out (Middleton, 1998) that the density of defects decreases approximately exponentially with the defect core energy Ec. However, this calculation was for defects placed in specified (nonoptimal) positions. By looking at the energy of these dislocations, Middleton (1998) argued that, at weak disorder, the superrough phase is stable with respect to dislocation formation, in agreement with Gingras and Huse (1996). However, Zeng et al. (1999) considered dislocations placed in optimal positions, and they found that at arbitrarily weak disorder dislocations are favorable. They thus conclude that, in two dimensions, dislocations proliferate and destroy the Braggglass state. As shown in Fig. 8.10, for the fully packed loop model, the energy for a fixed dislocation pair, on average, increases with the distance of the two defects, however, in the optimal position their energy decreases with distance as (In L) 3/2 in agreement with the above mentioned results for the SOS model on a disordered substrate (Pfeiffer
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c)
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(see lefthand side), a) The probability distributions
of the energy of afixed dislocation pair with separation L/2 for sample sizes from L  12 to L  384. The solid lines are Gaussian fits. b) The corresponding average defect energy
Ea (solid circle) and the rootmeansquare (rms) width ~r(Ea) (solid square) are found to scale with system size as In L. The solid lines are linear fits. Energetics of optimized defects (see righthand side), c) The defect energy probability distributions for sample sizes from
L

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The solid lines are guides to the eyes. d) The average
defect energy plotted as [IE~ninl]2/3 vs. In L and the rms width plotted as [or(E~nin)] 2 vs. In L. Solid lines are linear fits. (From Zeng et al., 1999).
and Rieger, 1999b). The state which results due to this proliferation is not yet understood (Zeng et al., 1999).
9 9.1
Rigidity theory and applications Introduction
Connectivity percolation became popular in the late 1960s and its application to the conductivity of practical materials, such as composites and porous media, has
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further emphasized its importance. The simplest model of charge transport in random networks is the randomresistor network in which each present bond is a resistor. It is natural to consider a second simplemodel in which each present bond is a Hooke spring which obeys the force law F  k/zr (IF[  r0), where r0 is the spring's natural length, II means modulus, and #r is the unit vector in the direction of ?, i.e. #r ~/1~1. Note that we have written this in its vector form to emphasize the fact that the restoring force is in the same direction as the radial vector. There is no restoring force for angular distortions or twists. Networks of Hooke springs and other systems in which there is no restoring force for angular deflection or twists are called centralforce systems. A connected network of resistors is able to transport a current. A connected network of Hooke springs is not always able to carry a force. The problem of rigidity percolation or network rigidity is to find the configurations of springs which are able to transmit stress. Related questions were raised by James Clerk Maxwell (1864) due to his interest in the stability of truss structures (e.g. truss bridges) and he developed a lower bound on the number of zero energy ("floppy") modes in these networks. However, his bound is poor near the rigidity threshold (this is the critical volume fraction of Hooke springs needed to support a stress). An exact theory relating the rigidity of a structure to its connectivity is elegant and quite sophisticated (Asimov and Roth, 1978, 1979; Sugihara, 1980; Imai, 1985; Tay and Whiteley, 1985; Hendrickson, 1992). This rigidity theory is developed in Section 9.2. Within the framework to be developed here, usual (scalar) connectivity is just a particular case of rigidity. Although we will discuss "rigidity", the concepts and methods apply to connectivity as well. A set of minireviews of this area is now available (Thorpe and Duxbury, 1999). The key practical tool in using rigidity theory is a bipartite matching algorithm (Hendrickson, 1992; Moukarzel, 1996; Jacobs and Hendrickson, 1997) (see Section 4.3.1). Rigidity theory compares the number of degrees offreedom which a set of nodes has to the number of constraints which exist in the graph. In the case of Hooke springs connecting sites in d dimensions, each node has d degrees of freedom (d translations), while each Hooke spring is a constraint. Bipartite matching is used to "match" or assign constraints to sites (see Section 9.2.3). This bipartite matching procedure is exact for the rigidity of graphs in the plane. It is also applicable to the special case of connectivity percolation in arbitrary dimensions. The use of bipartite matching has markedly improved understanding of rigidity percolation in two dimensions (Section 9.3) and has also provided high precision results for the connectivity backbone in all dimensions (Section 9.4). Although the general rigidity problem in three and higher dimensions does not have a welldeveloped rigidity theory, several important subclasses do. In particular if there is only one unconstrained angle in a threedimensional structure, then the bipartite matching procedure works. This applies to the rigidity of glasses and proteins where the bondbending forces are typically much larger than the 

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azimuthal twist forces (which are treated as the unconstrained degree of freedom). This is discussed briefly in Section 9.5. It is important to note that the bipartite matching methods, and indeed much of rigidity theory, is devoted to generic rigidity (Laman, 1970; Gluck, 1975; Asimov and Roth, 1978, 1979; Crapo, 1979; Whiteley, 1979; Tay, 1985; Tay and Whiteley, 1985; Hendrickson, 1992) (Section 9.2.3). This concept is foreign to the connectivity problem because all graphs are generic when it comes to connectivity. However, in the rigidity case, some constraints may be degenerate and these constraints violate the counting rules that apply in generic networks. This is analogous to the difference between a regular bead pack (which is not genetic) and random packing (which is generic) and geneticrigidity concepts have been shown to have important consequences for granular media due to this observation (Moukarzel, 1998b).
9.2 Rigidity theory 9.2.1
Rigidity and bar independence
Consider a framework F made of n pointlike joints in ddimensional space, connected by b bars of arbitrary fixed length. A framework can be formally represented by means of a graph G(V,E) plus a set X = {Xl,X2 . . . . . Xn} in which xi is the spatial location of joint i in ddimensional space. We call X a realization of G. Clearly ~ contains the topological information about F , while X contains the geometric information. We now look for possible transformations of X which leave all bar lengths unchanged. A continuous transformation satisfying this condition is called a flex. Clearly, if the number b of bars is small, it will be possible to continuously deform F while keeping all bar lengths constant. We say in this case that .T" is flexible (Fig. 9.1a). On the other hand if F has a "large enough" (to be defined later) number of bars, the only possible flexes will be isometries, i.e. translations and rotations in Euclidean space. In this case we say that ~ is rigid (Fig. 9. lb). A second important notion is the concept of infinitesimal rigidity, which amounts to the nonexistence of nontrivial infinitesimal flexes. More precisely, an infinitesimal flex V is defined by a set of instantaneous velocities {Vl, v2 . . . . . Vn} satisfying (Xi  X j )
9 (l)i  V j ) = 0
V
pair (i, j) connected by a bar.
(9.2.1)
That is, the relative velocity of each pair of joints connected by a bar must be normal to it. If the only vectors V satisfying (9.2.1) are infinitesimal isometries (i.e. trivial), F is infinitesimally rigid. Otherwise it is infinitesimally flexible.
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a
a)
b)
c)
Fig. 9.1 a) Flexible; b) Rigid; c) Degenerate frameworks.
Let us now write (9.2.1) in matrix form as MV   0 ,
(9.2.2)
where M is a dn x b matrix whose rows are in onetoone correspondence with the bars of U. We will say that a subset of bars is independent in a given realization, if their associated rows in M are linearly independent. Clearly, infinitesimal motions of U belong to a vector space of dimension dn. Within this space, infinitesimal isometries span a subspace of dimension Gd  d + d ( d  1) / 2 (d translations plus d (dL1)/2rotations), and they are always solutions of (9.2.2). Therefore the rank K (M) of M can be at most dn  Gd. If K (M) < n d  Gd, there are additional, nontrivial, solutions of (9.2.2), and .T" is flexible. Definition A framework is infinitesimally rigid if and only if K (M)  n d  Gd.
Since K equals the number of independent rows of M we have the following important equivalence Definition A framework is infinitesimally rigid if and only if it has nd  Gd independent bars.
By definition, removing an independent bar reduces K by one. In other words, each independent bar effectively eliminates one degree of freedom from the space of motions. Dependent bars, on the other hand, (also called redundant bars) can be removed without modifying K i.e. they are essentially useless for rigidity. In order to decide whether a given framework .T" is rigid, all we need to do is to identify dependent bars. Once these are removed, if there are n d  G a independent bars remaining, then U is rigid. If the remaining number of bars is smaller (it cannot be larger), 5 is flexible.
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To illustrate the difference between rigidity and infinitesimal rigidity, consider the twodimensional example shown in Fig. 9.1 c, which is rigid, since no continuous transformation (other than rototranslations) leaves all bar lengths constant. But it is not infinitesimally rigid, since an infinitesimal displacement of joint b as depicted in the figure satisfies (9.2.1). This situation (i.e. a framework which is rigid but not infinitesimally rigid) is rare, and can only happen for particular choices of joint locations, which are called degenerate. In the particular example of Fig. 9.1 c, a degeneracy occurs because a, b, c are on the same line. Otherwise this framework would be both rigid and infinitesimally rigid. Degeneracies have zero probability if the joint locations are randomly chosen, and therefore one can safely ignore them if random locations are assumed. The distinction between rigidity and infinitesimal rigidity leads us to the concept of generic properties, i.e. properties which are valid for most choices of X. In this sense, a graph is generically rigid, if almost all its realizations are infinitesimally rigid, with the exception of a zeromeasure subset of degenerate ones. Notice that generic rigidity is a graph property, i.e. it does not depend on geometry but on topology alone. The claim that generic properties are "typical" and degeneracies an exception can be justified by observing that IMI is a polynomial of the joints' locations. Therefore K can be less than its maximum value over all locations, only on a subset of zero measure, which are the zeros of IM]. The complement of this subset (i.e. the generic configurations) is therefore an open dense subset. This is contained in an important result due to Gluck (1975). Theorem (Gluek) If G has a single infinitesimally rigid realization, almost all its realizations are infinitesimally rigid. A statement similar to Gluck's theorem holds for bar (or, in graph terminology, edge) independence: if the edge set E of G is independent in a single realization, then it is independent in almost all realizations. Therefore we will in the following only be concerned with generic properties, which only depend on the topological information contained in G, relying on the fact that they will be valid for most frameworks generated from G. We now seek a graphtheoretic method to identify redundant and independent bars in ~. Because of what we have discussed, such a method would also constitute a graphtheoretic characterization
of generic rigidity.
9.2.2
Graphtheoretic characterizations of rigidity
Because generic properties depend on topology and not on geometry, it should be possible to decide whether or not a certain framework is (generically) rigid just by looking at its graph, i.e. without any knowledge of the nodes' positions. A graphtheoretic characterization of rigidity is a combinatorial method to decide
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g
[ a)
A
]
b)
Fig. 9.2 a) This threedimensional structure has n = 8 joints and 3 x 8  6 = 18 well distributed bars, yet they are not independent and therefore it is flexible, as can be seen by considering relative rotations of its two halves around the dashed axis. b) A bodyjointbar framework.
whether a graph is generically rigid. Because of our previous discussion, deciding whether a graph is rigid is entirely equivalent to being able to identify redundant bars. We now discuss methods to detect bar redundancy. It is easy to give a sufficient condition for bar redundancy (equivalently, a necessary condition for bar independence). Because our considerations about the maximum rank of M apply to any subset of Y, if any subgraph {7' with n t nodes has m o r e than n t d  G d bars, some of them are necessarily redundant. If on the other hand no subgraph has too many bars, i.e. if b' < d n t  G d for all subgraphs, we will say that the bars are w e l l distributed. Correct distribution in the sense above is a n e c e s s a r y condition for bar independence, in all rigidity problems that we will be discussing, but regretfully it is not always a sufficient condition. In dimensions greater than two, it is possible to build j o i n t  b a r structures with welldistributed yet dependent bars. A wellknown example of a threedimensional structure with welldistributed yet dependent bars is shown in Fig. 9.2a. In two dimensions, correct distribution is a necessary and sufficient condition for bar independence (Laman, 1970). The matching algorithm (MA) to be developed in this section only works for "rigidity problems" in which correct distribution is a necessary a n d sufficient condition for bar independence, since it is simply a method to decide (efficiently) whether bars are correctly distributed or not. As already advanced, the rigidity problem is not only meaningful for barjoint
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frameworks, but also for more generally defined structures, in which the number of degrees of freedom of the objects represented by the nodes have arbitrary values. These objects are still connected by bars, each fixing the distance between two points and thus acting as one constraint. In many of these cases, correct distribution of bars is a sufficient condition for bar independence, and thus the algorithm to be presented here can be applied to it. We next advance some examples. We start with twodimensional jointbar frameworks, for which Laman (1970) has shown the following theorem. Theorem (Laman) A graph ~ contains no dependent bars in two dimensions if and only if all nontrivial subgraphs G', with n I joints and b f bars, satisfy b ~ _< 2n t  3. If no subgraph contains too many bars, we can be sure that all bars are independent. This is the nontrivial part of Laman's theorem, which, as said, does not hold in d > 3. Moreover, Laman's theorem does not immediately suggest a good combinatorial algorithm. A naive implementation would imply testing all subgraphs, of which there are an exponentially large number. But Sugihara (1980) showed that Laman's theorem can be recast in different terms, involving a matching on an associated bipartite graph. This equivalent restatement gave rise to the first polynomialtime algorithm for rigidity. Hendrickson (1992) later used Sugihara's restatement as a base to provide an algorithm for twodimensional rigidity with an O(n 2) timecomplexity. It is on variants (Moukarzel, 1996; Jacobs and Hendrickson, 1997) of this algorithm that most recent largescale investigations (Moukarzel and Duxbury, 1995; Jacobs and Thorpe, 1995, 1996; Moukarzel et al., 1997a; Jacobs and Thorpe, 1998; Moukarzel and Duxbury, 1999) of rigidity percolation are based. In its original form, Hendrickson's algorithm performs close to its worstcase complexity on typical problems. Moukarzel (1996) proposed a modification of this algorithm with an improved average behavior (almost linear in n). This modified algorithm is based on the idea of successively condensing (shrinking) rigid subsets of the framework as they are identified, and treating them as single rigid objects called bodies. The correctness of the condensation procedure relies on the observation that Laman's theorem also holds for Bframeworks, that is, structural barlinkages which, in addition to joints, also include bodies (Fig. 9.2b). Bodies are extended rigid objects and thus have three degrees of freedom in two dimensions. Bars can be connected between two arbitrary points on different bodies, and fix the distance between these two points. The topological structure associated with Bframeworks is a multigraph since several bars can join two bodies, but we will ignore this technicality and simply call it a graph. For twodimensional
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Bframeworks a graphtheoretic characterization of rigidity also exists, and it is similar to Laman's theorem (Moukarzel, 1996).
Theorem (body Laman) A graph G with n joints, m bodies and b bars contains no dependent bars in two dimensions if and only if all nontrivial subgraphs G', with n ~joints m z bodies and b f bars, satisfy b ~ < 3m ~ + 2n ~  3 There are still at least two further cases in which a complete characterization of rigidity in graphtheoretic terms can be given. The first is bodybar rigidity in arbitrary dimensions (though without pointlike joints). In ddimensional space, bodies have Gd  d ( d + 1)/2 degrees of freedom, which correspond to d (d  1)/2 rotations plus d translations. For these bodybar linkages, Tay (1985) demonstrated that
Theorem (Tay) A graph G with m bodies and b bars contains no dependent bars in d dimensions if and only if all nontrivial subgraphs G', with m ~bodies and b ~ bars, satisfy b ~ Gdm' + grin' The matching algorithm builds a maximum independent subset of edges El, and this is done by testing all edges in E one at a time. Each time a new edge e is considered, counting condition IIB is used to decide whether e is independent from the existing set El. If e is independent it is added to El, otherwise e is marked redundant and removed from the graph. For simplicity we will refer to the process of adding Gd copies of an edge e to a graph as replicating edge e. In these terms, the independence test involves replicating each edge in a graph, and looking for subgraphs with too many edges. This has to be done after each new edge is added, and is still very timeconsuming. The following observation, due to Hendrickson (1992), is key for reducing the complexity of the procedure: if one adds a new edge enew to an independent set
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El, then in order to decide whether EI tO enew is still independent one only needs to replicate enew and look for subgraphs with too many edges. If none exists, then enew is independent.
Theorem (Hendrickson) Let EI be an independent edge set and consider the graph G whose edge set is EI tO enew. The following are equivalent: A. EI to enew is independent.
B. When enew is replicated obtaining ~enew' no subgraph {7etn e l / ) of Genew has more than G d m ~ + gdn ~ edges.
Proof: A =~ B is a consequence of counting condition II. To prove B =~ A assume E1 to enew is dependent. Then because of counting condition II, there must be some subgraph ~ of Gg (obtained by replicating edge g,) with "too many" edges. But ~ must necessarily include enew since El is independent. Therefore {7~ would contain the same number of edges if enew were replicated instead of ~, and the result is proven. [] Therefore, edges are added one at a time in order to build an independent set. Each time a new edge is added and replicated, we have to check whether any subgraph has too many edges. This checking can be done efficiently by mapping Hendrickson's condition B onto a bipartitematching problem (Sugihara, 1980; Csima and Lovfisz, 1992; Hendrickson, 1992). In order to illustrate this mapping, we first introduce the following notions. Given a graph {7 = (V, E) we define an auxiliary directed graph D(G) by assigning to each edge e 6 E a direction, i.e. transforming it into an arrow. Arrow orientations are arbitrary, so that D({7) can in principle be in one out of many "states", each corresponding to a configuration of arrow orientations. We say that a configuration of arrows, one per edge in E, is satisfying, if no node i 6 V has more than Yi incoming arrows, where Fi is the number of degrees of freedom of node i, that is, Fi = gd if node i is a joint, and Fi = G d if it is a body. If node i has exactly Fi incoming arrows, we will say that it is saturated (See Fig. 9.3). From now on, only satisfying configurations will be allowed on D(~). Consider now a bipartite graph B with node set (V1, V2), where V1  E and V2 is composed of ?'i copies of each node i 6 V. Nodes Vl 6 V1 and v2 6 V2 are connected by an edge if the corresponding objects (resp. an edge in E and a node in V) are incident to each other in ~. Satisfying arrow configurations on D are in onetoone correspondence with complete matchings from V1 to V2, that is, matchings that leave no node in V1 exposed. Because of this, we say that an arrow is matched to a node when it points to it in a satisfying arrow configuration. We use the bipartite matching techniques described in Section 4.3.1 in order to
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IB 1
c
I 1 VV~
A a)
b)
Fig. 9.3 a) A Bframework in two dimensions (bodies A and B have Y = 3 and joint c has F  2). b) The associated satisfying arrow configuration when one bar is replicated. All nodes are saturated in this example.
construct satisfying arrow configurations on 79. The relevance of the matching problem for our independence characterization (Hendrickson's condition B) is a consequence of the following equivalence. T h e o r e m ( S u g i h a r a  H e n d r i c k s o n ) The following are equivalent. A. A satisfying arrow configuration exists on "D(~e .... )" B. ~enew contains no subgraph with too many edges. Proof:
A ::~ B is a trivial consequence of the definition of ~)(~enew)" To prove that B =:~ A assume that no satisfying arrow configuration exists on ~:)(~enew)' and let C be a maximally satisfying arrow configuration, i.e. one with the maximum possible number of matched arrows and all unmatched arrows removed. Let ~ be one of the arrows that cannot be matched. Both its end nodes must therefore be saturated. Send both nodes to a queue Q and start a BFS from Q, following arrows in reverse direction only. This means, from each node x we only explore arrows which are pointing to x. All nodes visited in this BFS must be saturated, otherwise an augmenting path would have been discovered and g, could be matched, contradicting our maximality hypothesis. Therefore, for each jointnode visited during the BFS, we simultaneously find gd unvisited edges (those matched to it), and for each bodynode we find Gd unvisited edges. So when the BFS ends, we have visited a subgraph with a total of m bodies, n joints and exactly gdn + Grim edges. Now taking into account ~, we have a subgraph with at least one excess edge. t3
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We now sketch the matching algorithm for the general rigidity problem. We are given a graph G(V, E) and we want to determine a maximal independent edge set EI c_ E, or equivalently: we want to identify and discard redundant edges. We start with 79 having no edges. Then for each edge enew E E, we add (Gd + 1) arrows between the corresponding two nodes in 79 and: 9 If we are able to find a matching (a configuration of arrow orientations that is satisfying), then enew is independent and is added to EI. In this case, the Gd "extra" copies of enew are removed from 79. 9 If on the other hand no satisfying configuration exists, enew is redundant and all copies of it are removed from 79. In this way, the edge set of 79 is kept, at all times, independent. In practice a matching in 79 is kept from previous stages of the algorithm, and we only have to enlarge it to include the (Gd + 1) copies of each new edge enew. The enlargement is done by BFS from any of the nodes incident to enew, as described in Section 4.3.1. If the matching on 79 cannot be enlarged we learn that enew is redundant and it is discarded. This procedure enables us to build a maximal subset of independent bars. If this subset contains exactly ngd + mGd  Gd bars, then our Bframework is rigid. If it contains less bars then our Bframework is not rigid. But this knowledge alone is not always very useful since in most physical applications (e.g. on disordered systems), the framework under consideration will almost certainly contain nonrigid parts, so that this naive procedure would only give us an obvious piece of information: that the system as a whole is not rigid. In practical applications we would more generally be interested in decomposing the system into rigid clusters. Fortunately this is also possible within the matching algorithm. When a matching fails, we learn that enew is redundant. In this case we also obtain an important piece of information from the algorithm: a new rigid cluster is identified, as the following two results show. Theorem If only Ga copies of any new edge enew are added to 79 whose edge set EI is independent, a matching is always possible. Proof: Assuming no matching is possible, start a BFS as described in the proof of the SugiharaHendrickson theorem and a subgraph will be identified with bI > grin' + Gdm', involving the Gd copies of enew. If now the Gd copies of enew are replaced by Gd copies of any other edge in this subgraph, a matching would also not be possible, but this is in contradiction with the assumed independence of Ei. []
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So the first Gd copies of any new edge can always be matched if the edgeset in 79 is independent. Therefore it is the result of the (Gd + 1)th search for an augmenting path that decides whether enew is independent. If this search fails, the visited subgraph (without enew) is minimally rigid, as we now show. Theorem (duster) When the (Gd + 1)th search for an augmenting path fails for a new edge enew, the failed BFS spans a minimally rigid subgraph of U, i.e. one with exactly ngd + mGd  Gd well distributed edges. Proof:
By the same reasoning as in the SugiharaHendrickson theorem, we can see that the failed BFS visits a subgraph with exactly ngd + mGd edges, and that the Gd copies of enew are in this subgraph. Remove the Gd copies of enew, and the result is proven. [] This property of the algorithm allows us to identify rigid clusters, even when the system as a whole may not be rigid. This is very important in practical applications as already mentioned. In Hendrickson's original algorithm, newly discovered rigid clusters are given a common label. Using these labels one can save some CPUtime since, if a new bar enew has both ends connected to the same rigid cluster (the same label), then no test is necessary. In this case, enew is trivially redundant. The resulting algorithm can be shown to need O(n 2) time in the worst case (Hendrickson, 1992). Experimental results using this algorithm show (Moukarzel, 1996) that its average timecomplexity is not much better than this. We now describe a modification of Hendrickson algorithm, which has much improved average behavior.
9.2.4
Condensation
As already discussed, each time a B FS fails to find an exposed node to which the (Gd + 1)th copy of a bar (a, b) can be matched, the set of nodes and bars visited forms a minimally rigid subgraph C(a, b). In Hendrickson's algorithm, these clusters are given a common rigid label and left in the system. It is possible to dramatically improve the time and memoryrequirements of the algorithm by condensing these rigid clusters (Moukarzel, 1996), i.e. replacing them on 79 by a single bodynode with Gd degrees of freedom (An alternative to condensation consists in changing the internal connectivity of these rigid clusters in order to speed up subsequent searches, as described by Jacobs and Hendrickson (1997). This has the drawback of being more complicated than condensation (which consists in simply deleting all nodes forming the rigid cluster).). Subsequent searches are then done much faster since now one node must be checked instead of a whole subgraph.
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,d ,
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':~~' b) b c
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Fig. 9.4 An example of condensation in a twodimensional Bframework (gd  2, Gd = 3). a) A redundant bar is found (dashed), and a new rigid graph is identified (shaded). The framework in this example consists of two separate rigid clusters which can rotate around node d. b) Direct reconnection: the rigid graph is condensed to a new bodynode N', and bars (e, d), (g, d) and (c, d) are directly reconnected to N'. Direct reconnection of bars which are incident to a jointnode is not correct in this case. c) In order to preserve the rigid properties of the system after condensation, an auxiliary structure must instead be used to reconnect bars which are incident to a jointnode. This auxiliary structure is composed of the original jointnode (d in this figure), rigidly connected to N" by gd auxiliary bars (dashed). Condensation is implemented by first erasing all objects (nodes and bars) visited during the failed search, that is, all objects in C(a, b), and then reconnecting all external bars incident to them to a newly created bodynode N'. In this reconnection step, a distinction is to be made between bars which are incident to a bodynode in C(a, b) and those incident to a jointnode in C(a, b): the former are directly reconnected to N" while bars incident to joints in C(a, b) must be again connected to their joints (which are not deleted), and these in turn rigidly fixed to N" through an auxiliary structure made of gd bars. In order to see why auxiliary structures are necessary, consider the example shown in Fig. 9.4. Consider what would happen if all bars which are incident to node d were directly reconnected to N" (Fig. 9.4b): if next a bar (a, b) were added to this structure, a unique rigid cluster would be formed. But this does not reflect the rigid properties of the original structure (Fig. 9.4a), which would still be composed of two rigid clusters with a common joint d, after adding (a, b). The correct procedure for reconnecting bars which are incident to a joint (bars (c, d), (g, d) and (e, d)) is illustrated in Fig. 9.4c, and it is easy to see that the rigid properties of the system are unchanged by condensation in this case (Moukarzel, 1996).
9.2.5
Extracting the physics
As we have seen, the matching algorithm provides a way to identify redundant bonds or constraints. Here we will see how it is possible to obtain other physically important pieces of information from this algorithm.
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Each time a redundant bond is found, a new rigid (or connected) cluster is also identified. This rigid cluster is the set of bonds with respect to which the new bond is redundant, which is to say the bonds that would be stressed if the new bond were replaced by a spring. In the connectivity case, the matching algorithm identifies the loop which would carry a current if the redundant bond were a battery. Clearly, a naive application of this algorithm does not ensure that all rigid clusters will be identified. According to what we have discussed, only overconstrained clusters will be discovered for sure, i.e. those which have bonds in excess of rigidity. Thus condensation only happens for overconstrained clusters, and our labeling (each condensed set has a unique label) corresponds to a classification of the system into disjoint overconstrained clusters. Minimally rigid clusters (those rigid but not overconstrained) would thus in principle go unnoticed, since they have no redundant bonds, and the MA is only able to detect a rigid cluster when a redundancy is found. If one is interested in identifying all rigid clusters, i.e. given any pair of points one would like to know whether they are rigidly connected, there is a straightforward way to do so. The trick is simply to test a fictitious, auxiliary bond between these two nodes in the graph. If this auxiliary bond is found redundant (its last copy cannot be matched) then the nodes are rigidly connected. The fictitiousbond trick also serves to identify the backbone, i.e. the subset carrying the load between two given points, or, in connectivity, the currentcarrying subset. This is so because of the property already mentioned, that the subset visited during the failed search is the set of bonds that carry stress if the last, redundant, bond is replaced by a spring. A fictitious bond tested between two ends of a rigid sample will be redundant, and the set searched during the last (failed) BFS is the backbone. In the infinite percolating cluster, the red bonds are particularly important, as removal of any one of them disconnects the sample. The MA algorithm allows an easy identification of these critical bonds. When implementing the fictitiousbond trick across the sample, the rigid subgraph identified (the backbone) will, in general, be found to consist of overconstrained clusters (clusters that have been already condensed at least once) rigidly connected by bonds. These bonds necessarily provide a minimally rigid connection, otherwise some would be redundant and they would have been condensed. Therefore all bonds in this rigid subgraph are red bonds, also called cutting bonds. When studying the structure of glasses and amorphous solids one is often interested in their vibrational properties (Caprion et al., 1996; Dove et al., 1997; Fabian, 1997; Oligschleger and Sch6n, 1997). An important quantity is the number of soft modes orfloppy modes (Maxwell, 1864; Thorpe, 1983; Cai and Thorpe, 1989; Dove et al., 1997; Hammonds et al., 1997); the number of null eigenvalues of the dynamic matrix. Floppy modes correspond to infinitesimal deformations which cost no energy. Clearly, what we have discussed about infinitesimal rigidity
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in Section 9.2, is the static counterpart of these concepts, since floppy modes are simply nontrivial flexes. The number of floppy modes F equals the number of remaining degrees of freedom once all independent constraints have been taken into account and thus for jointbar networks one has (9.2.4)
F=dnGdb+R,
where b is the total number of bonds and R is the number of redundant bonds. Once we know the number of redundant bonds, the number of floppy modes is easily found from (9.2.4) (Franzblau and Tersoff, 1992; Jacobs and Thorpe, 1996; Duxbury et al., 1999). Early methods for counting the number of floppy modes were similar to those used to calculate the full density of states. In general, one relied in diagonalizing the dynamic matrix and then counting the number of zerofrequency eigenstates (He and Thorpe, 1985; Cai and Thorpe, 1989). Numerically this is an extremely hard task. A much better alternative consists in finding the number of redundant constraints with the help of the MA (Jacobs and Thorpe, 1995, 1996; Jacobs and Hendrickson, 1997; Jacobs, 1998; Duxbury et al., 1999) and using (9.2.4).
9.3 9.3.1
Rigidity percolation on triangular lattices Introduction
One of the first attempts to describe the mechanical properties of disordered systems using percolation ideas is due to de Gennes (1976), who suggested that the elastic modulus E of gels close to the gelation point might behave in the same way as the conductivity ]e ~ (p  pc) t of disordered systems near a percolation point (Stauffer and Aharony, 1994; Sahimi, 1995). It became clear later that these two problems are not equivalent, due to their differing tensorial characters. In the elastic problem vectors (forces or displacements) rather than scalars (electric charge) must be transmitted. In fact E behaves as (p  Prig) f , with f ~ t, where Prig is the rigidity threshold. It was numerical work by Feng and Sen (1984) which first suggested that the elasticmodulus exponent f is in general different from the conductivity exponent t. In this work the Born model (Born and Huang, 1954) of elasticity,  Z
+
(9.3.1)
(i,j)
was used, where ( U i   /~j)[[,_k is the relative displacement of neighbors i j, respectively in the directions parallel and perpendicular to bond (i, j), and Kij is 1 with the stiffness of bond (i, j). Randomly bonddiluted lattices have Kij =
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probability p and K i j   0 with probability 1  p. The present authors notice that conductivity and elasticity are only equivalent if a = b, in which case the elastic problem decouples into d independent scalar problems (A similar decoupling happens in centralforce elastic systems which have zerolength springs (Note 2, p 317). This particular case of "scalar elasticity" is for example relevant for the mechanical properties of polymers and rubbers, in which centralforces are of entropic origin and can in some cases be similar to zerolength springs.). But elastic modulus and conductivity are not proportional to each other in general, as can be seen by taking b  0: this gives a centralforce (CF) potential, in which case a higher rigidity threshold was found, and there was also evidence (Feng and Sen, 1984) that fCF > t. Later Kantor and Webman (1984) considered a potential including bondbending terms as well as central forces.
V = a Z
Kij(~li 
{tj)~ + b ~
KijKjk(6Oijk) 2
(9.3.2)
When b # 0, a change in the angle Oijk between a pair of adjacent bonds costs energy, as if the bonds had to be "bent". In two dimensions, the existence of BB forces makes the rigidity problem simple; any connected path is rigid, and therefore the geometry of the rigid clusters is dictated by simple connectivity. Therefore Prig  Pc when BB forces are nonzero. For any nonzero b, the elastic properties near criticality are dominated by bondbending forces (Kantor and Webman, 1984) because the energy involved in a typical bondbending contribution scales down to zero faster with size, than that due to bondstretching forces. Using the nodeslinksblobs model (de Gennes, 1976; Skal and Shklovskii, 1975; Coniglio, 1981 ; Pike and Stanley, 1981), Kantor and Webman showed that f8 8 >_ dv + 1 where v is the correlationlength exponent. This lower bound is definitely larger than t ~ 1.3 (in two dimensions), proving that elasticity and conductivity are not equivalent. Similar ideas were later considered by various authors (Feng et al., 1984; Roux, 1985, 1986; Sahimi, 1986), who refined the above bound to f88 < 2v + t, conjecturing that it might be an equality. This conjecture is consistent with the most accurate available simulations (Zabolitsky et al., 1986). Therefore, bondbending (Kantor and Webman, 1984; Feng et al., 1984; Bergman, 1985; Sahimi, 1986; Feng, 1985; Zabolitsky et al., 1986; Roux, 1985, 1986; Wang, 1996b) elasticity is fairly well understood in two dimensions. In d > 2, bondbending constraints are not enough to provide rigidity at the connectivity percolation point (Phillips and Thorpe, 1985), so that the situation might be different. The more subtle centralforce rigidity problem has been studied by several authors (Feng and Sen, 1984; Day et al., 1986; Thorpe and Garboczi, 1987; Burton and Lambert, 1988; Roux and Hansen, 1988; Hansen and Roux, 1989; Guyon et al., 1990; Arbabi and Sahimi, 1993; Knackstedt and Sahimi, 1992; Jacobs and
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Thorpe, 1995; Moukarzel and Duxbury, 1995; Obukov, 1995; Jacobs and Thorpe, 1996; Moukarzel et al., 1997a; Jacobs and Thorpe, 1998), but the most basic issues such as the universality class have been controversial for some time. Early results indicate that the CF elastic problem belongs to a new universality class (Feng and Sen, 1984; Day et al., 1986), i.e. that both the geometrical (/3, v, y, or) and elastic exponents ( f ) are different from those of BB elasticity. On the other hand, there have been also claims on the contrary (Roux and Hansen, 1988; Hansen and Roux, 1989; Guyon et al., 1990; Hansen, 1990; Wang, 1996a), based on the argument that effective bondbending forces would be prevalent at large scales. Other authors even suggested that the bonddiluted and sitediluted CF cases might behave differently (Arbabi and Sahimi, 1988, 1993; Knackstedt and Sahimi, 1992), i.e. that CF elasticity would not be universal in a broad sense. Some of the reasons why these early studies have been so inconclusive are as follows. 9 Lacking a combinatorial algorithm to identify rigid clusters, one had to rely on solving the force equations, which is extremely timeconsuming and restricted the numerical analysis to relatively small systems (e.g. 80 • 80 (Hansen and Roux, 1989)). 9 The lack of good independent estimates for pC F seriously affects the measurement of critical indices (Hansen and Roux, 1989). 9 It has been suggested (Day et al., 1986) that CF rigidity is a particularly difficult task for conjugategradient iterative solvers, since there is an unusually large accumulation of roundoff errors. It should be noted that the centralforcerigidity problem is an idealization. Real materials have bondbending terms and there are effective bondbending and twistenergy terms introduced by temperature (Plischke and Jo6s, 1998; Jo6s et al., 1999). Nevertheless the centralforce limit provides an important limiting case. In the regime between the centralforcerigidity threshold and the connectivity threshold "soft" materials are held together by bondbending and twistenergy terms which can be quite weak.
9.3.2
Results from matching methods
The introduction of matching algorithms (MA) (Hendrickson, 1992; Moukarzel, 1996; Jacobs and Hendrickson, 1997) in the study of CF rigidity percolation has alleviated all of the problems described in the previous subsection. The geometrical properties of rigid clusters can now be obtained without knowing the stresses, which has enabled a large increase in the sizes accessible to numerical simulation. Matching algorithms were used in studies of twodimensional systems containing 106 sites (Jacobs and Thorpe, 1995; Moukarzel and Duxbury, 1995; Jacobs and
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Thorpe, 1996; Moukarzel et al., 1997a) and more recently up to 1.6 x 107 sites (Moukarzel and Duxbury, 1999). It is possible to implement the MA in such a way that the exact percolation point is identified for each sample (Moukarzel and Duxbury, 1995; Moukarzel, 1996). Thus, measurements exactly at Pc can be done (Moukarzel and Duxbury, 1995; Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999) at all sizes, and one does not have to rely on datacollapse to estimate it together with the critical indices. This removes an important source of error (Hansen and Roux, 1989) from critical indices estimates. Regarding the convergence problems with iterative solvers, if the rigid backbone is identified with the MA, and subsequently the elastic equations are solved on the backbone alone, it has been found (Moukarzel and Duxbury, 1995) that the conjugate gradient performs much better, enabling the study of elastic properties on systems as large as 512 x 512. As a result of these recent studies, it has became clear that twodimensional CF rigidity percolation does not belong to the universality class of connectivity percolation (Moukarzel and Duxbury, 1999) (and, consequently, of BB rigidity). Precise estimates for the correlationlength exponent v in CF rigidity percolation have been obtained from studies of bond and sitediluted triangular lattices (Moukarzel and Duxbury, 1995, 1999): v cF  1.16(3) and (Jacobs and Thorpe, 1995, 1996) v cF = 1.21(6), which have demonstrated that CF rigidity is in a different universality class than connectivity percolation(where v  4/3). This difference is even more striking in tree models and in infinite range models where rigidity percolation is first order (Moukarzel et al., 1997b; Duxbury et al., 1999), whereas connectivity percolation is second order. In addition to CF rigidity, we also discuss a twodimensional rigidity model in which each site is a rigid body with three degrees of freedom, and contiguous bodies are "pinned" at a common point by a universal joint, allowing the possibility of relative rotation. This sitediluted bodypin (BP) model (Moukarzel and Duxbury, 1999) is depicted in Fig. 9.5. Each joint or pin provides two constraints, and can therefore be represented by two bars, thus allowing the use of the MA. Although forces are not "central" in this model, we have found that CF and BP are in the same universality class. This is consistent with the idea (Adler et al., 1987; Guyon et al., 1990; Moukarzel et al., 1997b) that it is incomplete transmission of information what makes CF rigidity different from scalar connectivity. We also present results from a twodimensional CF rigidity model, which we call the "braced square net" (BSN) and which has a firstorder rigidity transition (Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999). An undiluted square net with open boundary conditions is "at the verge of rigidity": having L 2 sites and 2L 2 _ 2L bars, there is a total of 2L  3 "floppy modes" or infinitesimal flexes. This system can be rigidified by randomly adding diagonal bonds. Since the number of remaining floppy modes is not extensive, the density of diagonals Pd at
2 o
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o
I i i r i i I i I I
o
o
o
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Fig. 9.5 Sitediluted square lattices, a) The jointbar rigidity model. Each site is a point with two degrees of freedom, and each bond is a rotatable bar which fixes the distance between them, providing one constraint, b) The bodypin rigidity model. Each site contains a rigid body with three degrees of freedom, and neighboring bodies are pinned at a common point. Each such pin provides two constraints. the rigidity threshold goes to zero in the limit of large lattices. A similar behavior is obtained if the boundary conditions are periodic in one direction, or if rigid busbars are located on a pair of opposite sides (while keeping free boundaries in the other). The nongeneric version of this problem has been studied by many authors (Bolker and Crapo, 1977; Dewdney, 1991; Obukov, 1995). Our numerical simulations correspond to the busbar boundary condition, and to generic rigidity. In numerical studies of percolation, when pc is not known exactly it is usual to take numerical measurements at closely spaced values of p, from which Pc, and also other information at Pc, are inferred later, through, for example, finitesize scaling. Since in the MA it is necessary to add bonds one at a time (in arbitrary order), we have found the following procedure more convenient: add bonds, one at a time, at random locations (this is equivalent to slowly increasing the bond occupation density p by the smallest possible steps). We check, after each bond addition, whether rigidity percolates. This check is not time consuming within the MA (Moukarzel, 1996), and serves to exactly detect the percolation point for each sample, thus allowing very precise measurements of critical properties. In order to detect when there is a systemspanning rigid cluster, we connect an additional fictitious bar or spring between the upper and lower busbars on opposite sides of the system. This auxiliary spring mimics the effect of an external load, and therefore the first time that a macroscopic rigid connection exists, a globally stressed region (the backbone) appears. We take an initially depleted lattice and add bonds (in the bonddiluted case) or sites (in the sitediluted case) to it one at a time as described above, and use the MA to identify the rigid clusters that are formed in the system. For the
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case of bond dilution, p is the density of present bonds, while in site dilution it indicates the density of present sites. We have used two distinct definitions of the critical point (Moukarzel and Duxbury, 1995). First, we determine whether an externally applied stress can be supported by the network, which we call applied stress (AS) percolation. Secondly we studied the percolation of internallystressed (IS) regions. At the AS percolation point, an applied stress is first able to be transmitted between the lower and upper sides of the sample. The stressed backbone is detected as a selfstressed cluster within the MA, because of the existence of the fictitious bond mentioned above. The IS critical point is defined as the bondor site density at which internal stresses percolate through the system. This means that the upper and lower sides of the system belong to the same selfstressed cluster (Moukarzel and Duxbury, 1995). This is easily detected within the matching algorithm (Moukarzel, 1996). The AS and IS definitions of percolation are in principle different, but we found (Moukarzel and Duxbury, 1995) that the average percolation threshold and the critical indices do not depend on the definition used for large lattices. Most of our simulations were done using the AS definition of the critical point. The rigidity thresholds can be estimated with good accuracy: pCsF = 0.6975(3) on sitediluted triangular lattices (Moukarzel and Duxbury, 1995" Jacobs and Thorpe, 1996), and pCF _ 0.6602(3) on bonddiluted triangular lattices (Jacobs and Thorpe, 1995, 1996). For the bodypin model on sitediluted square lattices, our numerical estimate is Pc  0.74877(5). We define the spanning cluster (Fig. 9.6) as the set of bonds that are rigidly connected to both sides of the sample. However, only a subset of these bonds carry the applied load. This subset is called the backbone. At the AS critical point, the backbone will always contain some cutting bonds (Stanley, 1977; Coniglio, 1981, 1982; Moukarzel and Duxbury, 1995), so named because the removal of any one of them leads to the loss of global loadcarrying capability. Cutting bonds attain their maximum number exactly at pc (Coniglio, 1981, 1982). The backbone bonds which are not cutting bonds are parts of internally overconstrained blobs (selfstressed clusters). The spanning cluster also contains bonds which are rigidly connected to both ends of the sample but which do not carry any of the applied load. These are called dangling ends. This classification is standard in connectivity (scalar) percolation. The MA allows efficient identification of all these spanningcluster subsets (dangling ends, backbone and cutting bonds). We analyze the size dependence of the following densities associated with the spanningcluster: Ps, the backbone density, PD, the danglingend density and Pec, the infinitecluster density, exactly at the percolation threshold of each sample. In Fig. 9.7ac, this data is presented for the three different cases in 3200, (b) L max 4096, Figs. 9.6ac. Maximum sizes studied were" (a) L max 1024. and (c) L max 





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Fig. 9.6 Infinite percolation clusters, a) CF rigidity percolation on a sitediluted triangular lattice, b) Bodypin rigidity percolation on sitediluted square lattice, c) CF rigidity percolation on the braced square net. The system size is L = 64 and rigid busbars are set on the upper and lower ends of the sample. The backbone, is composed of "blobs" of internally stressed bonds (thick black lines), rigidly interconnected by cutting bonds (gray lines). Cutting bonds are also called red bonds. Removing one of them produces collapse of the system. Dangling ends (thin lines) are rigidly connected to the backbone, but do not add to the ability of these networks to carry an external load (current in the connectivity case). (From Moukarzel and Duxbury, 1999). 100 r
10 ~
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/ 10
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.............. iiii1 100
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r
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Fig. 9.7 Density of backbone bonds (circles), dangling bonds (squares) and infinitecluster bonds (diamonds) at the AS critical point, a) Rigidity percolation (g = 2, G = 3) on a sitediluted triangular lattice, b) Body rigidity (G  g = 3) on a sitediluted square lattice. c) Rigidity on a randomlybraced square lattice. (From Moukarzel and Duxbury, 1999).
It is clear from Fig. 9.7 that the BSN (Fig. 9.7c) has a qualitatively different behavior than the other cases. For the BSN, P s , P ~ and PD all have a finite density at large L, indicating that the rigidity transition is first order in this case (Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999). In contrast, in both rigidity cases (Figs. 9.7a,b), P8 and P ~ are decreasing in a power law fashion over the available size ranges. However, the behavior of PD is more complex. Let us first discuss the behavior of the backbone density P s .
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1.0 /i
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b)
Fig. 9.8 The spanning cluster density exponent/~/v as numerically estimated, a) Rigidity percolation on a triangular lattice (L max   3200). b) Bodypin rigidity on a square lattice (L max   4096). These plots show  l n P ( L ) / l n L vs. 1/lnL, which intercept the ordinate axis at a value equal to the estimated leading exponent ~/v. Solid lines are fits using P(L)  C1L~/v(1 + C2LC~ Different estimates result in each case from fitting the scaling of spanning cluster density P~ (triangles) and dangling end density PD (circles). This indicates that finitesize effects are still too large. (From Moukarzel and Duxbury, 1999).
Taking into account corrections to scaling, the fit PB  C I L  e ( 1 + C2 L~~ I to the numerical data for the CF and BP cases lead to ~gr/V  0.22(2). In consequence the CF and BP rigid backbones are fractal at Pc, with a fractal dimension DB  1.78(2) (Moukarzel and Duxbury, 1995; Jacobs and Thorpe, 1995, 1996; Moukarzel et al., 1997a; Moukarzel and Duxbury, 1999). Now we consider Pc~ and PD. In the rigidity cases there are strong finitesize effects and even at sizes of L  3200 (jointbar rigidity, Fig 9.7a), and L  4096 (bodypin rigidity, Fig. 9.7b), it looks as though the dangling probability may be saturating, while the infinite cluster density continues to decrease. Since P ~ = PB + PD, and PB + 0 more rapidly than Pc~, then P ~ and PD must have the same asymptotic behavior. The analysis in Fig. 9.8 demonstrates that the current data is not sufficiently precise to illustrate that fact. Clearly the numerical results for the range of system sizes currently available are still influenced by strong finitesize effects, and the results depend on the analysis method chosen. The danglingend density and the spanningcluster density are not displaying their asymptotic behavior in Fig. 9.7. Either P o starts to go down, or P ~ levels up, for large lattices. A fit to the P ~ data of Fig. 9.7a,b yields f l / v  0.147(5) (see Fig. 9.8a,b). But a similar fit of the dangling end density gives f l / v ~ 0.03 for the jointbar rigidity case (Fig. 9.8a) and f l / v ~ 0.01 for the bodypin rigidity case (Fig. 9.8b).
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Similar investigations by Jacobs and Thorpe (1995, 1996) using equivalent numerical methods produced results consistent with ours, but they chose to interpret the infinitecluster probability as being key, disregarding the danglingend behavior. This led them to conclude that, at the CF rigidity transition the spanning cluster is fractal. If on the other hand one were guided by the Cayleytree results (Moukarzel et al., 1997b) for rigidity percolation, which indicate a firstorder jump in the infinitecluster probability, it is natural to interpret Fig. 9.7a,b as indicating a saturation of the infinitecluster probability at the danglingend value of about 0.1 (Moukarzel et al., 1997a) and thus/~  0. Clearly much larger simulation sizes are required to find ~ / v precisely. Now we turn to the calculation of the correlationlength exponent for rigidity percolation. At a secondorder transition there is a diverging correlation length ~ [ P  Pcl v. In rigidity percolation, the correlation length is of the order of the mean size of a rigid cluster. But instead of measuring cluster sizes, which is not computationally efficient, it is customary (Stauffer and Aharony, 1994) to find the exponent v of this divergence by measuring the width of the spanning probability distribution as a function of size L. The spanning probability pSpanning(p, L) is the probability that a spanning cluster exists on a system of size L and density p of present sites (or bonds), and its derivative with respect to p is the probability PL (pc) that a spanning cluster appears for the first time at density p. The width ~ r ( L )  ~ < p2 >L  < Pc >2) of this distribution, according to finitesize scaling, behaves as o(L) ~ L 1/v. In order to measure v, it is usual to do simulations at fixed values of p near the estimated critical point, and then extract the width of PL (pc) from the resulting estimate of pSpanning(p, L). We can do much better than this with help of the MA, since it allows us to determine Pc exactly for each sample. Therefore we measure ~r (L) as the sampletosample fluctuation in pc. An asymptotic analysis for o(L) is shown in Fig 9.9a for jointbar rigidity and in Fig. 9.9b for bodypin rigidity. From these figures we estimate 1/v = 0.85(2) for CF and BP rigidity percolation. This result is consistent with other recent studies of CF rigidity percolation (Jacobs and Thorpe, 1995, 1996). In the case of the firstorder rigidity on the braced square net, the variations in Pc behave as L 3/2, in accordance with analytical results for this model (Moukarzel, unpublished data). The MA also identifies the red bonds at the percolation point, for the case of AS percolation. The number NR of red bonds scales at Pc as L x. Coniglio (1981, 1982) has shown that x = 1/v exactly, for scalar percolation. Numerical evidence suggesting that x  1/v also in rigidity percolation was first presented by Moukarzel and Duxbury (1995). It is possible to extend Coniglio's reasoning to the case of centralforce rigidity percolation (Moukarzel, unpublished data). It turns out that x  1/v has to be rigorously satisfied also in this case, and therefore or(L) and 1/NR(L) must have the same slope in a loglog plot. Analysis of the
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number of cutting bonds is also presented in Fig. 9.9, and yields values of 1/v consistent with the analysis of variations in percolation thresholds described in the previous paragraphs. It has recently been demonstrated that the number of floppy modes behaves as a free energy for the rigiditypercolation problem, and of course also for the connectivitypercolation problem (Duxbury et al., 1999). The specificheat exponent can then be extracted from the second derivative of the number of floppy modes. Data from the analysis of floppy modes in the bonddiluted triangular lattice are presented in Fig. 9.10. From this data, the specific heat exponent =  0 . 4 8 ( 5 ) was extracted. This in combination with the exponent v = 1.16(3) violate the hyperscaling relation (dr = 2  c~), though only by a small amount. Possible rational values which are close to the numerical estimates and which obey hyperscaling are: (v = 6/5, ot =  2 / 5 ) ; (v = 5/4, ot =  1 / 2 ) ; (v = 7/6, ot =  1 / 3 ) . Assuming that rigidity percolation is conformal invariant it would be useful to find its central charge and hence to determine whether the exponents can be expected to be simple rationals. These results provide further strong evidence that rigidity percolation is not in the same universality class as scalar percolation. It is interesting to notice that CF and BP are in the same universality class, although the forces involved in BP rigidity are not at all central. It therefore seems appropriate to say that what characterizes the universality class of rigidity percolation (at least in two dimensions) is not the central character of the forces, but whether a bond provides complete information or not. CF and B P models both have incomplete transmission of
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9.4
Connectivity percolation
As mentioned in Section 9.2.2, the simplest case in which the matching algorithm is useful is s c a l a r connectivity. This corresponds to gd = G d = 1 since now the only degree of freedom is the "electric potential" of the sites, and it is easy to see that a Lamantype theorem trivially holds. Therefore everything we
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derived for rigidity percolation can be applied, without conceptual modification, to connectivity percolation. Of course, for the much simpler problem of connectivity percolation there exist extremely efficient integer algorithms (Hoshen and Kopelman, 1976; Leath, 1976b; Wilkinson and Willemson, 1983; Wilkinson and Barsony, 1984; Hoshen et al., 1997) for the identification of connected clusters, so it would hardly make sense to apply the relatively more complex MA for this task. But the same cannot be said if we are interested in backbone properties or red bonds. In the physics community the burning algorithm (Herrmann et al., 1984; Herrmann and Stanley, 1984) is the most popular backbone algorithm, although this procedure is relatively slow and does not allow the study of large systems unless careful optimization is done (Rintoul and Nakanishi, 1992, 1994; Porto et al., 1997). For the particular case of planar graphs, a very efficient algorithm has been implemented (Grassberger, 1992a, 1999). There are also other efficient algorithms available for higher dimensions (Tarjan, 1972), though they have not been applied to percolation as yet. The MA is particularly well suited for backbone identification in general, and it has been used for connectivity percolation in two to five dimensions (Moukarzel, 1998a). These studies provided backbone exponents which are among the most precise presently available. Also very good precision is obtained for the thermal exponent 1/v and threshold values Pc in dimensions three, four and five. The MA for connectivity percolation follows from the procedures described in Sections 9.2.3 and 9.2.5 for the particular values Gd = gd = 1. But since the algorithm takes a simple form in this case, it is worth explaining it in the following simplified terms, which are specific to the connectivity case (Moukarzel, 1998a): consider a system of n sites i  1. . . . . n connected by an arbitrary set E of b bonds (i, j) connecting sites i and j. The matching algorithm can be thought of as a way to identify and remove loops, and is implemented using a directed graph D as an auxiliary representation of the system. In this directed graph, lattice sites are represented by nodes i and bonds by directed edges (i, j). We may think of each edge as an arrow. These can be pointing in either direction, subject to the constraint that no node has more than one arrow pointing to it. A node with an incoming arrow will be said to be covered by the corresponding edge. A node with no incoming arrows is uncovered. The directed graph D will be kept loopless. In order to do this, each time a closed loop (a cycle, or circuit) is identified it will be removed from D, and replaced by a loopnode (a bodynode in the terminology of Section 9.2.3). A loopnode is a node in 79 that represents a deleted, or c o n d e n s e d loop. Therefore, although initially there are as many nodes in 79 as there are sites in our system, as the algorithm proceeds we will delete some nodes and create loopnodes. These loopnodes will be given a looplabel, which is the minimum of all edge and nodelabels contained in the loop.
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We start from a graph 79 initially containing n nodes and no edges, and add edges one at a time. Before adding an edge ab ~ E to 79, the following test is done in order to know if a loop is closed by ab. We attempt to reorient the existing arrows on 79, in order to uncover both a and b, without ever pointing two arrows to the same node. Since, by hypothesis, 79 without edge ab has no loops, it must always be possible to uncover one of them. Let us then assume that a is uncovered first. If after doing this b can also be uncovered (while keeping a uncovered), then the new edge ab does not close a loop, and therefore it is definitively added to 79. This means: we add an arrow between a and b, pointing to either of them (Fig. 9.1 1). If on the other hand it is not possible to uncover b while keeping a uncovered (Fig. 9.12), this necessarily means that a loop would be formed on 79 by the addition of ab. In this case, edge ab is not added to 79 but the following is done instead: starting from b (covered), we follow the arrows backwards. This will necessarily lead to a, thus identifying the new loop. All edges in this loop are
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given a common loop label Imin, which is the minimum amongst all node and edgelabels in the loop (including nodes a and b and edge ab). Next all nodes and edges in the loop are removed from 79 and replaced by a node with label Imin. This is the condensation step. We now present results for scalar percolation on sitediluted hypercubic lattices in two to five dimensions (Moukarzel, 1998a). The largest sizes studied were L = 4096 in two dimensions, L = 256 in three dimensions, L = 60 in four dimensions and L = 26 in five dimensions. These upper limits correspond to approximately 1217 million sites, and stem from the need to hold (in the present implementation of the algorithm) the whole system at once in memory, i.e. the algorithm is memory limited. The results described here were obtained on an Alpha500 workstation with 512Mb RAM. Since sites are added one at a time until the percolation point is reached, it is possible to measure, for each sample, the critical density pc of occupied sites. We
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assume that finitesize corrections in sampleaverages Pc (L) behave as Pc (L) = pc(OO) [1 + a L w~ (1 + bLW2)] and fit this expression to our data with five free parameters. In doing so it is found that the leading correction exponent w] is consistent with the values of 1/v estimated by us (see later). So if now wl is fixed to be l/v, we obtain: pc = 0.59273(2) in two dimensions; pc = 0.31162(3) in three dimensions; pc = 0.19682(5) in four dimensions; pc = 0.1408(2) in five dimensions. These values are consistent with previous work (Grassberger, 1992a; Ziff, 1992; Lorenz and Ziff, 1998; Ballesteros et al., 1999). The fluctuation crt` = < p2 > t` _ < Pc >2, where < > t. indicates averages over samples of size L, is expected to scale as at. ~ L 1/v with system size. On the other hand Coniglio (1981, 1982) has shown that the number Rt` of red bonds grows at pc as L 1/v. Therefore by measuring the number of red bonds Rt` (as described in Section 9.2.5) and CrL we get two independent estimates for the thermal exponent 1/v. Using this method, we find values of v which are also consistent with previous work, though the memory requirements of the matching method mean that invasion methods (see Section 6.2) are more effective in finding pc, v and/~. However, the matching method is very effective in finding the backbone. Data for the fraction B ( L ) of bonds on the backbone at pc are shown in Fig. 9.13. Assuming powerlaw corrections to scaling of the form B ( L ) = )~LIY/v(1 + aL~176
(9.4.1)
we estimate (Moukarzel, 1998a) db = 1.650(5) (two dimensions), 1.86(1) (three dimensions), 1.95(5) (four dimensions) and 2.00(5) (five dimensions), which are consistent with results found using the burning algorithm and depthfirst search (Grassberger, 1992a; Rintoul and Nakanishi, 1992, 1994; Grassberger, 1999).
9.5
Applications to soft materials
As stated in Section 9.2, there is no analog of Laman's theorem for barjoint graphs in arbitrary dimensions. However, for bodybar systems an extension of Laman's theorem is proven and an extension of the matching algorithms has also been demonstrated (Tay, 1985; Tay and Whiteley, 1985; Moukarzel, 1996). In addition, and of practical significance, it has been proposed that glasses and proteins which have weak "twist forces" but strong "flex" forces in three dimensions can be analyzed using matching methods (Jacobs, 1998; Jacobs et al., 1999; Thorpe et al., 1999). In these materials, it is assumed that the twist force is negligible and is the cause of the flexibility of the molecule. To illustrate the concept, we present (Fig. 9.14) a segment of a typical biomolecule. Bonds around which there is a circulating arrow indicate that the twist energy around that bond is very low.
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In contrast, the centralforce and bondflex energies are comparatively large. In the rigidity algorithms, the twist energy is assumed to be zero, and we seek to determine which bonds are rigid with respect to each other. Often systems of this sort are called "bondbending" networks. As a point of reference a ring of six twofold atoms (with zero twist energy) bonded in a ring is still rigid. However, a fivefold ring is stressed (overconstrained) while a sevenfold ring is floppy. The bondbending force can be represented by nextnearestneighbor bars. The presence of these nextnearestneighbor bars eliminates configurations such as Fig. 9.2a which cause a violation of Laman's theorem in three dimensions. For this reason, bondbending networks in three dimensions can be studied using matching methods. Jacobs (1998) has developed a matching algorithm for this problem and it has been applied to glasses (Thorpe et al., 1999) and proteins in three dimensions (Jacobs et al., 1999). In chalcoginide glasses (e.g. SexGelx) it has been suggested (Phillips, 1979, 1981; Thorpe, 1983; Phillips and Thorpe, 1985) that the rigidity transition point is correlated with their glassforming ability and with a change in properties as a function of composition. The limiting cases, Se (which is twofold) and Ge (which is fourfold) are "floppy" and "rigid", respectively. The rigidity transition predicted by constraintcounting meanfield
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Fig. 9.14 An amino acid sequence starting at the Nterminus of a polypeptide chain. The side groups (residues) are denoted by R. (D. J. Jacobs and M. F. Thorpe, unpublished).
theory is x = 2.4. The actual transition is very close to this point, but although the meanfield and Bethelattice calculations indicate a firstorder transition, while the numerical analysis indicates that the transition is second order. The combinatorial algorithms provide unprecedented precision in testing the properties of chalcogenide glasses as a function of composition. Proteins and enzymes are often close to their rigidity threshold. Hydrogen bonds determine whether they are rigid or floppy and hydrogen bonding also determines much of the tertiary structure such as orhelices and /Jsheets. A systematic study of the effect of hydrogen bonding on the rigidity of biomolecules can be carried out using the rigidity algorithm. Typically, as hydrogen bonds are added the structure changes from floppy to rigid. An example of the rigid cluster structure of an important protein near the rigidity transition is demonstrated in Fig. 9.15. The practical consequences of this intriguing application of matching methods are currently being pursued (Jacobs et al., 1999).
10
Closing remarks
We have given an introduction to some very interesting algorithms which have wide applications in quenched disordered systems. There are several ways in which this area of research is attractive from an esthetic point of view. First, it is possible to find exact solutions to problems that seem pretty hard. That is always nice. Secondly, in this area there is very close contact between combinatorics, computer science and physics which makes the analysis of algorithms rigorous and complete, where possible. Finally, there is still a great deal to be done, both in finding and applying new techniques from the computer science community and also expanding the physics applications of the algorithms described here. Anyone studying a problem concerning the lowtemperature equilibrium
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Fig. 9.15 Rigid regions of HIV protease. Only the main chain is shown for clarity. There is a single large rigid region shown in black and various flexible pieces shown in alternating light and dark grey colors. Hydrogen bonds are shown as dashed lines. (D. J. Jacobs, A. J. Rader, L. A. Kuhn and M. E Thorpe (unpublished)).
phases of disordered systems should know about and should seriously consider using optimization methods. Many of the algorithms that are needed are standard packages which can be downloaded from the intemet (see Note 1 at the end of the References) (e.g the LEDA library). It used to be assumed that models of "nontrivial" glassy behavior required NPcomplete models such as spinglass models in three dimensions. But that is clearly not correct as the randommanifold, randomsurface and multipledirectedpolymer problems are "glassy" (e.g. chaotic) but they are solved by polynomial algorithms. It is thus important to search for polynomial models of other glassy problems. Even more importantly it would be useful to compare polynomially solvable models with those that are NPcomplete in order to identify the factors which make the problem computationally hard. Once those factors were identified it would be possible to decide if they are "relevant" from a physics perspective. It is likely that in some cases the features which make a problem computationally hard are not important physically, while in other, perhaps more interesting problems, the opposite would be true. An unexpected, to us, area of opportunity which results from the mappings described herein, and the much more extensive body of results in the mathematics
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literature (Lov~isz and Plummer, 1986; Ahuja et al., 1993a, b; van Lint and Wilson, 1996), is to develop new rigorous results for disordered systems. The flow and matching representations of disordered systems introduce a vast array of theorems and results, which are largely unused in the disorderedsystems community. These results form the basis of many theorems in combinatorics, number theory and in combinatorial optimization. It is likely they can also have a broad impact in rigorous and analytic approximations to the physics of disordered systems. Finally, it is likely that ideas generated by solving these physics problems will, in turn, affect the way computer scientists think about bounds on algorithmic convergence. For example in physics we know that proximity to a critical point strongly affects the convergence of many algorithms. It is then natural to ask whether computational time is sometimes a critical variable, with a singular behavior near critical points. Similarly the structure of the growth front during greedy algorithms should be related to computational efficiency. Several fruitful ideas along these lines are already in the literature (see Monasson et al., 1999).
Acknowledgments
MA thanks Eira Sepp~il~i for providing several figures and Attilio Stella and the Dipartimento di Fisica, Universita degli studi di Padova for their kind hospitality. V. Pet~ij~i and M. Stenlund are thanked for collaborations, the Academy of Finland for several grants, and the Nordic Research Institute for Theoretical Physics (NORDITA) for hospitability and support during 19961998. PMD thanks Bruce Hendrickson for stimulating conversations about graph algorithms, and Paul Leath, Sorin Bastea, Don Jacobs and Mike Thorpe for profitable discussions and collaborations. He thanks Jan Meinke for providing several figures and the DOE for support under contract DEFG0290ER45418. CM acknowledges financial support from FAPERJ. HR wishes to express his special thanks to U. B lasum, J. Kisker, F. Pfeiffer, G. Schrrder, T. Knetter, N. Kawashima, V. Pet~ij~i, M. Diehl, M. Jiinger, G. Reinelt, and G. Rinaldi for a very fruitful collaboration on various topics treated in this review. He is also grateful to the J. von Neumann Institute for Computing (NIC) c/o Forschungszentrum Jiilich and the Institut fur Theoretische Physik of the Universit~it zu Krln for generous hospitality and support. Moreover, he acknowledges important financial support from the Deutsche Forschungsgemeinschaft (DFG).
References
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internet.
Note 2 A similar decoupling happens in centralforce elastic systems which have zerolength springs. This particular case of "scalar elasticity" is for example relevant for the mechanical properties of polymers and rubbers, in which centralforces are of entropic origin and can in some cases be similar to zerolength springs.
Index ,circuit, 345 ,clusters, 44 ,path, 34 Abrikosov lattices, 2357, 238, 240, 261 Admissible arcs, 164, 165 Agreement percolation, 914, 10011 AizenmanHiguchi theorem, 98 AizenmanWehr theorem, 214 Algorithms, complexity of, 14950, 163 AlmostMarkov property s e e quasilocality Amenability, 38 Applied stress (AS) percolation, 284 AshkinTeller model, 104 Assignment problem, 179 Augmenting path algorithms, 163, 179 Augmenting path theorem, 162, 1767 Automorphisms of lattices, 8 Axial nextnearestneighbour Ising model (ANNNI), 197 s e e a l s o frustrated magnets Backbones, 152 BakSneppen evolution model, 195 Bernoulli percolation, 316, 424 bond percolation, 36, 478 site percolation, 326 s e e a l s o dependent percolation Bethe lattice s e e regular tree Biconnected paths, 152 Bipartite graphs, 1779 Bipartite matching algorithm, 265 Blossom, 180 BoltzmannGibbs distributions, 10 s e e a l s o Gibbs distributions, finite volume Bond diluted Ising model (BDIM), 113116, 204 Bond percolation, 36, 478 Boolean models, 124 Born model, 27980 Braced square net model (BSN), 2827 Bragg glasses, 235, 241,25861 Breadthfirst search (BFS) algorithm, 151 Burning algorithm, 152, 1889, 290 BurtonKeane uniqueness theorem, 434 Capacitated network, 160 Cayley tree s e e regular tree Chaos, 21617, 2301,24952, 259 Chemical distance, 152
Chinese postman problem, 225 Cluster algorithms, 679 Cluster size, 356 Cluster theorem, 276 Complete analyticity, 84 Configuration, 78 Connectivity percolation, 152, 18890, 28993 Connectivity percolation problem, 188 Connectivity theorem, 2712 Continuum percolation, 1235 Contour loops (CLs), 260 Convex cost problem, 16875 Convex cost programming, 1826 Counting condition, 272 Coupling, 213 optimal, 223 Coupling inequality, 22 Crandom cluster measures, 60, 625 Critical behaviour, 325, 37, 1956 Cutting plane algorithm, 2267 Cycle cancelling algorithm, 171 Dependent percolation, 3742 Ising model, 3941, 656 s e e a l s o Bernoulli percolation; randomcluster model Depth first search algorithm, 152 Dijkstra's algorithm, 1546, 172 Diluted antiferromagnets in a field (DAFF), 197, 20712, 21621 s e e a l s o randomfield Ising magnets (RFIM) Directed cycles, 154 Directed polymer in random media (DPRM), 187, 1905,2047, 234, 237, 2416 Disagreement percolation, 738, 1213 Disordered elastic media, 241,25264 Disordered flux arrays, 2354 1 Disordered systems, 11223 Distance labels, 153 DobrushinLanfordRuelle (DLR) states, 1011 s e e a l s o Gibbs measures Dobrushin's uniqueness condition, 778 Domain wall renormalization group (DWRG), 232, 248, 24950 Dual linear programming problem, 183 Edmonds' algorithm, 1801,233 Edmonds' theorem, 181 EdwardsAnderson model, 2213, 228
Index
EdwardsSokal coupling, 489, 53, 589 Elastic glasses, 2345, 23941,2525, 25864 Euclidean matching problem, 197, 2334 Exact simulation, 69 Exponentially weakmixing, 72, 77, 8491, 11623 Extremal dynamics, 1956 Flow algorithms, 15874 Flow augmentation lemma, 1734 Flow network, 1602 Flux lattice, 2356, 23941 Fluxline lattice s e e Abrikosov lattice Forest fire algorithm, 1889 FortuinKasteleynGinibre (FKG) inequality, 256 FortuinKasteleyn model s e e randomcluster model Free boundary condition, 523 Frustrated magnets, 197 Fully packed loops (FPLs), 2601 Gauge glasses s e e vortex glasses Gibbs distributions, finite volume, 12 Gibbs measures, 913 correlation decay, 72 extremal, 1314 infinite volume, 11, 1213 multiple, 1314 as phases, 14 structure of set of, 1314 uniqueness, 728 Gibbs systems, computer simulation of, 679 GinzburgLandau theory (GLtheory), 2367 Gluck's theorem, 268 Graph, locally finite, 6 Graphical distribution, 7982 Graph theory, terminology, 68, 1489 Greedy algorithms, 1501,155, 1956 Grey measure s e e graphical distribution Griffiths regime, 11617 Growth algorithms, for percolation, 189 Hamiltonian, 9 relative, 10 XY, 238 Hard core (lattice gas) model, 1820, 77, 108 agreement percolation, 103 stochastic domination, 301 Hendrickson's matching algorithm, 270 modified, 2769 Hendrickson's theorem, 273 Holley's inequality, 245 HoshenKopelman algorithm, 189 Hottestbond algorithm, 195 Imbalance, 171 Independent percolation s e e Bernoulli percolation Indicator function, 9 Infinite cluster, 325 Infinite clusters, number of, 427 Infinitesimal rigidity, 2668, 267 Integer linear programming (ILP), 182 Internal stress (IS) percolation, 284 Invasion algorithms, 18990 Invasion percolation, 69
319
Ising model antiferromagnetic, 1718, 301, 4755, 1078 agreement percolation, 1023 application of random cluster measures to percolation, 65 computer simulation, 679 dependent percolation, 401 ferromagnetic, 4, 1517, 559, 757, 878, 1067 agreement percolation, 94100 diluted, 115, 116 stochastic domination, 2630 random, 11723 s e e a l s o randombond magnets; randomfield Ising magnets (RFIM); random Ising magnets; triangular Ising solidonsolid model (TISOS) Ising spin glasses, 197, 22134 KardarParisiZhang (KPZ)equation, 187, 192, 194 Kruskal's algorithm, 158 Label correcting algorithms, 153 minimal path, 1567 Label setting algorithms, 153 Laman's theorem, 270, 28990, 2935 body Laman theorem, 271 Lattices, 68 Law of large numbers, 14 Leath algorithm, 189 Lily pond model, 124 Linear programming, 1813 Loops, 2601 Loops s e e contour loops (CLs); fully packed loops (FPLs) Markov chain Monte Carlo method, 68 Markov random field, 11 s e e a l s o quasilocality Matching algorithms, 17581,2726 Matching problems, history of, 175 Matchings, terms, 175 Maximumcardinality bipartitematching algorithm, 179 Maximumcut problem, 2257 Maximumflow problem, 1607, 199200, 20811, 2556 Maximum matching on general graphs, 17981 Maximumweight algorithm, 175 Maximumweight forest (MWF), 155 Maximumweightmatching problem, 179, 184 Minimalenergyinterface problem, 196 Minimal path, 1537 Minimalpath algorithm, 1925 Minimalpath problem, 1945 Minimalpath tree, 1534 Minimalspanning trees, 155, 1578 Minimumcostflow problem, 1679, 175, 183, 24852, 2545 Minimumcost path, 153 Minimumcut/maximumflow theorem, 1612 Minimumcut problem, 226 Minimumenergy interface problem, 196, 199200 Mixed integer linear programming (MILP), 1856
320 Monte Carlo method, 68 Negative cycle cancelling algorithm, 16970 Negative cycle cancelling theorem, 171 Nondeterministic polynomial (NP) algorithms, 150 Nondeterministic polynomialcomplete (NPcomplete) algorithms, 150 Observables, 89 quasilocal, 8 Open cluster, 32, 36 infinite, 32, 427 Open path, 32, 36 Order parameter, detection of phase transition, 13 Overlap, 217 Papangelou intensities, 1278 Percolation theory, 45 see also Bernoullis percolation Perfect simulation, 69 Periodic elastic media, 2345 Phase transitions, 13, 1314, 289, 55, 924, 10511, 11316, 1259 PirogovSinai theory, 5, 88, 923 Plaquettes, 10911 frustrated, 2225 Poisson blob model, 124 Poisson processes, 1235 Poisson random edge model, 1245 Polynomial (P) algorithms, 14950 Positive correlation, 25, 28, 47 Potential lemma, 173 Potts model, 18, 559, 108, 109 computer simulation, 679 constant spin clusters in, 1002 random, 11315 and random cluster model, 4759 stochastic domination, 30 Power law singularities, 116 Predecessor labels, 153 Preflow, 164 Preflowpustdrelabel algorithm, 1656 Preprocess procedure, 165 Primal linear programming problem, 1823 Prim's algorithm, I578, 190 Probability measures finite energy, 43 monotone, 25 with positive correlations, 25, 28, 47 weak topology of, 9 ProppWilson algorithm, 69 Pseudoflow, 171 Pseudopolynomial algorithms, 163 Pushrelabel algorithms, 1636 Quasilocality, 11 Quenched magnetization, 11415 Randombond magnets interfaces in, 197207 see also randomsurface model Randomcluster model, 4755, 5965, 702, 7884, 1002, 104, 11314, 119, 126
Index
Randomconnection model, 1245 Randomfield Ising magnets (RFIM), 1967, 20721,20727 see also Ising model Random fields, 9 Random Ising magnets, 11316, 196234 see also diluted antiferromagnets in a field (DAFF); Ising model Randomphase sineGordon model, 2394 1,2523 Randomsurface model, 241,2525, 2614 Randomsurface problem, 2556 Reduced cost optimality theorem, 173 Reduced costs, 153, 172 Regular trees, 6, 667 Residual costs, 168 Rigidity percolation, 27989 Rigidity theory, 26495 Roughness exponent, 200 Sandwiching inequality, 278, 61 Scaling algorithm, 166 Search algorithms, 1512 for percolation, 1889 Selforganized criticality, 18990 Sensitivity analysis, 166 Shortestpath algorithms, 1537 Singlesite heatbath algorithm, 689 Site percolation, 326, 427 Site randomcluster measure, 702 Solidonsolid model (SOS), 41,2525 continuum, 198 see also triangular Ising solidonsolid model (TISOS) Spanning clusters, 284 Spanning trees, 153 Spin glasses, 197, 22134 Stochastic domination, 21, 2331, 7884 Strassen's theorem, 234 Strong duality lemma, 183 Successive shortestpath algorithm, 163, 1714, 246 SugiharaHendrickson theorem, 274 Superconductors, 2354 1 Superrough phase, 2436, 25861 SwendsenWang algorithm, 69 Tail tralgebra, 8, 1314 Tay's theorem, 271 Transfer matrix algorithm, 1934, 200 Travelling salesman problem, 1856 Triangular Ising solidonsolid model (TISOS), 2568 see also solidonsolid model (SOS) Twolevel systems (TLS), 21820 Uniqueness theorem, 434, 769 Vortex glasses, 234, 2389, 2468 WidomRowlinson model continuum, 1259 lattice, 20, 30, 702, 77, 1034, 108 Wiredboundary condition, 523 XY spin glass, 239