Selected Works of Oded Schramm (Selected Works in Probability and Statistics)

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Author: Itai Benjamini | Olle Häggström

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Selected Works in Probability and Statistics

For other titles published in this series, go to www.springer.com/series/8556

Itai Benjamini · Olle Häggström Editors

Selected Works of Oded Schramm

123

Editors Itai Benjamini Weizmann Institute of Science Department of Mathematics Rehovot 76100, Israel [email protected]

Olle Häggström Department of Mathematical Statistics Chalmers University of Technology Göteborg, Sweden [email protected]

Printed in 2 volumes ISBN 978-1-4419-9674-9 e-ISBN 978-1-4419-9675-6 DOI 10.1007/978-1-4419-9675-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011931468 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface to the Series

Springer’s Selected Works in Probability and Statistics series offers scientists and scholars the opportunity of assembling and commenting upon major classical works in statistics, and honors the work of distinguished scholars in probability and statistics. Each volume contains the original papers, original commentary by experts on the subject’s papers, and relevant biographies and bibliographies. Springer is committed to maintaining the volumes in the series with free access of SpringerLink, as well as to the distribution of print volumes. The full text of the volumes is available on SpringerLink with the exception of a small number of articles for which links to their original publisher is included instead. These publishers have graciously agreed to make the articles freely available on their websites. The goal is maximum dissemination of this material. The subjects of the volumes have been selected by an editorial board consisting of Anirban DasGupta, Peter Hall, Jim Pitman, Michael Sörensen, and Jon Wellner.

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Preface

Oded Schramm was born on December 10, 1961, in Jerusalem, and died at the age of 46 in a climbing accident on Guye Peak, WA, on September 1, 2008. In between, he made profound and beautiful contributions to mathematics that will have a lasting influence. In these two volumes, we have collected some of his papers, supplemented with three survey papers by Steffen Rohde, Olle Häggström and Cristophe Garban that further elucidate his work. Despite the seemingly generous size of the collection, spatial considerations neverthelss forced us to omit most of Oded’s papers, and the mere fact that all of them are inspiring pieces of work led to some difficult issues on what to include and what to omit. The reader should not view our choices as an attempt to separate his best works from those that are merely great. Rather, we have tried to put together a representative collection that shows the breadth, depth, enthusiasm and clarity of his work. Others may have diverging opinions on what should or should not have been included, but we do hope that Oded himself would not have been too displeased by our choices. The papers we have included speak for themselves; let us just say a few words about how we have organized them into five sections. Oded began his mathematical career as a geometer, making GEOMETRY the natural topic for Section 1. This section opens with Oded’s first two papers from his Master’s thesis in 1987 under Gil Kalai at the Hebrew University. We then move on to circle packing and conformal geometry, including examples of his extraordinarily fruitful collaboration with Zheng-Xu Hu, and end the section with the joint paper with Mario Bonk on embeddings of hyperbolic spaces. Of course geometric aspects permeate also all of the following sections. In fact, Oded once mentioned to one of us that in order to be able to think about a problem he always liked it to have a geometric component. In the mid 1990’s, Oded became interested in the topic of probability theory, which dominates Sections 2-5 of this collection. Section 2 deals in particular with the study of NOISE SENSITIVITY, pioneered in a joint paper with Itai Benjamini and Gil Kalai that opens the section. Noise sensitivity turned out to be a rich topic with applications ranging from voting systems to percolation. In the two other papers of this section, the first coauthored with Jeff Steif and the second with Christophe Garban and Gabor Pete, Oded went progressively deeper into noise sensitivity in a percolation setting, arriving at surprisingly detailed insights. In Section 3 we have collected some of Oded’s papers on RANDOM WALKS AND GRAPH LIMITS. This includes (i) a paper with Itai Benjamini establishing recurrence of random walks on suitably defined limits of finite planar graphs, (ii) the joint paper with his student Omer Angel on distributional limits of triangulations, (iii) a singly-authored paper on compositions of random transpositions, (iv) a collaboration with Yuval Peres, Scott Sheffield and David Wilson on the remarkably fruitful interplay between the infinity Laplacian and certain board games, and (v) the concise 2008 paper on so called hyperfinite graph limits. One of the probabilistic objects that caught the strongest grip on Oded’s imagination was PERCOLATION, which appeared prominently in Section 2 as a major testing ground for noise sensitivity. Papers on other aspects of percolation are collected in Section 4. Here we will see how Oded joined forces with Itai Benjamini, Russ Lyons and Yuval Peres in order to uncover many of the new vii

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and interesting phenomena that happen when we move beyond the usual setting of percolation in a Euclidean geometry, and instead study what happens on hyperbolic lattices and other nonamenable graph structures. Most of the papers in this section are joint work with (subsets of) this team of coauthors, plus Harry Kesten who joined them in establishing a beautiful result on uniform spanning trees. We end the section on a different note, namely Oded’s joint paper with Itai Benjamini and Gil Kalai making progress on the important open problem of determining the order of magnitude of fluctuations of first passage percolation on the Euclidean lattice. Despite strong competition from Oded’s other works, there seems to be a consensus view that the most important of all his contributions to mathematics is his discovery and subsequent study of SCHRAMM-LOEWNER EVOLUTION (or stochastic Loewner avolution as Oded himself preferred to call it; conveniently, the abbreviation SLE works either way), which is the topic of Section 5. SLE is a family of conformally invariant random processes in the plane that turn out to appear as the scaling limit of percolation and a variety of other critical models. We begin this section with the famous paper, published in 2000 in the Israel Journal of Mathematics, where Schramm singlehandedly discovered SLE and obtained the first preliminary results on scaling limits. Then followed a series of papers with Greg Lawler and Wendelin Werner in which SLE was exploited to deduce deep results on intersection properties of random walks; here we include only some of the highlights. We furthermore include important joint papers with Steffen Rohde, Scott Sheffield, David Wilson and Stanislav Smirnov, plus Oded’s contribution to the International Congress of Mathematicians in Madrid, 2006, in which he gives a survey of the field with an emphasis on open problems. There were no signs of a decrease in creativity or productivity on Oded’s part until the untimely and tragic end, and we can only guess what further discoveries we miss because of it. Substantial parts of his work, joint with others, is still not completely written up and will appear in the coming years. However, Oded will be missed not just because of his mathematics, but even more because of the gentle, warm-hearted and generous person that he was. Of course, the loss of him is felt most strongly by his wife Avivit, his daughter Tselil and his son Pele. But he was very much loved by the mathematical community and by everyone who knew him, as amply witnessed on the memorial blog which was set up shortly after his death: http://odedschramm.wordpress.com/ We hope that this collection will contribute, however modestly, to keeping the memory of Oded alive, and to nourishing the mathematical heritage he left for all of us. Yehi zichro baruch - may the memory of him be a blessing. Itai Benjamini Olle Häggström

Acknowledgements

This series of selected works is possible only because of the efforts and cooperation of many people, societies, and publishers. The series editors originated the series and directed its development. The volume editors spent a great deal of time organizing the volumes and compiling the previously published material. The contributors provided comments on the significance of the papers. The societies and publishers who own the copyright to the original material made the volumes possible and affordable by their generous cooperation: American Institute of Mathematical Sciences American Mathematical Society Duke University Press Hebrew University Magnes Press Institute of Mathematical Statistics Institute of Mathematical Studies Institut des Hautes Ètudes Scientifiques International Mathematical Union The London Mathematical Society Princeton University and the Institute for Advanced Studies Springer Science+Business Media Yuval Peres

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Contents

Volume 1 Part 1: Geometry Commentary: Oded Schramm: From circle packing to SLE, by Steffen Rohde. To appear in Ann. Probab. Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

O. Schramm. Illuminating sets of constant width. Mathematika Vol. 35 No. 2, 180–189 (1988). Reprinted with permission of The London Mathematical Society . . . . . . . . . . . . . .

47

O. Schramm. On the volume of sets having constant width. Israel J. Math. Vol. 63 No. 2, 178–182 (1988). Reprinted with permission of Hebrew University Magnes Press . . . . . . .

57

O. Schramm. Rigidity of infinite (circle) packings. J. Amer. Math. Soc. Vol. 4 No. 1, 127–149 (1991). Reprinted with permission of the American Mathematical Society . . . . .

63

O. Schramm. How to cage an egg. Invent. Math. Vol. 107 No. 3, 543–560 (1992). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . .

87

Zheng-Xu He and O. Schramm. Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2) Vol. 137 No. 2, 369–406 (1993). Reprinted with permission of Princeton University and the Institute for Advanced Study . . . . . . . . . . . . . . . . . . . . . . .

105

Zheng-Xu He and O. Schramm. Hyperbolic and parabolic packings. Discrete Comput. Geom. Vol. 14 No. 2, 123–149 (1995). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

O. Schramm. Circle patterns with the combinatorics of the square grid. Duke Math. J. Vol. 86 No. 2, 347–389 (1997). Reprinted with permission of Duke University Press. An electronic version is available at doi: 10.1215/S0012-7094-97-08611-7 . . . . . . . . . . . . . . . . .

171

Zheng-Xu He and O. Schramm. The C∞-convergence of hexagonal disk packings to the Riemann map. Acta Math. Vol. 180 No. 2, 219–245 (1998). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

M. Bonk and O. Schramm. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. Vol. 10 No. 2, 266–306 (2000). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part 2: Noise Sensitivity Commentary: Oded Schramm’s contributions to noise sensitivity, by Christophe Garban. To appear in Ann. Probab. Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287

I. Benjamini, G. Kalai, and O. Schramm. Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. No. 90, 5–43 (1999). Reprinted with permission of Institut Des Hautes Études Scientifiques . . . . . . . . . . . . . . .

351

O. Schramm and J. E. Steif. Quantitative noise sensitivity and exceptional times for percolation. Annals of Mathematics, Pages 619–672 from Volume 171 (2010), Issue 2. Reprinted with permission of Princeton University and the Institute for Advanced Study

391

C. Garban, G. Pete, and O. Schramm. The Fourier spectrum of critical percolation. Acta Math. 205 (2010), 19–104. Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

445

Part 3: Random Walks and Graph Limits I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. Vol. 6, No. 23, 13 pp. (electronic) (2001). Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

533

O. Angel and O. Schramm. Uniform infinite planar triangulations. Comm. Math. Phys. Vol. 241 No. 2–3, 191–213 (2003). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

547

O. Schramm. Compositions of random transpositions. Israel J. Math. Vol. 147, 221–243 (2005). Reprinted with permission of Hebrew University Magnes Press . . . . . . . . . . . . . . .

571

Y. Peres, O. Schramm, S. Sheffield, and D. B. Wilson. Tug-of-war and the infinity Laplacian. Journal of the American Mathematical Society 22(1):167–210, (2009). Reprinted with permission of Yuval Peres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

595

O. Schramm. Hyperfinite graph limits. Electron. Res. Announc. Math. Sci. Vol. 15, 17–23 (2008). Reprinted with permission of the American Institute of Mathematical Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

639

Volume 2 Part 4: Percolation Commentary: Percolation beyond Zd : The contributions of Oded Schramm, by Olle Häggström. To appear in Ann. Probab. Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

649

I. Benjamini and O. Schramm. Percolation beyond Zd , many questions and a few answers. Electron. Comm. Probab. Vol. 1, No. 8, 71–82 (electronic) (1996). Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . .

679

I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. Vol. 27 No. 3, 1347–1356 (1999). Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . .

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R. Lyons and O. Schramm. Indistinguishability of percolation clusters. Ann. Probab. Vol. 27 No. 4, 1809–1836 (1999). Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

701

I. Benjamini and O. Schramm. Percolation in the hyperbolic plane. J. Amer. Math. Soc. Vol. 14 No. 2, 487–507 (electronic) (2001). Reprinted with permission of the American Mathematical Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

729

I. Benjamini, H. Kesten, Y. Peres, and O. Schramm. Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12, ... Ann. of Math. (2) Vol. 160 No. 2, 465–491 (2004). Reprinted with permission of Princeton University and the Institute for Advanced Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

751

I. Benjamini, G. Kalai, and O. Schramm. First passage percolation has sublinear distance variance. Ann. Probab. Vol. 31 No. 4, 1970–1978 (2003). Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

779

Part 5: Schramm-Loewner Evolution O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. Vol. 118, 221–288 (2000). Reprinted with permission of Hebrew University Magnes Press . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

791

G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. Vol. 187 No. 2, 237–273 (2001). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

859

G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents II: Plane exponents. Acta Math. Vol. 187 No. 2, 275–308 (2001). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

897

G. F. Lawler, O. Schramm, and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. Vol. 32 No. 1B, 939–995 (2004). Reprinted with permission of the Institute of Mathematical Statistics . . . . . . . . . .

931

S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2) Vol. 161 No. 2, 883–924 (2005). Reprinted with permission of Princeton University and the Institute for Advanced Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

989

O. Schramm and S. Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Mathematica, Vol. 202 No. 1, 21–137 (2009). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1031

O. Schramm, S. Sheffield, and D. B. Wilson. Conformal radii for conformal loop ensembles. Commun.Math.Phys.288: 43–53, (2009). Reprinted with permission of Springer Science+Business Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1149

O. Schramm. Conformally invariant scaling limits: an overview and a collection of problems. In International Congress of Mathematicians. Vol. I, 513–543, Eur. Math. Soc., Zürich (2007). Reprinted with permission of the International Mathematical Union

1161

O. Schramm and S. Smirnov. On the scaling limits of planar percolation. To appear in Ann. Probab. Reprinted with permission of the Institute of Mathematical Statistics . . . .

1193

Contributors

Itai Benjamini The Weizmann Institute of Science, Rehovot POB 76100, Israel, [email protected] Cristophe Garban Département de mathématiques, CNRS, 46, allée d’Italie, 69364 Lyon Cedex 07, France, [email protected] Olle Häggström Mathematical Sciences, Chalmers University of Technology, 412 96 Göteborg, Sweden, [email protected] Steffen Rohde University of Washington, Department of Mathematics, C-337 Padelford Hall, Box 354350 Seattle, WA 98195-4350, United States, [email protected]

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Author Bibliography

Oded Schramm’s publications, as of January 2011 *O. Schramm. Illuminating sets of constant width. Mathematika Vol. 35 No. 2, 180–189 (1988). *O. Schramm. On the volume of sets having constant width. Israel J. Math. Vol. 63 No. 2, 178–182 (1988). O. Schramm. Packing two dimensional bodies with prescribed combinatorics and applications to the construction of conformal and quasiconformal mappings. Ph. D. thesis. Princeton University (1990). *O. Schramm. Rigidity of infinite (circle) packings. J. Amer. Math. Soc. Vol. 4 No. 1, 127–149 (1991). O. Schramm. Existence and uniqueness of packings with specified combinatorics. Israel J. Math. Vol. 73 No. 3, 321–341 (1991). *O. Schramm. How to cage an egg. Invent. Math. Vol. 107 No. 3, 543–560 (1992). O. Schramm. Square tilings with prescribed combinatorics. Israel J. Math. Vol. 84 No. 1–2, 97–118 (1993). *Zheng-Xu He and O. Schramm. Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2) Vol. 137 No. 2, 369–406 (1993). G. Kuperberg and O. Schramm. Average kissing numbers for non-congruent sphere packings. Math. Res. Lett. Vol. 1 No. 3, 339–344 (1994). Zheng-Xu He and O. Schramm. Rigidity of circle domains whose boundary has σ -finite linear measure. Invent. Math. Vol. 115 No. 2, 297–310 (1994). Zheng-Xu He and O. Schramm. Koebe uniformization for “almost circle domains”. Amer. J. Math. Vol. 117 No. 3, 653–667 (1995). *Zheng-Xu He and O. Schramm. Hyperbolic and parabolic packings. Discrete Comput. Geom. Vol. 14 No. 2, 123–149 (1995). Zheng-Xu He and O. Schramm. The inverse Riemann mapping theorem for relative circle domains. Pacific J. Math. Vol. 171 No. 1, 157–165 (1995). O. Schramm. Transboundary extremal length. J. Anal. Math. Vol. 66, 307–329 (1995). O. Schramm. Conformal uniformization and packings. Israel J. Math. Vol. 93, 399–428 (1996). Z. Reich, O. Schramm, V. Brumfeld, and A. Minsky. Chiral discrimination in DNA-peptide interactions involving chiral DNA mesophases: a geometric analysis. Jour. Amer. Chem. Soc. Vol. 118, 6345–6349 (1996). Zheng-Xu He and O. Schramm. On the convergence of circle packings to the Riemann map. Invent. Math. Vol. 125 No. 2, 285–305 (1996).

∗ Indicates

article found in Vol.1

+ Indicates

article found in Vol.2

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Author Bibliography

+ I. Benjamini and O. Schramm. Percolation beyond Zd, many questions and a few answers. Electron.

Comm. Probab. Vol. 1, No. 8, 71–82 (electronic) (1996). I. Benjamini and O. Schramm. Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. Vol. 126 No. 3, 565–587 (1996). I. Benjamini and O. Schramm. Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. Vol. 24 No. 3, 1219–1238 (1996). *O. Schramm. Circle patterns with the combinatorics of the square grid. Duke Math. J. Vol. 86 No. 2, 347–389 (1997). I. Benjamini and O. Schramm. Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal. Vol. 7 No. 3, 403–419 (1997). Zheng-Xu He and O. Schramm. On the distortion of relative circle domain isomorphisms. J. Anal. Math. Vol. 73, 115–131 (1997). *Zheng-Xu He and O. Schramm. The C∞-convergence of hexagonal disk packings to the Riemann map. Acta Math. Vol. 180 No. 2, 219–245 (1998). I. Benjamini and O. Schramm. Exceptional planes of percolation. Probab. Theory Related Fields Vol. 111 No. 4, 551–564 (1998). I. Benjamini and O. Schramm. Conformal invariance of Voronoi percolation. Comm. Math. Phys. Vol. 197 No. 1, 75–107 (1998). I. Benjamini and O. Schramm. Recent progress on percolation beyond Zd. (1998). revised Jan 18 2000 [Web paper] I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Group-invariant percolation on graphs. Geom. Funct. Anal. Vol. 9 No. 1, 29–66 (1999). R. Lyons and O. Schramm. Stationary measures for random walks in a random environment with random scenery. New York J. Math. Vol. 5, 107–113 (electronic) (1999). O. Schramm and B. Tsirelson. Trees, not cubes: hypercontractivity, cosiness, and noise stability. Electron. Comm. Probab. Vol. 4, 39–49 (electronic) (1999). + I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. Vol. 27 No. 3, 1347–1356 (1999). I. Benjamini, R. Lyons, and O. Schramm. Percolation perturbations in potential theory and random walks. In Random walks and discrete potential theory (Cortona, 1997), Sympos. Math., XXXIX, 56–84, Cambridge Univ. Press, Cambridge (1999). + R. Lyons and O. Schramm. Indistinguishability of percolation clusters. Ann. Probab. Vol. 27 No. 4, 1809–1836 (1999). *I. Benjamini, G. Kalai, and O. Schramm. Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. No. 90, 5–43 (1999). I. Benjamini, O. Häggström, and O. Schramm. On the effect of adding ε-Bernoulli percolation to everywhere percolating subgraphs of Zd. J. Math. Phys. Vol. 41 No. 3, 1294–1297 (2000). *M. Bonk and O. Schramm. Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. Vol. 10 No. 2, 266–306 (2000). + O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. Vol. 118, 221–288 (2000). J. Jonasson and O. Schramm. On the cover time of planar graphs. Electron. Comm. Probab. Vol. 5, 85–90 (electronic) (2000). + I. Benjamini and O. Schramm. Percolation in the hyperbolic plane. J. Amer. Math. Soc. Vol. 14 No. 2, 487–507 (electronic) (2001). I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Uniform spanning forests. Ann. Probab. Vol. 29 No. 1, 1–65 (2001). + G.F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. Vol. 187 No. 2, 237–273 (2001).

Author Bibliography + G.F.

xix

Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents II: Plane exponents. Acta Math. Vol. 187 No. 2, 275–308 (2001). G.F. Lawler, O. Schramm, and W. Werner. The dimension of the planar Brownian frontier is 4/3. Math. Res. Lett. Vol. 8 No. 4, 401–411 (2001). *I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. Vol. 6, No. 23, 13 pp. (electronic) (2001). O. Schramm. A percolation formula. Electron. Comm. Probab. Vol. 6, 115–120 (electronic) (2001). G.F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents III: Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. Vol. 38 No. 1, 109–123 (2002). G.F. Lawler, O. Schramm, and W. Werner. Analyticity of intersection exponents for planar Brownian motion. Acta Math. Vol. 189 No. 2, 179–201 (2002). G.F. Lawler, O. Schramm, and W. Werner. Sharp estimates for Brownian non-intersection probabilities. In In and out of equilibrium (Mambucaba, 2000), Progr. Probab. Vol. 51, 113–131, Birkhäuser Boston, Boston, MA (2002). G.F. Lawler, O. Schramm, and W. Werner. One-arm exponent for critical 2D percolation Electron. J. Probab. Vol. 7, No. 2, 13 pp. (electronic) (2002). R. Lyons, Y. Peres, and O. Schramm. Markov chain intersections and the loop-erased walk Ann. Inst. H. Poincaré Probab. Statist. Vol. 39 No. 5, 779–791 (2003). + I. Benjamini, G. Kalai, and O. Schramm. First passage percolation has sublinear distance variance. Ann. Probab. Vol. 31 No. 4, 1970–1978 (2003). *O. Angel and O. Schramm. Uniform infinite planar triangulations. Comm. Math. Phys. Vol. 241 No. 2–3, 191–213 (2003). G.F. Lawler, O. Schramm, and W. Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc. Vol. 16 No. 4, 917–955 (electronic) (2003). + I. Benjamini, H. Kesten, Y. Peres, and O. Schramm. Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12, ... Ann. of Math. (2) Vol. 160 No. 2, 465–491 (2004). + G.F. Lawler, O. Schramm, and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. Vol. 32 No. 1B, 939–995 (2004). G.F. Lawler, O. Schramm, and W. Werner. On the scaling limit of planar self-avoiding walk. In Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math. Vol. 72, 339–364, Amer. Math. Soc., Providence, RI (2004). I. Benjamini, S. Merenkov, and O. Schramm. A negative answer to Nevanlinna’s type question and a parabolic surface with a lot of negative curvature. Proc. Amer. Math. Soc. Vol. 132 No. 3, 641–647 (electronic) (2004). I. Benjamini and O. Schramm. Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality. In Geometric aspects of functional analysis, Lecture Notes in Math. Vol. 1850, 73–76, Springer, Berlin (2004). + S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2) Vol. 161 No. 2, 883–924 (2005). O. Schramm and S. Sheffield. Harmonic explorer and its convergence to SLE(4). Ann. Probab. Vol. 33 No. 6, 2127–2148 (2005). *O. Schramm. Compositions of random transpositions. Israel J. Math. Vol. 147, 221–243 (2005). I. Benjamini, O. Schramm, and D.B.Wilson. Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read. In STOC’05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 244–250, ACM, New York (2005). O. Schramm. Emergence of Symmetry: Conformal invariance in scaling limits of random systems. In European congress of mathematics (ECM), Stockholm, Sweden, June 27–July 2, 2004, Ari Laptev editor, 783–786, European Mathematical Society, Zurich (2005). O. Schramm and D.B. Wilson. SLE coordinate changes. New York J. Math. Vol. 11 :659–669, (2005).

xx

Author Bibliography

R. O’Donnell, M. Saks, O. Schramm, and R. Servedio. Every decision tree has an influential variable. In Proceedings of the 46th Annual Symposium on Foundations of Computer Science (FOCS) (2005). A. Naor, Y. Peres, O. Schramm, and S. Sheffield. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J. Vol. 134 No. 1, 165–197 (2006). R. Lyons, Y. Peres, and O. Schramm. Minimal spanning forests. Ann. Probab. Vol. 34 No. 5, 1665– 1692 (2006). Y. Peres, O. Schramm, S. Sheffield, and D.B. Wilson. Random-turn hex and other selection games. Amer. Math. Monthly Vol. 114 No. 5, 373–387 (2007). + O. Schramm. Conformally invariant scaling limits: an overview and a collection of problems. In International Congress of Mathematicians. Vol. I, 513–543, Eur. Math. Soc., Zürich (2007). R. Lyons, B.J. Morris, and O. Schramm. Ends in uniform spanning forests. Electron. J. Probab. 13 Paper 58 (2008), 1701–1725. R. Lyons, R. Peled, and O. Schramm. Growth of the number of spanning trees of the Erd˝os-Rényi giant component. Combinatorics, Probability and Computing,Volume 17 Issue 5, 711–726, (2008). I. Benjamini, O. Schramm, and A. Shapira. Every minor-closed property of sparse graphs is testable. Proceedings of the 40th annual ACM symposium on Theory of computing, May 17–20, 2008, Victoria, British Columbia, Canada. *O. Schramm. Hyperfinite graph limits. Electron. Res. Announc. Math. Sci. Vol. 15, 17–23 (2008). *Y. Peres, O. Schramm, S. Sheffield, and D.B. Wilson. Tug-of-war and the infinity Laplacian. Journal of the American Mathematical Society 22(1):167–210, (2009). + O. Schramm and S. Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Mathematica, Volume 202, Number 1, 21–137, (2009). + O. Schramm, S. Sheffield, and D.B. Wilson. Conformal radii for conformal loop ensembles Commun.Math.Phys. 288:43–53, (2009). Y. Peres, O. Schramm, and J.E. Steif. Dynamical sensitivity of the infinite cluster in critical percolation. Ann. Inst. H. Poincaré Probab. Statist. Volume 45, Number 2 (2009), 491–514. O. Schramm and W. Zhou. Boundary proximity of SLE. Probability Theory and Related Fields Volume 146, Numbers 3–4, 435–450, (2009). A.E. Holroyd, R. Pemantle, Y. Peres, and O. Schramm. Poisson matching. Ann. Inst. Henri Poincaré Probab. Stat., 45(1):266–287, (2009). *C. Garban, G. Pete, and O. Schramm. The Fourier spectrum of critical percolation. Acta Math. 205 (2010), 19–104. I. Benjamini and O. Schramm. KPZ in one dimensional random geometry of multiplicative cascades. Commun. Math. Phys, 289:46–56 (2009). I. Benjamini, J. Jonasson, O. Schramm, and J. Tykesson.Visibility to infinity in the hyperbolic plane, despite obstacles. ALEA Lat. Am. J. Probab. Math. Stat., Vol. 6 232–343 (2009). *O. Schramm and J.E. Steif. Quantitative noise sensitivity and exceptional times for percolation. Annals of Mathematics, Pages 619–672 from Volume 171 (2010), Issue 2. C. Garban, G. Pete, and O. Schramm. The scaling limit of the minimal spanning tree - a preliminary report. XVIth International Congress on Mathematical Physics (Prague, 2009), ed. P. Exner, pp. 475–480., World Scientific, Singapore, 2010. I. Benjamini, O. Schramm, and S. Sodin. Poisson asymptotics for random projections of points on a high-dimensional sphere. Israel J. Math., to appear. + O. Schramm and S. Smirnov. On the scaling limits of planar percolation. Ann. Probab., to appear. C. Garban, S. Rohde, and O. Schramm. Continuity of the SLE trace in simply connected domains. arXiv:0810.4327 I. Benjamini, O. Gurel-Gurevich, O. Schramm. Cutpoints and resistance of random walk paths. Ann. Probab., to appear.

Author Bibliography

xxi

I. Benjamini and O. Schramm. Lack of sphere packing of graphs via non-linear potential theory. Jour. of Topology and Analysis, to appear. N. Berestycki, O. Schramm, and O. Zeitouni. Mixing times for random k-cycles and coalescencefragmentation chains. Ann. Probab., to appear. I. Benjamini, O. Schramm, and A. Timar. Separation. Groups, Geometry and Dynamics, to appear. C. Garban, G. Pete, and O. Schramm. Pivotal, cluster and interface measures for critical planar percolation. arXiv:1008.1378 O. Schramm and S. Sheffield. A contour line of the continuum Gaussian free field. arXiv:1008.2447

Part I

Geometry

Oded Schramm: From Circle Packing to SLE Steffen Rohde* July 12, 2010

Contents 1

Introduction

2

2

Circle Packing and the Koebe Conjecture 2.1 Background . . . . . . . . . . . . . . 2.2 Why are Circle Packings interesting? 2.3 Existence of Packings . . . . . . . . 2.4 Uniqueness of Packings . . . . . . . . 2.4.1 Extremal length and the conformal modulus of a quadrilateral 2.4.2 The Incompatibility Theorem 2.4.3 A simple uniqueness proof. 2.5 Koebe's Kreisnormierungsproblem 2.6 Convergence to conformal maps. 2.7 Other topics . . . . . . . . . . . .

3 3 5 7 9

3

The Schramm-Loewner Evolution 3.1 Pre-history . . . . . . . . . . . . 3.2 Definition of SLE . . . . . . . . . 3.2.1 The (radial) Loewner equation 3.2.2 The scaling limit of LERW .. 3.2.3 The chordal Loewner equation, percolation, and the UST 3.3 Properties and applications of SLE . . . . . . 3.3.1 Locality . . . . . . . . . . . . . . . . . 3.3.2 Intersection exponents and dimensions 3.3.3 Path properties . . . . . . . . . . . . . 3.3.4 Discrete processes converging to SLE . 3.3.5 Restriction measures 3.3.6 Other results 3.4 Problems . . . . . . . . . . 3.5 Conclusion . . . . . . . . . 'University of Washington, Supported in part by NSF Grant DMS-OS0096S.

1

I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_1, 3

10

11 13 13

14 16 18 18

19 19 21 23 25 25 27

29 31 33 35 36 37

1

Introduction

When I first met Oded Schramm in January 1991 at the University of California, San Diego, he introduced himself as a "Circle Packer". This modest description referred to his Ph.D. thesis around the Koebe-Andreev-Thurston theorem and a discrete version of the Riemann mapping theorem, explained below. In a series of highly original papers, some joint with Zhen-Xu He, he created powerful new tools out of thin air, and provided the field with elegant new ideas. At the time of his deadly accident on September 1st, 2008, he was widely considered as one of the most innovative and influential probabilists of his time. Undoubtedly, he is best known for his invention of what is now called the Schramm-Loewner Evolution (SLE), and for his subsequent collaboration with Greg Lawler and Wendelin Werner that led to such celebrated results as a determination of the intersection exponents of two-dimensional Brownian motion and a proof of Mandelbrot's conjecture about the Hausdorff dimension of the Brownian frontier. But already his previous work bears witness to the brilliance of his mind, and many of his early papers contain both deep and beautifully simple ideas that deserve better knowing. In this note, I will describe some highlights of his work in circle packings and the Koebe conjecture, as well as on SLE. As Oded has co-authored close to 20 papers related to circle packings and more than 20 papers involving SLE, only a fraction can be discussed in detail here. The transition from circle packing to SLE was through a long sequence of influential papers concerning probability on graphs, many of them written jointly with Itai Benjamini. I will present almost no work from that period (some of these results are described elsewhere in this volume, for instance in Christophe Garban's article on Noise Sensitivity). In that respect, the title of this note is perhaps misleading. In order to avoid getting lost in technicalities, arguments will be sketched at best, and often ideas of proofs will be illustrated by analogies only. In an attempt to present the evolution of Oded's mathematics, I will describe his work in essentially chronological order. Oded was a truly exceptional person: not only was his clear and innovative way of thinking an inspiration to everyone who knew him, but also his caring, modest and relaxed attitude generated a comfortable atmosphere. As inappropriate as it might be, I have included some personal anecdotes as well as a few quotes from email exchanges with Oded, in order to at least hint at these sides of Oded that are not visible in the published literature. This note is not meant to be an overview article about circle packings or SLE. My prime concern is to give a somewhat self-contained account of Oded's contributions. Since SLE has been featured in several excellent articles and even a book, but most of Oded's work on circle packing is accessible only through his original papers, the first part is a bit more expository and contains more background. The expert in either field will find nothing new, and will find a very incomplete list of references. My apology to everyone whose contribution is either unmentioned or, perhaps even worse, mentioned without proper reference. Acknowledgement: I would like to thank Mario Bonk, Jose Fernandez, Jim Gill, Joan Lind, Don Marshall, Wendelin Werner and Michel Zinsmeister for helpful comments on a first draft. I would also like to thank Andrey Mishchenko for generating Figure 3, and Don Marshall for Figure 6.

2

4

2

Circle Packing and the Koebe Conjecture

Oded Schramm was able to create, seemingly without effort, ingenious new ideas and methods. Indeed, he would be more likely to invent a new approach than to search the literature for an existing one. In this way, in addition to proving wonderful new theorems, he rediscovered many known results, often with completely new proofs. We will see many examples throughout this note. Oded received his Ph.D. in 1990 under William Thurston's direction at Princeton. His thesis, and the majority of his work until the mid 90's, was concerned with the fascinating topic of circle packings. Let us begin with some background and a very brief overview of some highlights of this field prior to Oded's thesis. Other surveys are [Sa] and [Ste2].

2.1

Background

According to the Riemann mapping theorem, every simply connected planar domain, except the plane itself, is conformally equivalent to a disc. The conformal map to the disc is unique, up to post composition with an automorphism of the disc (which is a Mobius transformation). The standard proof exhibits the map as a solution of an extremal problem (among all maps of the domain into the disc, maximize the derivative at a given point). The situation is quite different for multiply connected domains, partly due to the lack of a standard target domain. The standard proof can be modified to yield a conformal map onto a parallel slit domain (each complementary component is a horizontal line segment or a point). Koebe showed that every finitely connected domain is conformally equivalent to a circle domain (every boundary component is a circle or a point), in an essentially unique way. No proof similar to the standard proof of the Riemann mapping theorem is known.

n c IC with finitely many connected boundary components, there is a conformal map f onto a domain n' c IC all of whose boundary components are circles or points. Both f and n' are unique up to a Mobius transformation.

Theorem 2.1 ([K01]). For every domain

Koebe conjectured (p. 358 of [K01]) that the same is true for infinitely connected domains. It later turned out that uniqueness of the circle domain can fail (for instance, it fails whenever the set of point-components of the boundary has positive area, as a simple application of the measurable Riemann mapping theorem shows). But existence of a conformally equivalent circle domain is still open, and is known as Koebe's conjecture or "Kreisnormierungsproblem". It motivated a lot of Oded's research. There is a close connection between Koebe's theorem and circle packings. A circle packing P is a collection (finite or infinite) of closed discs D in the two dimensional plane IC, or in the two dimensional sphere 8 2 , with disjoint interiors. Associated with a circle packing is its tangency graph or nerve G = (V, E), whose vertices correspond to the discs, and such that two vertices are joined by an edge if and only if the corresponding discs are tangent. We will only consider packings whose tangency graph is connected. Conversely, the Koebe-Andreev-Thurston Circle Packing Theorem guarantees the existence of packings with prescribed combinatorics. Loosely speaking, a planar graph is a graph that can be drawn in the plane so that edges do not cross. Our graphs will not have double edges (two edges with the same endpoints) or loops (an edge whose endpoints coincide). 3

5

,

b

, f g

Figul'c 1: A circle packing and its tangency graph .

T heore m 2.2 ([Ko2], [T], [AI]). For every finite planar gruph G , there is a circle packing in the plane with nerve G. The lJacking is unique (up to Mobius tmns/orlnations) jfG is a triangulation of 52.

See the following seetions for t he history of this theorem, and sketches of proofs. In particular, in Section 2.3 we wi!] indicate how the Circle Packing T heorem 2.2 call be obtained from t.he Koebe Theorem 2.1 , and COll\'crsc]y that the Kocbc theorem call be deduced from t. he Circle Packing Theorem. Eyery fillite planar gra ph can be extended (by adding vertices and edges as in Figure 3(c)) to a triangulation, hCIU.;e pad:(,\bility of triangulations implies paclmbility of finite planar graphs (there arc many ways to extend a graph to a triangulation, and uniqueness orthe packing is no longer true). T he situation is more complicated for infinite graphs. Oded wrote scyeral papers dealing with this case. T hurston conjectured that circle packings approximate conformal maps, in the following scnsc: Consider the hexagollal packing H~ of circles of radius € (a portion is visible in Fig. 2 and Fig. 3(11)). Let fl c C be a domain (a connes of fl n He, as in Fig. 2 and Fig. 3(a) (more precisely, take the connected component containing p of the union of those circles whose six neighbors are still contained in fl). Complete the nerve of this packing by adding one vertex for each connected component of the complement to obtain a triangulation of the sphere (there arc three new vertices VI , V2 , V:J in Fig. 3(c); t he three copies of 'lIJ are to be identified) . By the Circle Packing Theorem, there is a circle packing of t he sphere with the same tangency graph (Figures 2 lind 3(d) show these packings after steroographic projection from the sphere onto the plane; the circle corresponding to V 3 was chosen as the upper hemisphere and became thc outside of the large circle after projection). Notice that each of the complementary components now corresponds to one ("large" ) circle of p~, a nd the circles in the boundary of Pc are tangent to these complementary circles. Now consider the map Ie that sends t he centers of the circles of Pc to the corresponding centers in ~ , and extend it in a piecewise linear fash ion. Rodin and Sullivan proved Thurston's conjecture t hat It approximates

P:

6

?,\

Figure 2: A circle packing approximation t o a Riemann map.

t.he Riemann map, if !1 is simply connected (see Fig. 2): The ore m 2.3. {RSuj Let

n be

simply connected, P, (f E \1, and P: normalized such that the com-

plementary circle is the unit circle, and such that the circle closest to p (n:.sp. q) com:.sponds to a circle contai11ing 0 (resp . some positive n:al number). Then the above maps f~ converge to the

conformal map

f

subsets ol n as e:

-+

!1 -+ D that is 1W1malized by i(P) = 0 and /(q) O.

>

0, uniformly on compact

Their proof depends crucially Oll the non-trivial uniqueness of t.he hexagonal packing as the only packing in the plane wit.h nerve t.he triangular lattice. Oded found remarkable improvements and generali~at.ions of t.his theorem . See Section 2.6 for further discussion.

2.2

Why are C ircle Packings inte resting?

Despite their intrinsic beaut.y (see t.he book [Ste2] for stunning illustrat.ions and an elementary introduction), circle packings are interesting because they prOvide a canonical and conformally natural way to embed a planar graph into a surface. T hus they have applications to combinatorics (for instance the proof of Miller and Thurston [MT ] of t.he Lipton-Tarjan separator theorem, see e.g. the slides of Oded 's circle I)acking tal k on his memorial webpage) , to differential geometry (for instance the construction of minimal surfaces by Bobenko, Hoffmann and Springborn [BHS] and their references) , to gcometric analysis (for instance, the Bonk-Kleiner [BK] quasisymmetric parametrization of Ahlfors 2-regular LLC tOI)ological spheres) to discrete probability theory (for instance, through the work of Benjamini and Schramm on harmonic functions on graphs and recurrence Oil random planar graphs [BS 1],[BS2], [BS3]) and of course to complex analysis (discrete analytic functions, conformal mapping) . However, Oded 's work on circle packing did not follow any "main-stream" in confor·mal geomet.ry or geometric funct ion theory. I believe he cont.inued to work on thcm just because he liked it. His interest never wavered, and many of his numerous late contributions to Wikipedia wcre about Lhis topic. Existence and uniqueness are int.imately connected . Nevertheless, for better readability [ will discuss them in two separate sections. 5

7

{'-"

Vv

r

~

7\

V 7\ V'v

1\

(a)

(b)

(c)

(d)

Figure 3: A circle packing approximation of a triply connected domain, its nerve, its completion to a triangulation of 52, and a combinatorially equivalent circle packing; (a)-(c) are from Oded's thesis; thanks to Andrey Mishchenko for creating (d)

6

8

2.3

Existence of Packings

Oded applied the highest standards to his proofs and was not satisfied with "ugly" proofs. As we shall see, he found four (!) different new existence proofs for circle packings with prescribed combinatorics. Before discussing them, let us have a glance at previous proofs. The Circle Packing Theorem was first proved by Koebe [K02] in 1936. Koebe's proof of existence was based on his earlier result that every planar domain n with finitely many boundary components, say m, can be mapped conformally onto a circle domain. A simple iterative algorithm, due to Koebe, provides an infinite sequence nn of domains conformally equivalent to n and such that nn converges to a circle domain. To obtain nn+l from nn, just apply the Riemann mapping theorem to the simply connected domain (in IC U { 00 }) containing nn whose boundary corresponds to the (n mod m)-th boundary component of n. With the conformal equivalence of finitely connected domains and circle domains established, a circle packing realizing a given tangency pattern can be obtained as a limit of circle domains: Just construct a sequence of m-connected domains so that the boundary components approach each other according to the given tangency pattern. For instance, if the graph G = (V, E) is embedded in the plane by specifying simple curves Ie : [0, 1] ----> 8 2 , e E E, then the complement no of the set

U le[O, 1/2 -

E] U

eEE

U le[1/2 + E, 1] eEE

is such an approximation. It is not hard to show that the (suitably normalized) conformally equivalent circle domains n~ converge to the desired circle packing when E ----> O. Koebe's theorem was nearly forgotten. In the late 1970's, Thurston rediscovered the circle packing theorem as an interpretation of a result of Andreev [AI], [A2] on convex polyhedra in hyperbolic space, and obtained uniqueness from Mostow's rigidity theorem. He suggested an algorithm to compute a circle packing (see [RSu]) and conjectured Theorem 2.3, which started the field of circle packing. Convergence of Thurston's algorithm was proved in [dV1]. Other existence proofs are based on a Perron family construction (see [Ste2]) and on a variational principle [dV2]. Oded's thesis [Sl] was chiefly concerned with a generalization of the existence theorem to packings with prescribed convex shapes instead of discs, and to applications. A consequence ([Sl], Proposition 8.1) of his "Monster packing theorem" is, roughly speaking, that the circle packing theorem still holds if discs are replaced by smooth convex sets. Theorem 2.4. ([51), Proposition 8.1) For every triangulation G = (V,E) of the sphere, every a E V, every choice of smooth strictly convex sets Dv for v E V \ {a}, and every smooth simple closed curve C, there is a packing P = {Pv : v E V} with nerve G, such that Pa is the exterior of C and each Pv , v E V \ {a} is positively homothetic to Dv' Sets A and B are positively homothetic if there is r > 0 and s E IC with A = rB + s. Strict convexity (instead of just convexity) was only used to rule out that three of the prescribed sets could meet in one point (after dilation and translation), and thus his packing theorem applied in much more generality. Oded's approach was topological in nature: Based on a cleverly constructed

7

9

Figure 4: A packing of convex shapes in a Jordan domain, from Oded's thesis spanning tree of G, he constructed what he called a "monster". This refers to a certain IVIdimensional space of configurations of sets homothetic to the given convex shapes, with tangencies according to the tree, and certain non-intersection properties. Existence of a packing was then obtained as a consequence of Brower's fixed point theorem. Here is a poetic description, quoted from his thesis:

One can just see the terrible monster swinging its arms in sheer rage, the tentacles causing a frightful hiss, as they rub against each other. Applying Theorem 2.4 to the situation of Figure 3, with Dv chosen as circles when v 1and arbitrary convex sets D vj , Oded adopted the Rodin-Sullivan convergence proof to obtain a new proof of the following generalization of Koebe's mapping theorem. The original proof of Courant, Manel and Shiffman [CMS] employed a very different (variational) method. {VI, V2, V3},

Theorem 2.5. (fSl), Theorem 9.1; [eMS)) For every n + I-connected domain n, every simply connected domain D c IC and every choice of n convex sets D j , there are sets Dj which are positively homothetic to D j such that n is conformally equivalent to D \ Uf Dj. Later [S7] he was able to dispose of the convexity assumption, and proved the packing theorem for smoothly bounded but otherwise arbitrary shapes. As a consequence, he was able to generalize Theorem 2.5 to arbitrary (not neccessarily convex) compact connected sets D j , thus rediscovering a theorem due to Brandt [Br] and Harrington [Ha]. Oded then developed a differentiable approach to the circle packing theorem. In [S3] he shows Theorem 2.6. (fS3), Theorem 1.1) Let P be a 3-dimensional convex polyhedron, and let K C R3 be a smooth strictly convex body. Then there exists a convex polyhedron Q C R3 combinatorially equivalent to P which midscribes K. Here "Q midscribes K" means that all edges of Q are tangent to oK. He also shows that the space of such Q is a six-dimensional smooth manifold, if the boundary of K is smooth and has 8

10

positive Gaussian curvature. For K = 8 2 , Theorem 2.6 has been stated by Koebe [K02] and proved by Thurston [T] using Andreev's theorem [AI], [A2]. Oded notes that Thurston's midscribability proof based on the circle packing theorem can be reversed, so that Theorem 2.6 yields a new proof of the Circle Packing Theorem (given a triangulation, just take K = 8 2 , Q the midscribing convex polyhedron with the combinatorics of the packing, and for each vertex v E V, let Dv be the set of points on 8 2 that are visible from v). One defect of the continuity method in his thesis was that it did not provide a proof of uniqueness (see next section). In [S4] he presented a completely different approach to prove a far more general packing theorem, that had the added benefit of yielding uniqueness, too. A quote from [S4]: It is just about the most general packing theorem of this kind that one could hope for (it is more general than I have ever hoped for). A consequence of [S4] (Theorem 3.2 and Theorem 3.5) is

Theorem 2.7. Let G be a planar graph, and for each vertex v E V, let:Fv be a proper 3-manifold of smooth topological disks in 8 2 , with the property that the pattern of intersection of any two sets in :Fv is topologically the pattern of intersection of two circles. Then there is a packing P whose nerve is G and which satisfies Pv E :Fv for v E v. The requirement that :Fv is a 3-manifold requires specification of a topology on the space of subsets of 8 2 : Say that subsets An C 8 2 converge to A if lim sup An = lim inf An = A and AC = int(limsupA~). An example is obtained by taking a smooth strictly convex set K in R3 and letting :F be the family of intersections H n oK, where H is any (affine) half-space intersecting the interior of K. Specializing to K = 8 2 , :F is the familiy of circles and the choice :Fv = :F for all v reduces to the circle packing theorem. The proof of Theorem 2.7 is based on his incompatibility theorem, described in the next section. It provides uniqueness of the packing (given some normalization), which is key to proving existence, using continuity and topology (in particular invariance of domains).

2.4

Uniqueness of Packings

I was always impressed by the flexibility of Oded's mind, in particular his ability to let go of a promising idea. If an idea did not yield a desired result, it did not take long for him to come up with a completely different, and in many cases more beautiful, approach. He once told me that if he did not make progress within three days of thinking about a problem, he would move on to different problems. Following Koebe and Schottky, uniqueness of finitely connected circle domains (up to Mobius images) is not hard to show, using the reflection principle: If two circle domains are conformally equivalent, the conformal map can be extended by reflection across each of the boundary circles, to obtain a conformal map between larger domains (that are still circle domains). Continuing in this fashion, one obtains a conformal map between complements of limit sets of reflection groups. As they are Cantor sets of area zero, the map extends to a conformal map of the whole sphere, hence is a Mobius transformation. Uniqueness of the (finite) circle packing can be proved in a similar 9

11

fashion. To date, the strongest rigidity result whose proof is based on this method is the following theorem of He and Schramm. See [Bo] for the related rigidity of Sierpinski carpets.

Theorem 2.8 ([HS2], Theorem A). If n is a circle domain whose boundary has u-finite length, then n is rigid (any conformal map to another circle domain is Mobius). For finite packings, there are several technically simpler proofs. The shortest and most elementary of them is deferred to the end of this section, since I believe it has been discovered last. Rigidity of infinite packings lies deeper. The rigidity of the hexagonal packing, crucial in the proof of the Rodin-Sullivan theorem as elaborated in Section 2.6 below, was originally obtained from deep results of Sullivan's concerning hyperbolic geometry. He's thesis [He] gave a quantitative and simpler proof, still using the above reflection group arguments and the theory of quasiconformal maps. In one of his first papers [S2], Oded gave an elegant combinatorial proof that at the same time was more general:

Theorem 2.9 ([S2], Theorem 1.1). Let G be an infinite, planar triangulation and P a circle packing on the sphere 8 2 with nerve G. If 8 2 \ carrier(P) is at most countable, then P is rigid (any other circle packing with the same combinatorics is Mobius equivalent). The carrier of a packing {Dv : v E V(G)} is the union of the (closed) discs Dv and the "interstices" (bounded by three mutually touching circles) in the complement of the packing. The rigidity of the hexagonal packing follows immediately, since its carrier is the whole plane. The ingenious new tool is his Incompatibility Theorem, a combinatorial analog to the conformal modulus of a quadrilateral. To fully appreciate it, lets first look at its classical continuous counterpart, and defer the statement of the Theorem to Section 2.4.2 below.

2.4.1

Extremal length and the conformal modulus of a quadrilateral

If you conformally map a 3x1-rectangle to a disc, such that the center maps to the center, what fraction of the circle does the image of one of the two short sides occupy? Despite having known the effect of "crowding" in numerical conformal mapping, I was surprised to learn of the numerical value of 0.0114". from Don Marshall (see [MS].) Of course, the precise value can be easily computed as an elliptic integral, but if asked for a rough guess, most answers are around 1/10 (the uniform measure with respect to length would give 1/8). Oded's answer, after a moments thought (during a tennis match in the early 90's), was 1/64, reasoning that this is the probability of a planar random walker to take each of his first three steps "to the right".

An important classical conformal invariant, masterfully employed by Oded in many of his papers, is the modulus of a quadrilateral. Let n be a simply connected domain in the plane that is bounded by a simple closed curve, and let Pl,P2,P3 and P4 be four consecutive points on an. Then there is a unique M > 0 such that there is a conformal map f : n --+ [0, M] x [0,1] and such that f takes the Pj to the four corners with f(Pl) = 0 (by a classical theorem of Caratheodory, f extends homeomorphic ally to the boundary of the domains). There are several quite different instructive proofs of uniqueness of M. Each of the following three techniques has a counterpart in the circle packing world that has been employed by Oded. Suppose we are given two rectangles and a conformal map f between them taking corners to corners. 10

12

One method to prove uniqueness is to repeatedly reflect J across the sides of the rectangles. The resulting extention is a conformal map of the plane, hence linear, and it follows that the aspect ratio is unchanged. This is similar to the aforementioned Schottky group argument. A second method is to explicitly define a quantity A depending on a configuration (D,Pl, ... ,P4) in such a way that it is conformally invariant and such that one can compute A for the rectangle [0, M] x [0,1]. This is achieved by the extremal length of the family r of all rectifiable curves 'Y joining two opposite "sides" [PI,p2] and [P3,p4] of D. The extremal length of a curve family r is defined as (inf'Y J'Y pldzl? _ (1) A(r) - sup f 2d d ' P JCP x y where the supremum is over all "metrics" (measurable functions) p: C --; [0,(0). For the family of curves joining the horizontal sides in the rectangle [0, M] x [0, 1], it is not hard to show A(r) = M. This simple idea is actually one of the most powerful tools of geometric function theory. See e.g. [Po2] or [GM] for references, properties and applications. Discrete versions of extremal length (or the "conformal modulus" I/A) have been around since the work of Duffin [Du~. In conformal geometry, they have been very succesfully employed beginning with the groundbreaking paper [Can]. Cannon's extremal length on a graph G = (V, E) is obtained from (1) by viewing non-negative functions p : V --; [0, (0) as metrics on G, defining the length of a "curve" 'Y C V as the sum I:vE'Y p( v), and the "area" of the graph as I: p( v? See [CFPI] for an account of Cannon's discrete Riemann mapping theorem, and for instance the papers [HK] and [BK] concerning applications to quasiconformal geometry. Oded's applications to square packings and transboundary extremal length are briefly discussed in Section 2.7 below. A third and very different method is topological in nature and is one of the key ideas in [HSI]. Suppose we are given two rectangles D, D' with different aspect ratio and overlapping as in Fig. 5, and a conformal map J between them mapping corners to corners. Then the difference J(z) - z is i= on the boundary aD. Traversing aD in the positive direction, inspection of Fig. 5 shows that the image curve under J(z) - z winds around in the negative direction. But a negative winding is impossible for analytic functions (by the argument principle, the winding number counts the number of preimages of 0).

°

2.4.2

°

The Incompatibility Theorem

Again consider the overlapping rectangles D, D' of Fig. 5, and two combinatorially equivalent packings P, P' whose nerves triangulate the rectangles, as in Fig. 6. Assume for simplicity that the sets Dv and D~ of the packings are closed topological discs (except for the four sides Dl, ... D 4, Di, ... , D~ of the rectangles, which are considered to be sets of the packing). Intuitively, two topological discs D and D' are called incompatible if they intersect as in Fig. 5. More formally, say that D cuts D' if there are two points in D' \ interior(D) that cannot be connected by a curve in interior(D' \ D). Then Oded calls D and D' incompatible if D cuts D' or D' cuts D. As he notes, the motivation Jor the definition comes from the simple but very important observation that the possible patterns oj intersection oj two circles are very special, topologically. Indeed, any two circles are compatible. Theorem 2.10 ([S2], Theorem 3.1). There is a vertex v Jor which Dv and D~ are incompatible. 11

13

'"

"

,•

, q.

q.

f igure 5: Conformally inequivalent rectangles; from [HS1 J.

Figure 6: An incompat.ibility at t.he center.

12

14

Oded calls this result a combinatorial version of the modulus. However, it has rather little in common with the above notion of discrete modulus, except for the setup. Oded's clever proof by induction on the number of sets in the packing uses arguments from plane topology. An immediate consequence is that two rectangles cannot be packed by the same circle pattern, unless they have the same modulus M and hence are similar: if they could, just place the two packings on top of each other as in Fig. 6 and obtain two incompatible circles, a contradiction. In the same vein, it is not difficult to reduce the proof of the rigidity Theorem 2.9 to an application of the incompatibility theorem.

2.4.3

A simple uniqueness proof

To end this section, here is a beautifully simple proof of the rigidity of finite circle packings whose nerve triangulates S2. I copied it from the wikipedia (search for circle packing theorem), and believe it is due to Oded. As before, stereographically project the packing to obtain a packing of discs in the plane. This time, assume that the north pole belongs to the complement of the discs, so that the planar packing will consist of three "outer" circles and the remaining circles contained in the interstice between them.

"There is also a more elementary proof based on the maximum principle, which we now sketch. The key observation here is that if you look at the triangle formed by connecting the centers of three mutually tangent circles, then the angle formed at the center of one of the circles is monotone decreasing in its radius and monotone increasing in the two other radii. Consider two packings corresponding to G. First apply reflections and Mobius transformations to make the outer circles in these two packings correspond to each other and have the same radii. Next, consider a vertex v where the ratio between the corresponding radius in the one packing and the corresponding radius in the other packing is maximized. Since the angle sum formed at the center of the corresponding circles is the same (360 degrees) in both packings, it follows from the above observation that the radius ratio is the same at all the neighbors of v as well. Since G is connected, we conclude the radii in the two packings are the same, which proves uniqueness. "

2.5

Koebe's Kreisnormierungsproblem

Koebe's 1908 conjecture [Kol] that every planar domain can be mapped conformally onto a circle domain is still open, despite considerable effort by Koebe and others. Important contributions were made by Grotzsch, Strebel, Sibner and others. One difficulty is the aforementioned lack of uniqueness. Another problem is that Theorem 2.5 is not true in the infinitely connected case, as the following example from [S6] illustrates: If K = {x + iy : x = 0, ±1, ±~, ±~, ... , y E [-1, I]}, and if D = K, then there is no conformal map f of D, normalized by f (z) - z --+ 0 as z --+ 00, such that the component {iy: y E [-1, I]} of aD corresponds to a horizontal line segment (or a point) while the other complementary components of f(D) are vertical line segments. The same example also illustrates the fundamental continuity problem: There is a circle domain D' conformally equivalent to D, but the boundary component corresponding to {iy : y E [-1, I]} is just a point, so that the conformal map from D' to D cannot be extended to the boundary.

t\

The first joint paper of He and Schramm provided a breakthrough: 13

15

Theorem 2.11 ([HS1]). Iffl has at most countably many boundary components, then fl is conformally equivalent to a circle domain fl', and fl' is unique up to Mobius transformation. Essentially, this result is still the strongest to date. Oded later [S6] gave a conceptually different and simpler proof based on his transboundary extremal length, which also applies to certain classes of domains with uncountably many boundary components. The proof in [HS1] used transfinite induction and was based on the topological concept of the fixed-point index. I will illustrate the beautiful idea by sketching their proof of uniqueness. As it turned out, this argument for uniqueness had been given earlier by Strebel [Str]. The simple but crucial idea is to use the following (see [HS1], Lemma 2.2): If f is a fixed-point free orientation preserving homeomorphism between two circles G' and Gil, then the winding number of the curve f(z) - z, z E G', around 0 is non-negative (recall Fig. 5 for a situation where the winding number is negative). Let f : fl' -+ fl" be a conformal map and assume for simplicity that f extends continuously to the boundary (in case of finitely many boundary components this is immediate from the reflection principle, but in the countable case this step is non-trivial), and that f has no fixed points on the boundary. Composing with Mobius transformations, we may assume that 00 E fl' and that f(z) = z + adz + a2/ z2 + .... We want to show that f is the identity. If not, denote aj the first non-zero Taylor coefficient, then f(z) - z has winding number -j as z traverses a large circle Izl = R, because f(z) - z behaves like ajz- j . Moreover, each circular boundary component maps to a circular component. These boundary components are oriented negatively (to keep the domain to the left) and thus, by the above crucial idea, contribute a non-positive number to the winding of f(z) - z,z E o(fl n {Izl 00 as t -> T. Hence we can reparametrize "( so that T = 00 and g~(O) = et . Loewner's theorem says that

(3) for all t 2 0 and all z E G t , where the "driving term" (t

= gt("(t» 19

21

E 8JD)

'.

~

Figure 9: Conformal maps from slit discs onto discs. is continuous (a priori, 9t is only defined in G t, but it call be shown that Yt extends to i (t)) , A simple but crucial observation is that the driving term (T of the curve "IT = 91'(-r) (more precisely, the parametrized curve I'T(s) ; = YT b(T + 8))) is given by 0, consider the LERW on the graph Ii'£,2 n D, started at a point closest to a and stopped when reaching aD. Viewing the path of the LERW as a random subset of the sphere 8 2 = C u {oo}, its distribution is a discrete measure JLIj on the space of compact subsets of 8 2 . Equipped with the Hausdorff distance, the space of compact subsets of 8 2 is a compact metric space, and so is the space of its Borel measures. The existence of subsequential weak limits /1 = limj /18j follows at once. If the limit measure /1 = lim8--->O /18 exists, it is called the scaling limit of LERW from a to aD.

Theorem 3.1 ([89], Theorem 1.1). If each connected component of aD has positive diameter, then every subsequential scaling limit measure /1 of the LERW from a to aD is supported on simple paths. In other words, the measure of the set of non-simple curves is zero. This theorem is interesting in its own right. It has been known previously that, loosely speaking and under mild assumptions, random curves have uniform continuity properties that imply their (subsequential) scaling limits to be supported on continuous curves [AB]. However, the fact that the loop erased paths are simple curves does not directly imply that the limiting objects have no loops. Indeed, the limits of other discrete random simple curves such as the critical percolation interface or the uniform spanning tree Peano path are not simple. The proof uses estimates for the probability distribution of "bottlenecks", based on harmonic measure estimates and Wilson's algorithm, and a topological characterization of simple curves. Next, Oded formulated the conjecture of existence and conformal invariance of the scaling limit as follows.

21

23

Conjecture ([S9j,1.2) Let D ~ IC be a simply connected domain in IC, and let a E D. Then the scaling limit of LERW from a to aD exists. Moreover, suppose that f : D ---+ D' is a conformal homeomorphism onto a domain D' c IC. Then f*/La,D = /Lf(a),D', where /La,D is the scaling limit measure of LERW from a to aD, and /Lf(a),DI is the scaling limit measure of LERW from f(a) to

aD'. The most important and exciting result of [S9] was the insight that this conjecture implied an explicit construction of the limit in terms of the Loewner equation. By Theorem 3.1, the conjectural scaling limit /L induces a measure on the space of continuous real-valued functions (t via the correspondence ry f--+ ( = e i ( of the Loewner equation. Oded showed that the law of ( is that of a time-changed Brownian motion, B 2t : Theorem 3.2 ([S9], Theorem 1.3). Assuming the above conjecture, the scaling limit /L is equal to the law of the hulls K associated with the driving term ( = eiB2t , where B t , t ~ 0 is a Brownian motion started at a uniform random point in [0, 27f). In his characteristic way, Oded pointed out the simple idea behind the theorem. From his paper:

At the heart of the proof of Theorem 3.2 lies the following simple combinatorial fact about LERW. Conditioned on a subarc (3' of the LERW (3 from 0 to aD, which extends from some point q E (3 to aD, the distribution of (3\(3' is the same as that of LERW from 0 to a(D - (3'), conditioned to hit q. When we take the scaling limit of this property, and apply the conformal map from D - (3' to ill), this translates into the Markov property and stationarity of the associated Lowner parameter (. He also notes that "the passage to the scaling limit is quite delicate". The translation into the Markov property and stationarity is by means of the aforementioned principle that "conformally pulling down" a portion ry' of ry corresponds to shifting the driving term. Thus ( is a continuous process with stationary and independent increments. Now the theory of Levy processes (and the symmetry of LERW under reflection) implies that (t has the law of yfiiBt for some t;, > 0 and a standard Brownian motion B. It remained to determine the constant t;,. To this end, Oded gives the following Definition. The (radial) stochastic Loewner evolution SLE" with parameter t;, > 0 is the random process of conformal maps gt generated by the Loewner equation driven by (t = eiV;

°

Similarly, the half-plane exponent ( of the event that two independent motions do not intersect and stay in a halfplane is given by

JP'[B1[0, t] n B2[0, t] = 0 and Bj[O, t] c 1HI,j = 1,2] = (

1)'+0(1)

t

More generally, one considers exponents (p for the probability of the event that p independent motions are mutually disjoint, ((j, k) for the event that two packs of Brownian motions B 1U·· ·UBj and Bj+1 U ... U Bj+k are disjoint, and the corresponding half-plane exponents (p and ((j, k). So ( = ((1,1). Also relevant is the disconnection exponent 2r/j for the event that the union of j Brownian motions, started at 1, does not disconnect from 00 before time t. These and other intersection exponents have been studied intensively, and values such as ( = 5/8 had been obtained by Duplantier and Kwon [DK] using the mathematically non-rigorous method of conformal field theory. An extension of ((j, k) for positive real k > was given in [LWl], and some fundamental properties (in particular the "cascade relations") were established. In the series of papers [LSW2], [LSW3], [LSW5], [LSW7] (see [LSWl] for a guide and sketches of proofs), Lawler, Schramm and Werner were able to confirm the predictions, and they proved

°

°

Theorem 3.4. For all integers j 2': 1 and all real numbers k 2': 0,

r(' k)

= y'24JTI + y'24k + 1 96

s. Now Cardy's formula can be obtained llsing standard methods of stochastic calculus. For a simply connected domain D 1- C and boundary points p, q, chordal SLE from p to q in 0 is defined as the image of SLE in H under a conformal map of H onto D that takes 0 and 00 to p and q. Since the conformal map between H and D generally does not. extend to H, the continuity of the SLE t race in !) does not follow from Theorem 3.6. However, using T heorem 3.9 below and general properties of conformal maps, it can be shown to still hold t.rue, [GRS]. Another natural question is whether SLE is reversible, namely if SLE in D f!'Om p to q has the same law as SLE from q to p. T his question was recently answered positively for Ii ::::; 4 by Dapeng Zhang [Z 1] . It. is known to be false fo!' Ii ~ 8 [RoS], and unknown for 4 < Ii < 8. T he ore m 3 .8 ([ZI ]). For each

Ii

< 4, SL E" is reverSible, and for

Ii ~

8 it is not reversible.

The aforementioned derivat.ive expectations E[lfl(z)I"Jalso led to upper bounds for the dimensions of the trace a nd the frontier. T he technically more difficult lower bounds were proved by Vincent Beffara [Be] for the trace.

30

32

For K, > 4, notice that the outer boundary of K t is a simple curve joining two points on the real line. There is a relation between SLE", and SLE 16 /"" first derived by Duplantier with mathematically non-rigorous methods, and recently proved in the papers of Zhang [Z2] and Dubedat [Dub3]. Roughly speaking, Duplantier duality says that this curve is SLE16 /", between the two points. A precise formulation is based on a generalization of SLE, the so-called SLE(K" p) introduced in [LSW8]. As a consequence, the dimension of the frontier can thus be obtained from the dimension of the dual SLE. Based on a clever construction of a certain martingale, in [SZ] Oded and Wang Zhou determined the size of the intersection of the trace with the real line. The same result was found independently and with a different method by Alberts and Sheffield [AISh]. Summarizing: Theorem 3.9. For K,::::: 8, For K, > 4,

For 4

< K, < 8,

. K, dIm 1'[0, t] = 1 + S'

. dIm 8K t

2

= 1 + -. K,

. dIm 1'[0, t] n lR

8

= 2 - -. K,

The paper [SZ] also examined the question how the SLE trace tends to infinity. Oded and Zhou showed that for K, < 4, almost surely I' eventually stays above the graph of the function x f-+ x(logx)-,B, where (3 = 1/(8/K, - 2). 3.3.4

Discrete processes converging to SLE

In [LSW2], Lawler, Schramm and Werner wrote that ... at present, a proof of the conjecture that SLE6 is the scaling limit of critical percolation cluster boundaries seems out of reach ... Smirnov's proof [Sml] of this conjecture came as a surprise. More precisely, he proved convergence of the critical site percolation exploration path on the triangular lattice (see Figure 12(b)) to SLE6. See also [CN] and [Sm2]. This result was the first instance of a statistical physics model proved to converge to an SLE. The key to Smirnov's theorem is a version of Cardy's formula. Lennart Carleson realized that Cardy's formula assumes a very simple form when viewed in the appropriate geometry: When K, = 6, the right hand side f(s) of (6) is a conformal map of the upper half plane onto an equilateral triangle ABC such that 0, 1 and 00 correspond to A, Band C. Since SLE6 in ABC from A to B has the same law as the image of SLE6 in 1HI from to 00, the first point X' of intersection with BC has the law of f(X). It follows that X' is uniformly distributed on BC. (A similar statement is true for all 4 < K, < 8, where "equilateral" is replaced by "isosceles", and the angle of the triangle depends on K" [Dubl]). Smirnov proved that the law of a corresponding observable on the lattice converges to a harmonic function, as the lattice size tends to zero. And he was able to identify the limit, through its boundary values. The proof makes use of the symmetries of the triangular lattice, and does not work on other lattices such as the square grid, where convergence is still unknown.

°

The next result concerning convergence to SLE was obtained by the usual suspects Lawler, Schramm and Werner [LSW9]. They proved Oded's original Conjecture 3.2.2 about convergence of 31

33

LERW to 5LE2 , and the dual result (also conjectured in [S9]) that the UST converges to 5LEs , see Figures 10 and la. The hannonic explorer is a (random) interface dcfincd as follows: Givcn a planar simply connected domain with two marked boundary points that partition the boundary into black and white hexagons, color all hexagons in the interior of the domain grey, see Figure 16 (a) . T he (growing)

, (b)

(.j

FigU!"e 16: Definition of t.he Harmonic explorer pat.h, from [SSIJ. interface I starts at. one of the marked boundary points and keeps t.he black hexagons on its left. and the white hexagons on its right. It is (uniquely) determined (by turning left at white hegagons and right at black) until a grey hexagon is met. When it meets a grey hexagon It (marked by? in F igure 16) the (random) color of h is determined as follows. A random walk on the set of hexagons is started, beginning with the hexagon h. T he walk stops as soon as it meets a white or black hexagon, and h assumes I.hat color. Continuing in t his fashion , "f will evenwally reach the other boundary point. [n [SSl] , Oded and Scott Sheffield showed (distributional) convergence of "f to SLE4. T he overall strategy is again to directly analyze the Loewner driving term of the discrete path. The crucial property of 5LE4 is that, conditioned on the 5LE trace "f [0, t], the probability that a point z E H will end up on t he left of "f [0, 00) is a harmonic function of z (it is e O, so U E Nd xr· .

="

From now on, we work with an arbitrary. but fixed , set of constant width W e R". The lemma above shows that if E is a subsets of the boundary of

W, the n one direction can illuminate E, if, and only if,

n

."E

N w (x )' =

(UNw(X ») ' . E Ii:

is nonempty. The following proposition wilt help us find s ubsets of aW that are "easily" illuminated. For a subset A c 5 ,, - 1 define U w( A) to be the union of the sets N w (x ), x e aw. that intersect A :

U

N w l " J" A " 0

A direction in Uw ( A)+ illuminates every point x e W that satisfies N w (x) n A#- 0. In orde r to show that when A is chosen properly these points are " easily" illuminated, we want to prove that U w( A )'" is "large"'. Our means of doing so is by estimating the diameter of Uw(A ). (We view 5 "- 1 with the metric induced by the Euclidean metric in R". The diame ters of subsets of 5 "- 1 refer to this metric.) PROPOS IT ION 5 .

Lei A be a nQnemply subset of 5 "- 1. Then diameter Uw ( A) "'" 1 + diameter A.

Proof Since Uw ( A ) d oes not change when we replace W with a posilively homotheti c copy of itself, we may, and will, assume that W has conslant width 49

183

ILLU MINATING SE T S OF C O NSTANT WIDTH

1, and therefore also diameter 1. Let v" v2 be unit vectors in U w( A). By the definition of Uw{A) , there are points x .. X2 E aW such that Vi E N w( x;) and N W(x i ) ( \ A -,t. 12' for j = 1,2. Suppose H i is in Nw(x;) n A, j = 1,2. Since U i is an inward normal of W at X;, and since W has constant width I , the hyperplane {pe R ~ Ip . U i = Xi. U I + I} is a supporting hyperplane of W The only point on this hyperplane whose distiance from Xi is not greater than I is X i + U I. Since diameter W = I, we co nclud e that Xi + U i E aW for i := 1,2. Therefore I = ( diameter W)l ;;.1I( x, + U I) - xl f = Il x, - X2112 + 2u , . (x, -

X2)

+1

and 2

1 = (diameter W )2 ~ lI( x!+ ul)- x dI = Il x2- x, 1I2 +2u2. (x 2 - xd+ 1.

Summing these inequalities and rearranging we get (u , Because(u , So

U2 ) • ( X2 -

X i + U I,

II X 2 -

xdl2 .

x ,) ""-'; lI u, - u21111 x2 - x,ll, thi s implies IIu 1 - u 2 11 ~ Ilxl - x dl · II x! -

As with

x ,) ~

U2) . ( X l -

the points

Xi

x,lI,.,,-,; diameter A.

+ Vi lie

in

aw, and

(2. 1)

therefore

I ~ lI (x, + v,) - (x 2 + v2 )11 ~ II vl - v211 - li x1 - x dl·

Using (2.1 ), thi s implies

1+ diameter A ~ II v, - v211 . We use I./. to denote the standard p robability meas ure on 5 "- '. Define g( n, d ) = inr { I./. ( A + ) IA c 5 " - ', diameter A .,;; d }.

Let N(n , e) be the number of sets having diameter e that is required to cover S ~ - '. The core of thi s note is: PROPO SITION

6.

For 0 < e < ,f5. I (W )";; I +

1

we have

logN (n,e ) . - !og (l - g( n, l +e»

Proof It is easil y verified that 0 < g(n, 1 + e) < I (if 12' #- A c 5 "- ' then A + is contained in a hem isphere, so that J.L ( A "" ) :E;!. If also diameter A -= d

log N ( n, e) . -l og (1 - g ( n, I + e»)

It is sufficient to show that M directions can i1luminate W. Set N "", N(n, d , and let A " ... , A N be a covering of 5 " - 1 with sets of diameter £. By Proposition 5, we have diameter U w(A j ).,;; 1+ £, and therefore i = 1,2, . .. , N. 50

184

O. SCHRAMM

Pick M directions Ul •. . . ,UM at random, uniformly and independently distributed on S~ - l. Take any it}, I Of; i ~ N, I ~ j :!i M. The probability that IJ will be in Vw ( A ,),," is JL(Uw(A ;)"" ), which is at least g(lI, 1+,,). Therefore the probability that UW(A i( will contain none of the points u 1 ••••• UM is at most (1 - g(n, l +e »M. Thus the probability p that at least one Uw(A1)\ 1 0;;; l os;, N will contain no points of U l • • .. ,UM satisfies N

L

p :o;;:

•••

( I - g{ n, 1 + e))11-1 < N(1- g( n, 1+

n H £ ))1 0 8 N / - loll ( 1 - s 1 . . ))

=

L

This shows that one can choose M directions, so that each set Uw(A,r.. I = I, .. . • Nt contains at least one of them. Let "I, " " "/1-1 be such directions, and let x be a poine of aw. We claim that one of these directions illumi· nates x. Since Nw(x) is nonempty, and the sets AI , ... , AN cover S~ - I, one of them, say A, intersects Nw(x). By the definition of UW(A i ), we have Nw (x) c Uw(A j). So that Nw(x) + :;;> Uw(A i ) ~ ' Uw(A;t contains at least one of 'k

and therefore, by Lemma 4, Vi • •• • ' V"" illuminate W.

VI, ••• , VM,

say

Vk .

We have

E Uw ( A ;)+ c Nw(x)+ V~

illuminates x. This shows that the directions

In orderto deduce Theorem 1 from Proposition 6, we only have to estimate

g ( n, 1 + £) from below, and N(n, t:) from above. The former will be done in

the next section, and the latter is dealt with by the following well known fact. LEMMA 7.

N(n, e ):S;; (l+4/E}".

Let E be a maximal subset of 5 "- 1 having the property that lI u- 'II > ~£ for u ~ v in E. The maximality of E shows that the balls with radius !£ and centers in E cover S .. - I , therefore Proof

I E I~ N(n,d .

All the balls B(u, £/ 4), u E E, are disjoint and are contained B(O, 1+ t:/4). Comparing volumes gives

In

the ball

IE I(e / 4)"" (1 + e/4)",

Remark. Better estimates for N ( n, £ ) are known (see [9]), but do not seem to contribute any significa nt improvement to Theorem l.

§3.

In this section we give a lower bound for g(n, d), and prove Theorem 1. 51

185

ILLUMINATING SETS OF C ONSTANT WIDTH

PROPOS ITIO N 8. LeI d > 0 and leI A be a nonemply subset of S~ - I having diameter ,.., d. Suppose U E S ~- L, a > O and A is contained in the half-space {p E R ~l p . u ~ a }, then

A'" u TA'" => Do(u, arctan (2a ! d», where T : R

n

.....

R~

is the reflection through the line determin ed by u, - u :

Tp=:= 2( p.u )u - p, and DQ(u, tIJ) is the open spherical cap consisting of all unit vectors ha ving an angle with u, which is smaller than tIJ. Proof Suppose x is a point in S ~ - L but not in A "'u TA "' , and let 0 be the angular distance between x and u, 0 "" 8 "" -rr. Write

x =(cosO) u + (sin8)v,

(3. 1)

where v is a unit vector orthogonal to u ( we ignore the trivial case n = Si nce x l!: A"', there is a point y E A with O;;;o y . x =:= y . U cos O+ y . vsin O.

O.

(3.2)

Since T - 1 =:= T and x l!: TA '" we have TXl!: A .... Thus there is a poi nt ZE A with

o~ Z . Tx = z . u cos 0 -

z . v sin O.

(3.3)

Summing (3.2) and (3.3), and using Il y - zlI "" d, sin 8 ~ 0, we have O;;(y . u + z. u) cos 9-d sin O.

(3.4)

Temporarily suppose that 9 0, so by (3.4) tan 9 ?>

y.u+z . U 2a ;;
9.

I (3

t)d

2»)

2 - (2n + (2n + - (~ - on g( n d) ~ - - - + =:..:...=:..:..."O':7--~ , - J8 -rrn 2 4n +4 - 2d !n

for 0
Do( u, arctan ( 2a/ d ».

Now, since T is an orthogo nal transformation, we have IJ-(A "' ) "" J.L( TA .. ), and J.L(A -t) "" hJ.L( A +) + Ji,( TA +» ;:a, ~ J.L(A + u TA · )

2a)) -_! Vol~ _ L Do{ u,arctan 2a/d) SO->

;.;,1 ( (

.... zJ.L Do u, arctan d ~

2

VOI" _l

Vol.. _ 1 Do( u , a rctan 2a/d)

(3.8)

2nO"

Here 0" denotes the vol ume of the n-dimensional unit ball, and VOI"_1 is the (n - I) -dimensional volume.

Let D ' be the orthogonal projection of Do(u, arctan 2a/ d) to t he hyperplane {p e R" lp . u = O}. Obviously we have Vo l.. _ 1 Do(

D,

arctan 2:)

~ Vol

n_ L

D '.

(3.9)

D' is an ( n - I)-dimens ion al baH having radius

2) ( + d')

(

a si n arctan d =

1 4a-:

->I '

•

sa (3. 10)

Using (3.8), (3.9), (3 .10). we get +

On_l ( d ' ) - I" - 1)/ 2 1+ -2 . 2nO n 4a

,,(A) ; . - -

53

(3.11)

ILLUMINATING SETS OF CONSTANT WIDT H

187

Now n~ _l =

7J" 1..

On where

r

- 'I!2/r« 1+ n)/2)

7J"~ /2/ r(1

is lhe Gamma function. Since log

f(1

1'(1 + n/ 2) ';:;f«( 1 + n)/ 2)

+ n/2)

r

(3 .12 )

is convex ([1, p. 12J), we have

+ n/ 2)f( n/ 2) " n(1 + n )/ 2)',

and therefore

f ( 1 + n/2) f(1 + n/2) f((1 + n)/2)" f(( 1 + n)/2)

f«(1 + n )/2)'

/

Vf(1 + n/2)f(n / 2)

= /r(l+n/ 2) = ~ V f(n /2) V'i Using (3. 11 ), (3.12), (3.13), we gel I I ~(A +)~--

2n

J;

(3.13)

i;( 1+ -

d ' ) - '" - "" . 4a 2

-

2

And after substituling the value of a, I

J.£(A +)~ .j87J"n

(

d' ) - (~ - I )/2 1+ 4 -2d 2n/( n + l )

=_'_ (~+ (2 n+ l)d 2_ 2~1 -2 ) -1 " - 11/ ~.

J87J"n 2

4n+4 - 2d -n

Now proving Theo rem I is just a matter o f putting the pieces logether.

Pro%/Theorem I. Since 1 I then there are n +1 roots ofG, fo, ... , r ," such that every nonzero XE R " has a negative inner product with

at least one of them. At least when G is finite thi s follows from known results (see (6]).

Proofofthe Lemma. We will say thai a sel of vectors {vo,.'" v", } is almost independent if every proper subset of it is linearly independent but {Yo, . . . , V,.} is linearl y dependeryt. II is easily checked that {vo, ... , v"'} is almost indepen. dent, if, and only if, VI, .. • , v'" are linearly independent and Vo is a linear combination of VI , " ' , Vm with nonzero coefficients. Because of the hypotheses on G and because n ~ 2, G has at least one root, r. {f, - f \ is an almost independent set. Suppose {ro, . . . , r", } is the largest almost inde pendent set of roots of G. Let U be the subspace generated by fo, . . . , r ",. We claim that U = R" and therefore n = m. Let f be a root of G and suppose that U is not invaria nt under Sr. This implies r J! U and Srfi t- fi for some j = 0, 1, ... ,m. Assume, without loss of generality, that S,fo,t. f O' fO is a linear combination of r l , ••• , f", with nonzero coefficients. Since S,fo-fo is a nonzero multiple of f , S,fo is a linear combination of f , r 1 , •• . , r", with nonzero coe ffi cients. Because r, f1' . . . , f m are linearly independent, this means 55

I LLUM I NATI NG SETS OF CONSTANT WIDTH

189

that the set {S.fo, f, f l> ...• f",} is almost independent. co ntradicting the choice of {ro•... , rm}. This fo rces us to conclude that U is inva riant under t he reflections ge n era t~ ing G, and therefore under every transformation in G. G acts irreducibly on R" and U;e O. This implies U = R" and n = m. Because {r o,"" r,, } is almost independent, it is possible to write

with all coeffi cie nts nonzero. We rep/ace some of the roots rj by thei r negatives, to have all the coeffi cients Q ; positive. If x is any nonzero vector, we must have x . r l "'" 0 fo r some i = 0, I •...• n, because the f , span R ". Since O= x . O==I:7. 0 ai(x . r,) a nd a,> O, we must have x. r,< O fo r some i.

Proofof Th eorem 10. As mentioned in Sectio n I , I ( K ) ~ n + l holds for every convex body in R ", thus we only need to show that / ( K ).s;; n + l. The Theorem is obviously t rue when n = I , so we assume n > I. Let ro, ... , r" be t he roots of G gua ran teed by the lemm a and let x Ea K. First observe that th e origin is necessarily an in terior point o f K so x ¢ O. For some f " x . r, < O. We claim that this fi ill umi nates x. x - 2(x . fi)r, = S.;x E K, since S•. E G. For some positive I. x + IT, is an interi or point of K, because x, x-2( x .'fi)T; E K , K is stri ctl y convex and -2( x . r,» O. Th is verifies the claim and we see that To, .. . • r" ill uminate K. Remark. A set of constant width is stri ctly convex; therefo re Theorem 10 appli es to (sufficiently symmetric) sets of constant wi dth.

References 1. E. Artin. The Gamma F Wllc'ioll. translated from Ge rman (Holt, Rinehart and Winston. 1964). 1. V. G. Boltjansky. The problem of the illumination of the boundary of a con~e~ body. In Russia n. I:vf'sri)'a Moldavskogofiliala Akudemii Nauk SSS R, 10 (1960 ), 77-84. 3. V. G. Boltjansky and \. Ts. Gohberg. ResullS and problems ill combinarorial geomerry, translated from Russian (Cambridge Un iv. Press, Cambri dge, 1985). 4. G. D. Chakerian an d H. Groemer. Con~ex bodies of consta nt width, in "Convexiry alld irs applicariolls," ed. by P. M. G rube r and J. M. Wills (Birkhauser, Basel, 1983). S. H. G. Eggleston. ConV('xiry (Ca mbridge Univ. Press. Cambri dgf' . 1958). 6. L. C. Grove and C. T. Benson. Finile R ejlulion Groups (Springer. New York, 1985). 1. B. Griinbaum. Borsuk's probtem and related questions. Proc. S)'mp. Puff' Malh., 1 (196), 271 - 284. 8. B. Griinbaum. COllveX Poly/opts (Wiley, New York, 1967). 9. C. A. Roge rs. Covering a sphere with spheres. Mafhemalika, 10 (1963),157 - 164. to. C. A. Rogers. Symmetrical SetS of co nstant width and th eir parti tions. Mar/tf'malika, 18 ( 1911), 105-11 1. II . O. Schramm . On th e volu me of sets ha ving constant widt h. To appear, Israel J. Malh.

Dr. O. Sch ram m, Mathematics Department, Fine Hall. Princeton Uni~ersity, Princeto n NJ 08544. USA.

52A20.

CONVEX SETS AND RELATED GEOMETR IC TOPICS; Convex sets ill II dimf'Iu ions.

Rf'Cf'ived on Ihe 151h of FebrTIary. 1988.

56

ISRAEL lOURNALOF MATHEMATIcs. Val. 63, No.

1.

19&8

ON THE VOLUME OF SETS HAVING CONSTANT WIDTH! OY

ODED SCHRAMM

Mathemtltics Department, Fine Hall, Princeton UnilJl!rsity, Princeton, HJ 08544, USA

ABSTRACT

A lower bound is given fer the volume of sets of constant width.

I.

Introduction

A set of constant width d in Euclidean space R~ is a compact, convex set, such that the distance between any distinct, parallel supporting hyperplanes of

it isd(see [3.pp. 122-1311.[2]). The Blaschke-Lebesgue theorem states that of all planar sets having constant width d the Reuleaux triangie has the least area, Hn - jj)t:P. The problem of detennining the minimal volume of sets having constant width din R~, n > 2, seems considerably more difficult. Lower bounds for it have been given by Firey (4) and Chakerian (I]. Let Wbe a set of constant width d and circumradius r in R~. In this note we prove the lower bound

(1.1)

Vol

W;;:;( ~ - I)'VOIB(O.dI2).

which implies (1.2)

Vol

W;:;(

V+ n! 3

I -1)'VOIB(O.dI2).

I The research Clposed in this note was done while I was at the Hebrew University of Jerusalem,

as a student of Professor Gil Kalai. I would like to thank Prof. Kalai for his interest, encouragement and advice. Received February 22, 1988

178 I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_3, 57

Vol. 63, 1988

ON THE VOLUME OF SETS HAVING CONSTANT WIDTH

179

Here Vol denotes the n-dimensional volume in RII and B(x,p) is the ball having center x and radius p. This bound is, for n > 4, an improvement over those previously known. We also prove THEOREM I. Let K be a set 0/ constant width d and circumradius r in RII ha ving the origin 0 as the center ofits circumsphere, then K U - K contains the

ball of radius J S(dl2)' - r' - (dl2) around Ihe origin. This result can be seen as a relative to the well known theorem stating that the insphere of a set of constant width d is concentric to the circumsphere and its radius is d - r, where r is the circumradius (see [3, p. 125]). Arguments analogous to those below, but dealing with subsets of the unit sphere, are used in [5), where an upper bound is given for the number of directions sufficient to illuminate the boundary of sets having constant width, 2. For a set A C R" and for A > 0 we denote by A~ the intersection of all the balls of radius A, having centers in A:

A' -

n

B(x,.! ) - (pER" IB(p,.!):lA) .

• E'

We also use

,.,

(the support function of A),

h(A,x)-suPY'X

I

p(A, x) - inf(t > 0 Ix itA }.

Define

g(.!, r, t) ~ J.!' - r'

+ I' -I.

Notice thatg(A, r, t) is monotonic decreasing, positive and strictly convex as a function of I when A > r.

Let K bea nonemptyset contained in Ihe ball o/radius raround the origin in R" , then the relation LEMMA 1.

(2.1 )

p(K' , u) "' g(.!, r, h(K, - u»

is satisfied for every A ~ r and every UESII - 1• PROOF.

Let u be any unit vector, let A satisfy ).

~

r and let a be the right

hand sideof(2.1). We first showthatauEK~. Let x be any point of K. We have 58

180

1st. J. Matb.

O. SCHRAMM

IIxll ;:ir,

- x·u:i h(K, - u).

Using this and a ;;: 0 we obtain

II x- au ft' - II x II' - 2ax·u + a';:;; r'+ 2ah(K, - u)+ a' -A'. This means that auEB(x, J..) and since x is an arbitrary point of K, we have

auE

n

• ex

B(x,A)-K' .

The origin is also a point of K4 , because K C B(O, ,) and;' ;;: r. Kl is obviously

convex, so we have

{IU I0;:;; I;:; aJ C K'.

•

This shows that p(KA, u);: a, as needed.

In some contexts, a good way to present the volume of a set K c R" is to specify the radius of the ball having the same volume as K. We will call it the effective radius of the set K and denote it by er K:

Vol K - Vol B(O, er K). By J.I. we denote the n - I dimensional surface area measure on S" -

I,

the

boundary of the unit ball.

LeI K be a sel o[diameter d and circumradius r. Let 1 satisfy

THEOREM 2.

A>r.then

er K' and their interiors, together with the curved triangular regions in the complement of the packing. ) For example , the carrier of the hexagonal packing is the whole plane, and its complement with respect to the sphere consists of a single point, the point at infinity. Therefore the rigidity of the hexagonal packing follows from Theorem 1.1.

Similar results are proved for circle packings that almost fill an open geometric disk, and for packings consisting of convex shapes. See Theorems 5.1, 5.3, 5.4.

The original proof of the rigidity of the hexagonal packing had relied on some rather heavy machinery. Later, He [Hel] obtained a more direct proof with some accurate estimates. Both proofs use the theory of quasiconformal maps. The techniques of [RS] and [Hel] can be used to yield rigidity results for packings with bounded valence (that is, the nerve has bounded valence) that fill the plane, but it seems unlikely that they would be sufficient to prove the rigidity theorem above. Recently, Stephenson [Ste] announced a rigidity proof that uses probabilistic arguments, but the assumptions there are even more restrictive than bounded valence. The methods used here are mostly elementary plane topology arguments, together with simple manipulations with the Mobius group. You will find no formulae or inequalities in this paper. (But admittedly, some basic ideas of the theory of quasiconformal maps do playa role, in analogy, and in motivation. Conversely, the results here give a new perspective to the concept of the conformal modulus. ) Our main tool is a theorem that we call the Incompatibility Theorem (3.1 ). It is a theorem about packings of sets more general than circles. In fact, the theorem is purely topological - no geometry is present in its hypotheses or its conclusions. The generality of the Incompatibility Theorem might seem a little out of place to the reader, and warrants an explanation of the background and the motivation. In [Sch I] we generalized the existence part of the circle packing theorem to packings by more general shapes, in particular to packings of convex sets specified up to homothety, and of balls of Riemannian metrics. To give an illustration, we quote: Convex Packing Theorem. Let G = G( V , E ) be a planar graph. and for each

vertex v E V. let PI) be some smooch planar convex body. Then there exists a packing (P~ : v E V ) in the plane, with nerve G. and with each P~ positively homothetic to PI)' 64

RIGIDITY OF INFINITE (CIRCLE) PACKINGS

'29

The fact that ~ is positively homothetic to Pv means that P; is obtained from PI! by a positive homothety; i.e., a transformation of the form x _ ax+b for some b E )R2, a E lR, a > O. The incompatibility theorem was conceived in an effort to understand the extent to which these more general packings are unique. In fact, the incompatibility theorem does give a rather painless proof for uniqueness, as well as existence, for packings under very general conditions; see [Sch2]. As the question of uniqueness of infinite circle packings is now answered in Quite general circumstances, one is naturally led to the problem of existence. Consider an infinite planar triangulation T, and assume that T has just one end. (In the complement of any finite collection of vertices of T precisely one connected component is infinite.) It is easy to obtain a circle packing in the plane with nerve T by taking geometric limits of finite packings. But one can say more. There is such a packing whose carrier is either the plane or the open unit disk. For no such triangulation are both of these possibilities feasible and, in any case, the packing is unique up to Mobius transformations. The existence part will be proved elsewhere, and the uniqueness part follows from the results of this paper. (This proves a conjecture of Thurston [Th2].) 2. DEFINITIONS AND NOTATIONS

We consider the plane R2 = C as being contained in the sphere C = S2 . A circle in the plane is also a circle when thought of as a subset of S2. When we use the term 'circle', we usually mean 'the circle together with its interior'; that is, a closed round disk. For us, a packing means an indexed collection P = (Pv : v E V ) of compact connected sets in the sphere S2 with the property that the interior of each set Pv is disjoint from the other sets Pw ' w ¥- v. The nerve of the packing P = (PI! : v E V ) is a graph whose vertex set is V, and that is defined by the property that there is an edge between two distinct vertices, v, W, if and only if the corresponding sets, Pv ' Pw ' intersect. Note that there is at most a single edge between v, W in the nerve, even if PI! and Pw intersect in more than one place. If G is a graph, then we use the notation. G( V ) to mean that V is the vertex set of G. For any A c S2 let A C = S2 - A denote the complement of A. We deal with packings of sets that are well behaved topologically. Most of the sets we pack are disldike: a set A c S2 is disklike if it is the closure of its interior, its interior is connected, and the complement of any connected component of A C is a topological disk. The obvious example for a disklike set is a topological disk. A more general example is a closed ball for a path metric on S2 (a metric in which the infimum of the lengths of paths joining any two points is the distance between them). Let P = (PI! : v E V ) be a packing of disklike sets on the sphere, and let G be its nerve. If the intersection of any three of the sets PI! is empty, then 65

no

ODED SCHRAMM

the nerve of the packing is a planar graph; that is, it can be embedded in the plane. To find such an embedding one picks an interior point Pv in every Pv to correspond to the vertex v , and one chooses a simple path from Pv to Pm in Pv U Pw to correspond to an edge v ...... w in G. With a little care one makes sure that these curves intersect only at the vertices. If P satisfies the condition that the intersection of any three sets of P is empty, then P is called a nondegenerate packing. Otherwise, it is called a

degenerate packing.

An embedding of a connected locally finite graph in the sphere is called a tri-

angulation if the boundary of every connected component of the complement of the graph, with respect to the sphere, either consists of three edges of the graph or does not intersect the embedded graph at all I . The latter situation can occur only if the graph is infinite. It is easy to see that a connected locally finite graph embedded in the sphere is a triangulation if and only if the set of neighbors of every vertex is the set of vertices of a simple closed path in the graph. (We assume that there are no mUltiple edges or loops, and that the graph has more than three vertices. ) Because this condition does not depend on the particular embedding, it follows that a planar graph is a triangulation in every embedding if it is a triangulation in some embedding. Furthermore, the embedding is unique up to homeomorphisms of the sphere (using the fact that a cycle of an embedded graph separates the sphere). Thus we freely identify the graph with the embedded graph, and we say that the graph itself is a triangulation. A connected component of the complement of the embedded graph whose boundary consists of three edges is called a triangle of the graph. Generally we ignore triangulations with fewer than four vertices. Let G be a finite planar gmph, and let B be a simple closed path in G. We say that G is a triangulation with boundary B if G - B is connected and if G has an embedding I in the sphere S2 where all the connected components of S2 - I (G) are triangles (that is, bounded by three edges of I (G)) , except possibly for one component whose boundary is I (B ). If B has precisely four vertices, then G is called a triangulation of a quadrilateral. The vertices of B are the boundary vertices of G. A quadrilateral is a closed topological disk D in S2 with four distinguished points PO' PI ' P2 ' p) on its boundary that are oriented clockwise with respect to the interior of D. Di will be used to denote the arc of the boundary of this quadrilateral that extends clockwise from Pi- I to Pi' with P4 standing for po · Such a quadrilateral is denoted by (DI ' D2 , D), D4) , D(po' PI' P2' p)), or just D, depending on convenience. Similarly, a trilateral D = (D], D2 , D3) is defined as a topological disk with three distinct distinguished clockwise oriented points on its boundary. tIn standard terminology, the graph is the I-skelaton of a triangulation of a spherical surface without boundary. However, as explained shortly, it is justifiable to call the graph itself a triangulation, because the triangulation is reproducible from it. 66


co 00 0 O O@

III

FIGURE 2.1. Topologically there are eight ways in which two circles can intersect.

Let P be a nondegenerate packing in S2 whose nerve is a triangulation T, possibly with boundary. When we consider the sphere as oriented, the packing P naturally induces an orientation on its nerve T (that is, a 'clockwise' orientation for the triangles of T ). From now on, a triangulation (with boundary) means an oriented triangulation (with boundary), and when we say that a packing has a triangulation as its nerve, it is implicit that the orientation that it induces on the nerve is compatible with the orientation of the triangulation. The motivation for the following definitions comes from the simple but very important observation that the possible patterns of intersection of two circles are very special, topologically. See Figure 2.1. Let D be a subset of the sphere S2 and p , Q E S2 . We say that a curve y connects p and Q in D if the endpoints of y are p, q , and relint( y) c D. Here, and in the following, relint(y) means y - {its endpoints}. Note that p and Q do not have to be in D to be connected by a curve in D ; they may be in D-D. Let A, B be two topological disks in the sphere. We say that A cuts B if there are two points in B - interior(A) that are not connected by any curve in interior(B - A ) . A and B are incompatible if A ::/: B and A cuts B , or B cuts A. Otherwise, they are compatible. See Figure 2.2 on the next page. If A , Bare disklike, then they will be considered compatible if A = B or IC IC A and B are unequal and compatible for every connected component AI of A C and every connected component BI of B C • In other words, either A = B , or whenever you adjoin to A all but one of the connected components of its complement and do the same to B, the resulting topological disks are unequal and compatible. (This definition is, in turn , compatible with the definition for the case where A , B are topological disks. )

2.1. Examples. ( I ) Any two circles are compatible. (2) More generally, homothetic strictly convex bodies in the plane are compatible. (3) The closures of the complements of two compatible topological disks are also compatible. 67

'"

ODED SCHRAMM

(§D~

fj7@

~

(a) Some compatible pairs.

( 0 )

(b) Some incompatible pairs. FIGURE 2.2

(4) Homothetic convex shapes that are not strictly convex can be incompatible. (5) It is not hard to see that given a Riemannian metric on the sphere, any two metric balls of it are compatible. 1

And finally, some notation. Whenever Q = (Qv : v E V ) is a packing and U V€J Qu is abbreviated to Q/.

c V, the union

3. THE INCOMPATIBILITY THEOREM

This section is devoted mostly to the following theorem, which is our central tool. 3.1.

Incompatibility Theorem. Let T = T ( V) be a finite triangulation of a

quadrilateral with boundary vertices a, b, C, d in clockwise order with respect to the other vertices of T (if such exist ). Let D = (D" D2 , D3 , D4 ) be a quadrilateral in S2. Suppose that Q = (Qv : v E V ) and P = (Pv : v E V) are two nondegenerate packings in D. both having nerve T. Further suppose that Qv and Pv are disklike for v E V - {a , b , c, d}; that Pa C D! , Pb C D2 • Pc C D3' Q d :> D4 ; and that Pv isdisjoinlfrom Qd for v E V-{a, c, d } . Then there is some vertex V E V - {a , b, c, d} for which Qv and Pv are incompatible. See Figure 3.t. 3.2. Remarks. The quadrilateral D is not an imponant ingredient of the situation to which the theorem applies. It is merely a frame of reference that helps us describe the relative positions of the packings P and Q. In the situations where the theorem is applied the quadrilateral D is not mentioned. The incompatibility theorem is the packing-theoretic analogue of the concept of the conformal modulus of a quadrilateral. Lemma 5.2 is probably the corresponding analogue for the concept of the conformal modulus of an annulus. 3.3. Coalescing. Before the proof of the theorem, we describe a simple procedure that will be very useful to us. Let T = T ( V ) be some finite planar

68

RIGIDITY OF INFI NITE (CIRCLE) PACKINGS

J33

D,

b

D,

a

d

D,

FIOURE 3.1. An incompatibility is forced. (Find it. )

FIOURE 3.2. Coalescing a set of vertices. First some multiple edges are created, but these can be deleted, together with some of the vertices, 10 obtain a triangulation. triangulation, and let Q = (Qv : v E V ) be some packing of disklike sets with nerve T. Suppose that U is some nonempty connected subset of vertices, and suppose that z is some vertex that is not in U. We describe what we mean by coalescing Qu while keeping z. The idea is that we want to consider the packing Q that is the same as Q except for the fact that all the sets Qu' v E U are united into one, Qu ' Q might not be a good packing, however, because its nerve might not be a planar triangulation, and the set Q u might not be disldike. First, we look at the combinatorial picture. Coalescing the sets Qu' v E U is analogous to coalescing the vertices U into one. Consider a realization of T in the plane. Let t be a picture obtained from T by collapsing the vertices in U. See Figure 3.2. Generally, in t there are some loops and multiple edges, but every 2-cell determined by t has at most three edges of t on its boundary. We want to have a triangulation without any loops or multiple edges. Therefore we must remove the excess edges from t . and do this without having 2-cells with too many edges on their boundary. 69

134

ODED SCHRAMM

cO

(bl ~

(al

FIGURE 3.3. Initially, after coalescing, one may obtain a nondisklike set, but this is easily remedied.

Consider a loop I in t; that is, an edge having the same vertex at both its endpoints. If we just delete I from t, then we might have some resulting 2-cell, which is a quadrilateral, and we do not want that. The loop I determines two regions in t - J. If we delete I together with all the edges and vertices of t that are in one of these regions, then in the resulting picture all the 2-cells are still 1-. 2-. or 3-g005. Because we want to keep z . we delete with I all the stuff that is in a region that does not contain z. Similarly, we deal with multiple edges. If two edges of t have the same two vertices as endpoints, then we delete one of them, together with all the stuff in one of the two regions determined by these two edges that does not contain z. It is straightforward to verify that this procedure is consistent, and that one obtains a triangulation 1". (This is so provided that z neighbors with some vertex that is not in U. If all the neighbors of z are in U, then T' consists of two vertices and an edge between them.) In 1", z and the vertices neighboring with z are still present. Let V' be the set of vertices of T, and assume that the vertex corresponding to the coalesced set U is u. Let Q' = ( Q~ : v E Vi ) be the packing defined by C4. = Qv ' v E Vi - {u}, Q: = Qu ' Then the packing Q' has the nerve T', and the only remaining might not be disklike. See Figure 3.3(a). We then problem is that the set modify slightly to make it di sklike, as in Figure 3.3(b). Note that this can be done with an arbitrarily minute modification of while keeping QI as a packing with the same combinatorics. The packing obtained in this manner is called the packing obtained from Q by coalescing U while keeping z. This procedure is done similarly for triangulations with boundary, and for infinite triangulations, provided the boundary of U is finite.

Q:

Q:

Q:,

Proof of Incompatibility Theorem. First we make an easy reduction to the case where all the sets Qv' Pv ' V E V - {a, b, c , d} are topological disks. If Qv (v =F a, b , c, d ), say, is not a topological disk, then we can adjoin the connected components of its complement to Qv' except for the one that intersects the other sets Qw' W ::f:. v . Because Qv is disklike, the resulting set Q~ would be a topological disk. We apply this procedure to all the Qv' Pv ' V E V {a , b, c, d} that are not topological disks, and obtain two packings QI, pi 70

RIGIDITY OF INFI NITE (CIRCLE) PACKINGS

III

that satisfy the hypotheses of the theorem. If Q~ and P~ (W:f:. a, b, c, d ) are incompatible, then surely, the same holds for Qw and Pw ' We may, and do, therefore assume that the sets Qv ' Pv ' v E V - {a, b, c, d } are topological disks. The proof proceeds by induction on the numbtr of vertices in T. The base of the induction is the case in which there are no vertices in T other than a , b, C, d. Because DI and D3 are disjoint, and Pa C D I , Pe C D3 , there cannot be any edge between a and c in T. If there is an edge between band d, then Qd must intersect Qh C D2 • This would force Qd n Ph :f:. 121, because Qd is connected and intersects D2 and D4 • and Pb is connected and intersects DI and D 3 . By our assumptions, however, Qd n Ph = 121, and therefore there cannot be an edge between band d. As a result, since T is a triangulation, there must be some vertices other than a , b , c , d. This takes care of the base of the induction. (As a warm-up, the reader may want to examine the case where T has 5 vertices. ) Set J = V- {a , c, d}. Let a = vo ' V I ' ... , VII' v lI + I =c be the neighbors of din T in clockwise order. Note that b fJ. {va' V I ' ... , v/l+ I } ' If Qv _ and Pv, are incompatible for some i = 1 , 2 , ... , n , then we are done. So assume that Qv; and PVI are compatible for i = I , 2 , ... , n. Our method of proof is to find a nonempty set of vertices He { V I ' v 2 ' ... , vn} so that QH is disjoint from P' - H' After this is done, we examine the packings obtained by coalescing QH into Qd and PH into Pd ' Then the induction hypothesis establishes the theorem. The proof is divided into three sections. a. Constructing H. Let E be the connected component of D - (P, u Qd) whose boundary intersects Qd' See Figure 3.4 on page 136. P, is disjoint from Qd' and is connected. Therefore E is well defined. We think of E as a quadrilateral whose edges are, in clockwise order, Eo = 8E n D I ' Ep = 8E n P, . Ee = 8 E n D3 , EQ = BE n Qd' Note that there is no loss of generality in assuming that EQ is a simple curve, because we can easily modify Qd slightly so that it has a nice boundary without disrupting the hypotheses of the theorem. Thus we assume that this is the case. We think of EQ and Ep as being oriented from D] to D 3 , with E Q having E on its left, and E p having E on its right. With the intention of introducing some more notations, we walk along Ep. As we walk on E p , we visit, in consecutive order, the sets Pv ' Pv ' ... , Pv . This gives a sequence of arcs }'I' 1'2 ' ... , }'" . In other words, " Ep consists of these arcs in this order, and each arc lies on the boundary of Pv, • Let Pi be the terminal point of the arc Yi' i = I , 2 , ... , n (this is also the initial point of }'i+ ] for j < n ), and let Po be the initial point of f l' For i = 1, 2 , ... , n , Vi neighbors with d. Pick some arbitrary point qi in the intersection Q v_n Qd . It is important to note that the points Q] , Q2' .•• , QII appear in that order on E Q . ( E Q is oriented from D] to D 3 . ) Again, let i be some index in the range i = 1, 2 , ... , n. We say that V i is an invader

'i

71

.

ODED SCHRAMM

136

P

D,

P

P

"

"

Y,

'3

P

"

Y3

Y,

Y,

p

",

Y,

D3

FIGURE 3.4. The Quadrilateral E and its accessories.

Y,

Q, FIGURE 3.5.

Vi

is an invader.

if QUi intersects p,-{v; } - {Pi_ I ' pJ. See Figure 3.5. Consider some invader v i and some point P E Qu; n (P' -{v;} - {Pi- I' pJ } . As qj and p are in QV interior(Pv, )' there is some simple curve, say Pi ' connecting them in interior(Qvi - PI) ) i ' because Qu and P are compatible. We orient Pi from i VI qj to p. Let r be the first point of Pi that is in Pi ( Pi has some point of PJ , namely p ), and let O J be the part of Pi extending from qj to r . o. j is a cross·cut of the Quadrilateral E. It has one endpoint, qj' in EQ and one endpoint, r , in Ep. Note that r , the endpoint of 0. ; that is on E p , is not in f j ' If r -:f:. p , then this is clear, because the curve Pi avoids PI). , except perhaps at its endpoints. If r = p , then we know that r '" Pi- I ' Pi' and since P E p,- {v/ }' P rt 'Ii · For every invader Vi there are two possibilities. Either (Xi separates Yi from Ea in E , or it separates rj from Ee in E. (See Figure 3.5.) In the first case we call i a left invader, and in the second case we call i a right invader. Note that -

j

72


a

E

,

137

Y,

a, qj

Q,

q,

FIGURE 3.6. a: j and a:k must cross, but this is impossible. if j is a left invader, then OJ separates in E every one of the points qh' h < i from every one of the arcs }'m' m ~ i. Similarly, if i is a right invader, then a: j separates in E every one of the points qh' h > i from every one of the arcs '1m , m~i. For notational convenience, let 0 be considered a right invader, and let n + I be considered a left invader. Since 0 is a right invader, n+ I is left invader, and every invader is either right or left, there will be some indices a ~ j < k ~ n + I so that j is a right invader, k is a left invader, and there is no invader between j and k. (To find such j , k, start with j = O. k = n+ I . If there is no invader between j and k , stop. Otherwise pick some invader j in the range j < i < k . If j is a right invader, set j := j. Otherwise, set k := i. Continue in this manner until a situation is reached in which there is no invader j . j < i < k. ) Suppose that j < k are such indices, and let H = {Vj+ I' vj + 2 • · .• ,Uk_ I}' b. Showing that H ¥ 0 , and that QH is disjoint from P'-H' To prove H oj; 0 , it is sufficient to demonstrate that j + I ¥ k. Wishing to arrive at a contradiction, assume that j + I = k. Since n > 0 , either j > a or k < n + 1 (or both). Because these cases are symmetric, we assume that k < n + I. As k is a left invader, it is obvious that k > I and therefore, also j = k - I > O. Again, because k is a left invader, o k separates qj from }lk in E. Similarly, a: j separates qk from }lj in E. Since '1 j comes before Yk on Ep and qj comes before qk on E Q , this implies that the curves OJ and Ok must cross. (See Figure 3.6.) This is impossible, because vk ¥ v j ' and relint(a:,.) C interior(Qv ) , relint (a:k ) C interior(Qv, ) . This contradiction shows that k > j + I , thus establishing H ¥ 0 . Now we will see that QH n PJ - H = 0 ; i.e., that Qv. n P' - H = 0 for j < h < k. Let h be an index in the range j < h < k. If Qv. n p,- {v. } = 0, then obviously ~. n P'- H = 0. So assume otherwise, and let p be a point in Qv. n p, _{v.}· We need to show that p fi P' - H' Because v h is nOl an invader,

,

73

ODED SCHRAMM

we know that P must be Ph or Ph -I . Since these cases are treated similarly, we only consider the case P = Ph . The point Ph is in the intersection Pv n Pvh.1 . Because the intersection of any three sets in the packing P is empty, no other set Pu (u ¥ v h ' V h +1 ) contains Ph. Thus we only need to worry about Pv , . If h + 1 < k, then Vh+l E H , and if h + I = k = n + I , then Vh +l = c fJ. J. Therefore we may, and do consider only the case h + I = k < n + I . Now, qh ' Ph E Q v - interior(Pv ). • say 0:, that • and '4, and Pv are compatible. Therefore, there is some curve, • • connects qh and Ph in interior(Qv - Pv ) . • • interior(Qv ) and interior(Qv ) are The curve 0: cannot cross O:k' because • relint(o) is disjoint • disjoint. Similarly, relint(o ) is disjoint from EQ . Also, from E p , because it certainly is disjoint from '/h C Pv ' and v h is n01 an • invader. Thus relint(o) c E. But the two endpoints of 0:, qh ' and Ph = Pk -l are separated in E by G k , because v k is a left invader. This means that 0: must cross cx k ; a contradiction. This contradiction establishes Qv. n PJ _ H = 0, and therefore Q H n PJ - H = 0 . c. Applying the inductive hypothesis. Let Q' and pi be the packings ob· tained from Q and P , respectively, by coalescing H U {d} while keeping b. Let T = T ( V i) be the triangulation that is the common nerve of these two packings, and let d' be the vertex of T corresponding to the coalesced set H u {d }. Because of part b, we have Q~' n p~'-{a,c.d'} = 0, provided that the touch·up modification done in the coalescing procedure is not too big. The inductive hypothesis now applies to the packings Q' and p I, and establishes the theorem. 0 ~

..

4 . RIGIDITY OF INFINITE CIRCLE PACK.INGS THAT ALMOST FILL THE SPHERE

In this section we prove Theorem 1. 1. 4.1. Definitions. Let P = (Pv : v E V ) be a finite or infinite packing in S2. An interstice of p is a connected component of the complement of Pv = UVEv Pv whose boundary is formed by finitely many of the sets Pv ' The carrier of P , carrier(P) • is the union of all the interstices and all the sets Pv • A connected component of S2 - carrier(P ) is called a singularity of P. A singularity is parabolic if it consists of a single point. And singular(P ) denotes the union of the singularities of P: singular(P ) = S2 - carrier(P ) .

Proof of Rigidity Theorem 1.1. Let P = (Pv : v E V ) be another circle packing whose nerve is T , and let [a , b, c] be some triangle in T . By making an appropriate initial normalization, we assume that Pv = Q u for v = a , b, c. (The Mobius transformation that takes the three intersection points of the cir· cles Pa , PI)' Pc to the corresponding intersection points of Qa' QI) ,Qc also takes Pa , PI)' Pc to Qa' QI)' Qc ' respectively. 74

RIGIDITY OF INFINITE (CIRCLE) PACKlNGS

139

If Qv = Pv for each v E V , then we are done. With the intention of reaching a contradiction, we assume that this is not the case. Starting from [a, b, c] , we walk on triangles of T by moving from a triangle to an adjacent triangle. We can do this, and reach some first triangle where the three circles corresponding to the vertices are not the same in both packings. So, without loss of generality, we assume that Qd '" Pd ' where d is the vertex other than b that makes a triangle with a and c. We further assume that the point 00 is contained in the interstice corresponding to the triangle [a , b, c]. Thus our packings lie in the plane. There are two possibilities. Either Qd is smaller than P d , and the situation is as in Figure 4.1 (a) on the next page, or the other way around. Both cases are treated similarly, and we assume that the situation is as in Figure 4.1 (a). Let p be the point of intersection of Pa and Pc' which is also the point of intersection of Qa and Qc ' Pick some number p > I with P - 1 small, and expand the packing Q by a homothety with center p and expanding ratio p . Continue to denote the resulting packing by Q. The modified picture is given in Figure 4.1 (b). The boundary of Pv = UVE V P" is also the boundary of UVE v interior( P,,) . Therefore it is nowhere dense, and thus is of Baire category l. Because singular(Q) is countable, the set of translations S for which singular(S (Q)) intersects 8Py is also of Baire category I. By Baire's theorem, this shows that there is an arbitrarily small translation S for which singular(S( Q)) is disjoint from aPy • Apply such a translation to Q , and make it small enough so that qualitatively, Figure 4.1 (b) is still correct, except in a small neighborhood of p. We continue to denote the resulting packing by Q. The complement of Qa U Qb U Qc U Qd consists of a quadrilateral that contains the other sets of the packing Q , and two trilaterals (one of the trilaterals contains infinity). Let DQ be the (closure of the) quadrilateral region, and let

Q: ' Q~ , Q~, Q~ be the arcs of DQ that are on Qa' Qb' Qc' Qd' respectively. We have DQ = (Q:, Q~, Q~, Q~). Let the quadrilateral D p = (P;, P; ) in S2 - (Pb U Pb U Pc U Pd ) be defined in the same manner. These two quadrilaterals are indicated in Figure 4.2. Define new packings QI = (Q: : v E V ) and pi = ( P~ : v E V ), by Q: = Q", P~ = Pv for v '" a , b , c, d , and p~, Q: as defined above for v = a , h, c, d. Our two quadrilaterals D Q , Dp , and the two packings QI, p i are in the right relative position to apply the Incompatibility Theorem. (See Remark 3.2. ) If we could apply the theorem to them, then we could conclude that there is some v E V - {a, b, c, d} so that and P~ are not compatible. This is impossible, because these sets are circles, and this contradiction would finish the proof. The problem is that these packings are not finite. We overcome this problem by using the fact that singular(QI) is disjoint from 8P~. We use this fact to cook up finite packings from Q' and p'. Let SJ be the collection of all interstices of the packing p i, and let £: be the collection of all the singularities of the packing p l. The collection

p;, p;,

Q:

75

'"

ODED SCHRAMM

p

(a) Case Qd is smaller than

Pd '

p

(b) After expanding Q. FIGURE 4.1

{H , interior(P;), interior(L) : H E .f), v E V , L E .c} is an open cover for the compact set singular{Q'), because singular(Q' ) n O P~ = 0. Let / { H , interior ( P~), interior(L ) : H E i'/, v E Vi, L E .e } be a finite sub76

RIGIDITY OF INFINITE (CIRCLE ) PACKINGS

FIGURE 4.2.

cover; that is, fj ' C fj , V'

The quadrilaterals DQ and Dp.

c V,

G~ ( U interiOr(p;)) vE V'

141

£,'

c

U(

£,

are finite subsets with

U H) U ( U interiOr(L )) ,

He r:!,

Le.c'

containing singular( Q' ) . Let VI be the set of vertices v E V so that Q~ is not contained in G. VI is a finite set, because for any infinite sequence of distinct circles in the packing Q', the radii of the circles must tend to zero, and any accumulation point for the sequence is necessarily in singular(Q' ) , which is a compact subset of the open set G. Let V2 be some finite connected set of vertices that contains VI u {a . b , c •d } , and let P and Q be the packings obtained from p' and Q', respectively, by coalescing each connected component of V - V2 while keeping a. (RecaUlhe technique of coalescing from 3.3.) Because P and Q are finite, we can now apply the incompatibility theorem. We conclude from it that there are sets Pw and Qw' both corresponding to the same vertex w '" a , b . c, d that are not compatible. This vertex w cannot be in V2 , because the sets of P and Q that correspond to vertices in Vi are circles, and circles are always compatible. Let W be the connected component of V - V2 that coalesced to form w. The union Q:V = Uve w Q~ is a connected set whose closure is contained in G. The union defining G is a finite union of disjoint open sets. Because Q:V is connected, Q~ is contained in one set of this union, say Q~ C interior(F ), with FE fj' u or F = P~ for some U E V' . If F is not disjoint from Pw ' then F is contained in Pw ' Thus Q~ is either contained in the interior of Pw ' or is disjoint from Pw ' Pw and Qw are compatible, therefore, provided that the modifications done during the coalescing procedure are small enough. This contradicts our previous conclusion, and thereby completes the proof of the theorem. 0

r:

It is possible to obtain a generalization of this theorem to packings by convex

sets. This is given at the end of the next section. 77

ODED SCHRAMM

142

5. RIGIDITY OF PACKINQS THAT ALMOST FILL A DISK

S.l. Rigidity Theorem, hyperbolic case. Let T = T ( V ) be an infinite, planar triangulation. Let Q = (Qv : v E V ) be a circle packing in the open unit disk U with nerve T, and suppose that U - carrier(Q) is at most countable. Let P be another circle packing in U with nerve T. Further assume thai the singularity of P that corresponds to the singularity S2 - U of Q is also equal to S2 - U. Then Q and P are Mobius equivalent. We need the following lemma, which is topological in nature.

5.2. Lemma. Let T = T {V) be a finite triangulation of the sphere. and let a - b be an edge in T. Suppose that P = (Pv : v E V ) and Q = (Qv : v E V )

are two nondegenerate packings a/topological disks on 52 with nerve T. Further suppose that Pa c interior(Qa); Q b C interior(Pb); and Qz C interior(Pz) [or Pz C interior(Qz )] for some Z::f:. a, b in V . Then QI) and PI) are incompatible for some vertex V E V - {a, b, z }.

Proof. The idea is to take a double cover, and obtain a situation to which the Incompatibility Theorem can be applied. The statement of the lemma is symmetric in Q and P , and thus we consider only the case Qz C interior(Pz) . Pick some points sa E interior{Pa ) and Sz E interior{Qz) ' Consider a double cover M of S2, in the topological sense, branched over sa and sz. The Packings P , Q lift to packings p i = ( P~ : v E Vi ), Q' = ( Q~ : v E V i) in M , where to each vertex v E V - {a , z} correspond two vertices in V i, say VI' v 2 ' and to each of a and z corresponds one vertex in V i, say a' and z' , respectively. Figure 5.1 (a) illustrates the sets corresponding to the vertices a and b downstairs, and Figure 5.1(b) illustrates those corresponding to ai, b l ,b2 in the double cover M. For convenience, we assume that Pa n Ph contains precisely two distinct points, say p , p' , and that similarly Qa n Qh = {q, q'}. There clearly is no loss of generality in this assumption, because we can make slight modifications to Pa near Ph and to Q a near Qb ' Let PI ' P2' ql ' Q2' Q; ,q~ be the corresponding points in the double cover M. Let DQ be the closure of the connected component of M - ( Q~ , U Q:, u Q~) that contains the other sets of

p; , p; ,

,

,

,

Q' , and let Dp be the closure of the connected component of M -( P; UP;,UPh')

that contains the other sets of p'. We view DQ and Dp as quadrilaterals with • I, 1 1 ' I LetP a" d be vertices Q" Q2' QI' Q2 and PI ' P2' P" P2' respectlve y. an P" II

,

,

the two edges of the quadrilateral Dp that are on the boundary of P~I. Let ' respecttve . Iy. PhII and PhII be the two edges a f Dp that are on Ph' an d Ph' I

Q; ,

1

I

1

Similarly define the edges of Do' Q; I , 1 Q~I , Q~2 . See Figure 5.2. We now define modified packings p" = ( p~': v EVil), Q" = ( Q~: v EVil). Set V" = (Vi U {a" a 2}) - {a'} . P;' and Q~ have been defined above for v = a, ' a2 ' bl ' b2 . Set = P; and Q~ = Q~ for other v E V". The

p;'

78

RIGIDITY OF INFI NITE (CIR CLE) PACKlNGS

143

FIGURE 5.1 (a). The sets corresponding to a and b.

Q'

b,

P'

"

Q'

b,

Q'

"

FIGURE 5.1 (b). Sets corresponding to ai, b l , b2

In

FIGURE 5.2. The quadrilaterals Dp and D Q . 79

M.

144

ODED SCHRAMM

(b) After expanding and translating P. FIGURE 5.3

r'

packings Q" , pI! have as nerve a common triangulation of a quadrilateral with boundary Q ] , a z ' b l ,bz (provided the labeling in the branching v --+ VI' V z is done consistently for both packings), which can roughly be described as a double cover of T branched over the vertex z and the edge a . . . . b. It is clear that the incompatibility theorem applies to these packings and gives a vertex wE V" - {ai ' Qz' hi' bz} for which Q~ and P~ are incompatible. It is impossible that w = z', because Q~' c interior( P;~) . Thus W = vj ' say, with v E V - {a, b, z}, and j = 1 or j = 2. But this implies that Qv and Pv afe incompatible, as needed. 0

Proof 0/5. 1. After a few initial normalizations, we reach a situation where the above lemma can be applied. Let Qs be the singularity S2 - U of the packing Q , and let Ps be the corresponding singularity of P. (We assume that the symbol s is not a vertex of T. ) Pick some edge a ...... b in T , and let p be the point of intersection of Qa and Qb . After renormalizing by a Mobius transformation taking U onto U (i.e., a hyperbolic isometry), we may, and do, also assume that Pa n Pb = {p}, and that the outward unit normal of Q a at p is the same as that of Pa . There is some (unique) a > 0 so that expanding Pa by a homothety with center p and expansion a takes Pa to Qa • Apply this expansion to the pack· ing P, and continue to denote the resulting packing by P. Now Ps is either contained in Qs' or contains Qs' depending on whether (): ~ 1 or a ~ I , respectively. Case 1. Pb ~ Q b . See Figure 5.3{a). Expand the packing P by a homothety with center p. Make the expansion ratio P > 1 , but small enough so that afterwards Pa ~ Q a' Pb

above relations, but makes all the singularities of Q , except possibly Qs' be disjoint from the boundary of Pv = UVE V PV ' By Baire's category theorem, this can be done, as in the proof of 1,1. We assume that Ps C interior(Qs) ' The other possibility is dealt with similarly. Let Z be a set of vertices whose boundary is finite, so that all the sets Pv ' v E Z are contained in interior(Qs) ' and Pv-z is bounded away from Ps ' (To see that one can find such a set of vertices, let ~ be the set of vertices whose distance from a is at most n. Let Cn be the simple closed path in the boundary of ~ so that Pc separates Pv -c from Ps ' As n -+ 00 , Pc -+ 8 Ps ' Thus, for sufficiently "lrge n, one can take Z to be the connected co~ponent of V - Cn that does not contain a.) Let p i and Q' be the packings obtained by coalescing Z to some vertex, say z , while keeping a. We make further coalescings, as in the proof of 1.1 , but keep the vertices a , b , z. Then to each set in these packings we adjoin the connected components of its complement which do not intersect Qa (to make them into topological disks). Eventually, we obtain finite packings P" , Q" with a common triangulation as nerve; corresponding sets being compatible; . . (Q"z' ) QIIa C mtenor . . (P") . . (Q") ' an d PzII C mtenor a ' an d pil b C mtenor b ' Th IS contradicts the lemma, and thus Case I is ruled out. Case 2. Qb ~ Pb . We make the same argument in this case, only now take o < P < 1 and interchange the roles of a and b. Case 3. Q b t;/.. Pb and Pb t;/.. Q b ' This case is impossible, because the boundaries of Pb and Qb both pass through p and have the same unit outward normal there. Case 4. Qb = Pb . If Qv = Pv for all v E V , then we are done, so assume otherwise. We also have Qa = Pa , and therefore there is some triangle [c, d , e} in T with Qc = Pc' Qd = Pd ' and QI' #- PI" We then have either QI' bigger than PI' and Qe U Qc U Qd separates PI' from every Pv ' v #- c, d , e , as in Figure 5.4(a) (see p. 146), or the other way around. Both situations are dealt with similarly, so assume that QI' is bigger than PI" Let p' be the point of intersection of Pc and Pd (also Qc n Qd = {pi} ). Expand the packing P by a homothety with center pi and expansion ratio P > I with P-1 small. Then we have the sets Qv' Pv ' V = c , d , e as in Figure 5.4(b), and Ps c interior(Qs) , if 0" ~ I , or Qs c interior(Ps) ' otherwise. Now, if necessary, modify QI' so that its interior would contain PI" as in Figure 5.4(c), while keeping Q as a packing with the same combinatorics. (That is, Qt must still touch the sets it used to touch. It may perhaps no longer be a circle, though. ) A very slight translation would now give Qd C interior(Pd ), and a contradiction can be reached as in case I, with d replacing a and e replacing b . 0 In the proof above, except for the first normalization, all the transformations applied were translations and positive homotheties. Going through the proof, one sees that (except for this initial normalization), we did not use the fact that 81

146

ODED SCHRAMM

(a) Case 4, with Qe bigger than Pe .

P,

(b) After expanding P.

Q,

(c) After modifying Qe . FIGURE 5.4

82


147

a

a

u

b

c

c

(a) Cannot find a circumscribing convex U. FIGURE

(b) After a harmless modification, such a V exists. 5.5

we are working with circles and, in fact, strict convexity and smoothness are enough. Thus we have the following generalization of 5.1. 5.3. Theorem. Let T = T ( V ) be an infinite, planar triangulation. Let Q = (Qv : v E V ) be a packing with nerve T of smooth strictly convex sets in an open convex set U. Suppose that U - carrier(Q) is at most countable. Then Q is rigid up to three degrees of freedom, in the following sense. Let a .... b be some edge in T , and let P be another packing in U with nerve T. Further assume that Pv is positively homothetic to Qv for v E V; the singularity of P that corresponds to the singularity S2 - V of Q is also equal to S2 - V ; Pa n Pb = Qa n Qb ; and the ourward unit normal of Pa at this intersection point is the same as that of Qa . Then P = Q.

The only need for smoothness is to insure that the packings be nondegenerate. One could dispense with the smoothness hypothesis, if one restricts the word 'packing' to mean 'nondegenerate packing'. We also prove the following theorem, which can be seen as a generalization of 1.1. 5.4. Theorem. Let T = T (V ) be a (possibly infinite) planar triangulation. Let Q = (Qv : v E V ) be a packing with nerve T of smooth strictly convex bodies in the plane. Suppose that 00 is contained in one of the interstices of Q, say in the interstice corresponding to the triangle [a, b, c] of T. Further suppose that singular(Q) is at most countable. Then Q is rigid, in the following sense. Let P be another packing in the plane with nerve T; Pv positively homothetic to Qv for each v; and Pv = Qv for v = a, b, c. Then Q = P. Proof. First assume that there is some smooth convex body U that contains QaUQb UQc ' and each of Ov . v = a , b , c , intersects the boundary of V . Let T' be the triangulation obtained from T by adding one additional vertex, say s, 83

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ODED SCHRAMM

in the triangle [a, b, c), adjoining edges from s to a, b, c, and splitting the triangle [a , b , c] into three smaller triangles: [a, b , s1, [b , c, 5], [c , a , sl. Let p i and Q' be the packings obtained from P and Q by adding another set = P; = S2 - interior( U ). These packings obviously have nerve 1". Now, the proof of 5.1 Case 4 can be applied here, because the fact that = is not a singularity, but a packed set, only makes things easier. We get pi = Q' , which gives P = Q . There may be a situation where no such set U exists, as in Figure 5.5(a). However, we are free to manipulate the sets Pv ' Qu' v = a, b, c, provided we do not modify the parts of their boundaries that bound the component of S2 - (Qa U Qb U Qc) that contains the other sets of the packings. Thus we easily reduce the situation to the case where such a U exists. See Figure 5.5(b). 0

Q; Q; P;

6.

SOME PROBLEMS

One is naturally led to the following conjecture, which probably also occurred to other circle packers. 6.1. Conjecture. Let T be an infinite Iriangulation with at most countably many ends. Then there exists a circle packing P in 52 whose nerve is T and so that all the singularities oj P are either circles or single points. This packing is unique, up to Mobius transformations. With the additional assumption that T has bounded valence, the uniqueness part of this conjecture can be proved. The proof uses the techniques presented here, and the fact that in the bounded valence case, two packings with the same triangulation as nerve induce homeomorphisms between the boundaries of the corresponding singularities, provided these boundaries are simple closed curves. This fact follows from the analogous property for quasiconformal maps, and perhaps might also be true without the bounded valence restriction. As mentioned in the introduction, the conjecture holds when T has one end. The observant reader may have noticed that our rigidity results for packings of convex sets other than circles do not deal with the case where infinity is (on the boundary of) a singularity of the packing. The reason for this is that we need homotheties to perturb the singularities, and the point at infinity is a fixed point for the homotheties. However, this difficulty does not rule out some kind of rigidity for packings having {oo} as a singularity. For example, one may ask: 6.2. Problem. Let P and Q be packings of smooth strictly convex bodies in the plane, both having a triangulation T = T ( V ) as their nerve. Suppose that carrier(P ) = carrier(Q) = IR? ; PIJ is positively homothetic to QIJ for each v; and PIJ = QIJ for v =a , b ,c, where [a ,b,c) is some triangie of T. Does it follow that Q = P? Added in proof Conjecture 6.1 is true; the proof will appear in a joint work with Zheng·Xu He. 84


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ACKNOWLEDGMENTS

I am deeply thankful to my teachers Bill Thurston and Peter Doyle, and to Richard Schwartz, Burt Rodin, and Zheng·Xu He for stimulating discussions relating to packings. REFERENCES

(Ani] (An2J [BFP] [CR] lHe l] [He21 {RoiJ (Ro2]

IRSI {Sehl]

(Sch21 (Ste] (Thl l

ITh'I

E. M. Andreev, On com'ex polyhedra in LobaCevskir spaces, Mat. Sb. (N.S. ) 81 (1970), 44 5-478; English transl. in Math. USSR Sb. 10 (1970), 41 3-440. _ _ , On com'ex polyhedra of/mite l'olume in Lobatevski/ space, Mat. Sb. (N.S.) 8J (1970), 256-260; English trans!. in Math. USSR Sb. 12 (1970), 255-259. I. Barany, Z. Ftlredi, and J. Pach, Discrete convex functions and proof of the six circle conjeclure of Fejes TrJth, Ga nad. J. Math. J6.J (1984), 569- 576. I. Garter and B. Rodin, An inverse problem for circle packing and conformal mapping, preprint. Zheng-Xu He, An estimate for hexagonal circle packings, J. Differential aeom. (to appear). _ _ , Solving Beltrami equations by circle packing, Trans. Ame r. Math. Soc. (to appear). B. Rodin, Schwam's lemma for circle packings, Inve n!. Math. 89 (1987), 271 -2 89. _ _ , Schwartz's lemma fo r circle packings II, J. Differential aeom. JO (1 989), 539-554. B. Rod in and D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential aeom. 26 (1 987), 349-360. O. Schramm, Packing two-dimensional bodies with prescribed combina/orics and applications to the cons/ruction of conformal and quasicon/ormal mappings, Ph.D. thesis, Princeton, 1990. _ _ • Uniqueness and exiSlence o/packings with specified combina/orics, Israel J. Math. (to appear). K. Stephenson, Circle packings in the approximation of conformal mappings, Bull. Amer. Math. Soc. 23 (1990), 407-415. w. P. Thurston, The geometry and topology of 3-manifolds, Princeton Univ. Lecture Notes, Princeton, NJ. _ _ , The finite Riemann mapping theorem, invited talk at the Intemational Symposium in Celebration of the Proof of the Bieberbach Conjecture, Purdue University, March 1985.

DEPARTMENT OF MATHEMATtCS, PRINCETON UNIVE RSITY, PRIr-;"CETON, NEW JERSEY, 08544 Current address: Mathematics Department, University of California at San Diego, La Jolla, California 92093 E-mail address: [email protected]

85

Invent. math. 107,543- 560 (1992)

Inventiones

mathematicae

CI Springer-Verlag 1992

How to cage an egg Oded Schramm U niversity ofCalirornia, San Diego, Department of Mathematics, La Jolla, CA 92093, USA Oblalum 18-V-1991

Summary. This paper proves that given a convex polyhedron P e R. 3 and a smooth strictly convex body K c )R3, there is some convex polyhedron Q combinatorically equivalent to P which midscribes K; that is, all the edges of Q are tangent {Q K . Furthermore, with some stronger smoothness conditions on BK, the space of all such Q is a six dimensional differentiable manifold.

I Introduction

For any n= 3, 4, 5, one can find a convex n-gon in the plane so that all its vertices lie on the unit circle. In fact, the convex hull of any n distinct points on the unit circle gives such an n-gon. The situation in 3-space is rather different. Steinitz CSt] has shown that there are convex polyhedra PeR? so that there is no combinatorically equivalent polyhedron having all the vertices on the unit sphere. By duality, it follows that there are combinatorial types of convex polyhedra that are not realizable with all the (2-dimensional) faces tangent to the unit spherc. Schulte [Schu] has generalized the above result of Stcinitz, by showing that for pairs of integers (m, d) satisfying O;:;am < d, d> 2, (m, d)4: (1 , 3) there are combinatorial types of d-dimensional polytopes which are not (m,d)-scribable; that is, they cannot he realized in d-space in such a way that all the m-dimensional faces are tangent to the unit (d - 1)-sphere. The case (m, d) = (l , 3), is an exception. Koehe [Koe] has claimed that every 3 dimensional convex polyhedron is combinatorically equivalent to a polyhedron which midscribes thc unit sphere; that is, all its edges are tangent to the unit sphere. However, the proof in [Koe] is only for polyhedra which are simple or simplicial. The result for general polyhed ra follows from Andreev's Theorem [An i, An 2], as Thurston [Th, Chap. 13J observed. The midscribing polyhed ron is unique, up to projective transformations which preserve the sphere. Schulte also introduced the question of replacing the Euclidean sphere in the above by other convex bodies. It is the purpose of this paper to generalize the Koebe-Andreev-Thurston result in this d irection by proving : I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_5, 87

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1.1 Midscribability Theorem. Let P be a (3-dimensional) convex polyhedron, and let K c R 3 be a smooch strictly convex body. Then there exists a convex polyhedron

Qc R 3 combinarorially equivalent to P which midscribes K.

With stronger smoothness conditions on K , one can say more : 1.2 Theorem. In 11Ieorem 1.1, if for some integer k ;?; l iJ K is CH I-smooth and

has positive Gaussian curvature everywhere, then the space of such midscrihin g Q is a six dimension Ck-smooth manifold.

This also implies that the space of midscribing Q is C"". if iJK is C'" and has positive Gaussian curvature. A few words about the proof of Theorem l.l. The main part of the proof is establishing the theorem for the case where iJ K is C 2-smoo lh and has positive Gaussian curvature everywhere, and then the general case follows by approximations and convergence. We consider the space of all configurations ; where a configuration is, by definition, an indexed collection of planes and points in R) ; the points being indexed by the vertices of P, and the planes are indexed by the faces of P. A configuration corrresponds to a P-type K-mid scribing polyhedron if the obvious conditions hold: the point corresponding to any vertex v of P belongs to every plane corresponding to a face of P containing the vertex v, the line segment joining the points corresponding to two vertices which share an edge in P is tangent to K , etc. With the presence of the stronger smoothness assumptions on K, these different conditions determine differentiable subrnanifolds of configuration space, and so, each point in the intersection of these submanifolds corresponds to a P-type K-midscribing polyhedron. We use geometric observations and a combinatorial argument to show that any intersection of these submanifolds is transverse. Transversality then implies that if we have a P-type K-midscribing polyhedron Q, then for a sufficiently small perturbation K ' of K a K ' midscribing P-typc polyhedron Q' can be found. This enables us to transport midscribing polyhedra from one K to another, and the proof is then easily completed. Certainly, the central part of this argument is the proof of transversality. The method in which we establish transversality is in many respects reminiscent of Cauchy's approach to rigidity of polyhedra ([Ca, Ro]). In a forthcoming paper the author intends to elaborate on this matter, and to use some of the techniques presented here to obtain new proofs of infinitesimal Cauchy rigidity and related results. As noted above, Thurston's proof that the sphere can be midscribed by a polytope of given combinatorics is an application of Andreev's Theorem. As Thurston observed, Andreev's Theorem can be reinterpreted to obtain the Circle Packing Theorem, which says that any planar graph can be realized as the tangency graph of a circle packing on the sphere. In [Schr] the Circle Packing Theorem is generalized to packings of shapes other than circles. Using this generalization, Theorem 1.1 is proved there in the case that P is a simple or a simplicial polyhedron. Although this may not be apparent to the reader, the techniques of thi s paper are also closely related to the theory of packings in two dimensions. In fact, much of the arguments here were conceived as part of a differentiable proof giving uniqueness and existence of packings, in the spirit of [Schr]. Later, the method of [Schr] superseded the differentiable proof. Fortunately, this differentiable approach is most useful for the purposes of this work. 88

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It is worthwhile to note that Thurston's route from circle packings to sphercmidscribing polyhedra can be reversed, and so we obtain a new proof of the Circle Packing Theorem as an additional reward for our efforts. (See 6.1.) Acknowledgements. I would like to c1tpress thanks to Egon Shulte for suggesting to try and use my packing theorems to obtain polyhedra thai midscribe given conve1t bodies. I am also very grateful to Bill Thurston, who advised mc to try the diffeTCnliablc approach to obtain a uniqueness result for packings.

2 Preliminaries The main purpose of this section is to define some terms we later use, and to introduce notations. We freely use some of the most elementary notions in convcxity (good sources for these are, for example, [Eg, Gr]), and these are defined here only if some confusion is anticipated. Similarly, we assume some very rudimentary knowledge of differential topology, especially transversality of submanifolds. A convex body in IR 1 is a compact convex subset of R 3 which has in terior points. A smooth convex body is a convex body which has a unique supporting plane at every boundary point (this is equivalent 10 the boundary being CI-smooth), and a strictly convex body, is a convex body whose boundary does not contain any nontrivial line segment. Given points x, yel~.~. we usc the notation [x,y] to denote the line segment joining x and y. A polyhedron in lR J is the convex hull of finit ely many points in IR 3 which are not all coplanar. Let P be some arbi trary polyhedron in lR l, which will be fixed hencefo rth. We denote the set of vertices of P by V, the set of edges by E, and the sci of (2-dimensional) faces by F. A P-t ype polyhedron will mean a polyhedron which is combinatorical1y equivalent to P. Thc extended graph of P, G+ =G+(P), is the graph whose vcrtex set is Vu F, and whose edges are E u V F, where V F denotes the collection of all pairs ( v,J ) so that ve V, f e F and the vertex v belongs to the face f A vertex i in the graph G+ will be called a V- vertex if ie V, or an F-vertex if ieF. Hopefully, the fact that a face of P is a vertex of G+ will cause no confusion. An edge ( i, j> of G + is a v-v edge if (i,j>eE, and is termed a V-F edge if ( i,j> e VF. G+ has an embedding in S2, canonical up to homeomorphism, which can be visualized on the boundary of P by choosing an interior point in every face and connecting it to the vertices of that facc. We will always consider G+ with this embedding, which detcrmines a triangulation of S2. As above, if i,j are vcrtices of G+ which arc joined by an edge, then ( i,j> will denote that edge. Similarly, the notation ( i, j, will be used for a triangle of G+; that is, a triangle in the triangulation induced by G+. A polyhedron QcR 3 is said to midscribe a smooth strictly convex body KcR l if every edgeofQis tangent to K(that is, to oK).

k>

2.1 Obsenations. Let K c R l be a smooth and strictly convex body, and let Q be a K-midscribing polyhedron. (1) If m is a plane containing a face of Q, then the polygon m ("\ Q circumscribes the smooth convex set m ("\ K in the plane m. 89

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(2) If v,

II

are vertices of Q, then the line segment [v, u] intersects K.

Proof (I) is obvious. By definition, (2) ho lds if and Z,,=Y2 . respectively. Let E'= E-{, and the o ther open half space determined by m' With respect to tv we discriminate between three possibilities: tv eit her points above m (v .w), points below m(v ..,), or is parallel to m (M) ' It t" points above m' Noting that Zy and O=z" are in m O sufficiently small Z(S)i will

be above this plane. If also lj.l= +, then z(s)j will be above that plane for

s>O sufficiently small. But this would mean that the line segment [z(s)j, z(s);] joining z(s») and z(s); lies above this plane, which contradicts Z(S)ES( i.), because K does not intersect the half space above m{i. J>' Thus we see that two + labels cannot occur on the edge (i,j). If L= + , 11.;= 0, then, for sufficiently small s>O, the segment [z(s)j, z(s);] is above m(l.j), except possibly for points arbitrarily close to Zj' Again this gives a contradiction, because K is disjoint from a neighborhood of Zj and does not intersect the half space above m( l.j)' Thus we see that a + and a oare impossible. a nd a Because - l is also in TzS ( I.}), we see that two - labels, or a o are also impossible. The only remaining possibilities are + and - , or two O's, as required. The proof for the ease that (i,j) is a V-F type edge is similar (and simpler). 0 4.8 Lemma (Sign changes in a quadrila teral) Let (v, u) be a v-v type edge in G..j., and let the two faces of P which contain.~ this edge be e, fEF. Suppose that t ET" S (V. M) r1 T. S(~.,,)r1 T% S ( ~./)r1 T% S < ~.~ > r1 T% S(M ./) . If among v, u, e, f there is one live vertex, then the sum of the total number of sign changes in both triangles (v, u, e) and (v, II, f) of G..j. is at feast one. (See Definitio/l 4.5.) If among these four vertices there is more thall one live vertex, then this number of sign changes is af feast two.

Proof No new geometric arguments will appear in the proof - the lemma is a combi natorial consequence of the Orientation Lemma and Lemma 4.4. Consider some triangle (i, j, k) of G..j. , and assume that not all the labels in it are o and that reT" S{I.j)nT. S(J.k)r1T. S(k./). We will show, by a case by case analysis, that the total number of sign changes in this triangle is at least one. Suppose, without loss of generality, thal fi.J"FO, a nd suppose, by symmetry, that II.J= +. Then fj • l = - , by thc Orientation Lemma. We check thc three possibilities for the label fu. If fu= ~, then we already have onc full sign change in (i, j, k) at i. Consider the possibility lu=O. Then there is half a sign change at i, and Ik./=O, again by the Orientation Lemma. If Ik • j = +, - , then we also have half a sign change at k, a nd so only lk.j=O needs to be discussed. This however leads to IJ.k=O, and the half sign change appears at j. It remains to check lu = +, and, by sym metry, only the possibility fj • k = needs consideration. But then Il.I= - and ft . J= +, giving a sign change at k.

We return to the situation of the lemma. Assume that the total number of sign changes in (v, u, e) and (v, u, f) is less than 2. By the above, it follows that in at least one of these triangles, say in (v, u, e), all the labels are O. In particular, we have I~.M= /~. ~ =O. From Lemma 4.4 it then foll ows that 1~.f=O also, a nd by symmetry fw./ = O. From the Orientation Lemma we 96

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gel 11.~ = l,. u= O, and so all the labels on both triangles are O. If, for instance, v is li ve, than there arc two half sign changes at v, one in each triangle. Thus there is a tOlal of one sign change in our quadrilateral (the two neighboring triangles) at each live V-ve rtex.. Also, there clea rly is one sign change for each livc f-vertex. This completes the proof of the lemma. 0 Proof of Th eorem 4.1 Suppose now that t is in the intersection (4.2). As we pointed out in the subsection devoted to the codimension count, il is enough to prove t = 0. Consider the gra ph G+ with the labeling induced by f. Note that t eS(i.}) for every W) ~ L; codim(ll;, W), and that codim(V) =codim(U, W)+codim(W), if V , tJ,c W. Using the Iransversality of the intersection (4.5) and these properties, we have :

•

•

L codim(X )+ L codim(Y;) +codim(M) j

, "' I

=codim«n7_ I X I)n(ni.o! Y;)) + codim(M) =codim«n~_ l Xi)n(ni_ 1 Y;),M)+2codim(M) ~codim(n7", I XI' M)+codim(ni~ 1 ljnM, M)+2 cOdim(M) =codim(ni,'01 X/)+codim(n i" l ij n M) ~

•

•

L codim(Y,)+codim(M). ,.L,codim (-I,,) + ,., 98

How 10 cage an egg

This show that equality holds througho ut, and therefore codim(n;.ot = codim (M") +

•

,.I ,codim(Y;). That

proof is complete.

Y, n M)

is the required transversality at z, and the

0

Proof of Theorem 1.2 We do not yet prove the implicit claim that ZP.K is noncmpty, but prove the rest. Consider some point ZEZ P . K ' For showing tbat ZP . K is a six dimensional submanifold near Z there is clearly no loss of genera lity in assuming that Z EZ~. K ' because of the freedom in the choice of xo , X l ' Y2' Thus we will make this assumption. Consider the set M=Z ... n S(vo.v,)n S(vo,/l)n S(v,. /l) ' It is easy to see that M is a Ck-smooth submanifold having codimension 3, by showing that the in tersection defining M is transverse. (Recall that the S (v .u ) are C·-smooth, by Lemma 3.1). In fact, the transversality of that intersection follows from Theorem 4.1 , because we could have chosen other vertices and another face in place of Vo, VI ' f 2' We have (4.6) Since M :::) Z O n Z I n Z 2 and the intersection (4.1 ) is transverse, it follows from Lemma 4.10 that the intersection (4.6) is transverse at z. Therefore, in a neighborhood of Z, Z .... K is a C·-smooth submanifo ld whose codimension is equal to the sum of the codimensions in the intersection (4,6). So codim (Z .... K) = 3 + IE'I +IVF' I= codim(Z~,K) - 6, using our results from the cod imension count subsection. Thus dim (Zp .K) = 6, and this completes the proof. 0

R emark. It is probably true, and perhaps not too hard to prove, that the assumptio ns in Theorem 1.1 are sufficient to guarantee that Z .... K is a 6-manifold, pe rhaps not smooth.

S Existence of midscrihing configurations

In this section we will prove the Midscribability Theorem 1.1. The basic metbod is the following. We start with some realization of P, ZOEZp, and show that there is a convex bod y, KO, which has positively curved C 2-smooth boundary, a nd which P (zo) midscribes. Then we consider a curve of C 2 ·smooth convex bodies with positively curved boundary, K", 0 <s < 1, which joins KO to Kl = K . Take some SE[O, I), and assume that K" can be P-midscribed. It easily follow s from Theorem 4.1 applied to K" that K" can be P-midscribed for all s'e [O, 1] sufficiently close to s. This shows that the set of SE[O, I) for which K" is pmidscribable is a relatively open set. If we could show tbat the set of SE[O, I] for which K S is P-midscribable is a closed sct, then it would fo llow tbat this set is [0, IJ , and that K = K I is P-midscribable. To show closure, the natural approach is to take a limit of midscribing configurations, and to claim tbat this gives a midscribing configuration. The difficulty in doing tbis is that the points Zu may escape to infinity, or plunge into K' . That is where the proof needs a little care, and where the details come in. 99

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The following lemma will enable us to make sure that the vertices stay within a compact subset oflR 3 , 5.1 Lemma. For any smooth strictly convex body K and for any polyhedron Q rnidscribing K , there are at most two vertices of Q whose distances from K are greater than 2 diameter(K). Proof. Recall that if p, qeR3 arc two distinct vertices of Q. then the line segment joining p and q must intersect K. (Observation 2. 1(2).) Now consider three distinct vertices of Q, p, q, reRl, and assume, without loss of generality, that OeK. If the angles between the vectors p and q, and between the vectors p and rare ;:, 2n/3, then the angles between - p and q, and between -p and rare;;;; Tt/3. This would imply that the angle between q a nd r is ;;;; 2n/3. Therefore, at least one of the three angles determined by p, q, r at 0 is 1= 21[/3. Assume that this is the angle between p and q. Let x be the point closest to 0 on the line segment joining p and q. Because the angle between p and q is at most 2n/ 3, it follow s that 2 11 x ll ~minim um (ll q ll , Il pll ). But II x ll 1= diamt::ter(K), because Oe K , and x is the closest point to 0 on the segment joining p a nd q which contains some point of K. Thus minimum( ll q ll , Il plll 1=2diametcr(K), and the lemma follows. 0 5.2 Lemma. Let vo, VI be two neighboring vertices of P. For every constam C>O, there exists a realization of P, zeZ p , with the following properly. The edge joining z"" and z,,' has length C, but the distan ce from the midpoim of this edge to any vertex z". V=F VO , VI is less than t. Proof We start with an arbitrary realization Q= P(z) of P, a nd modify Q by a projective transformation. By applying an expansion, if necessary, assume that the edge e = [z"", z",J has length C. Pick a p lane Lo in R J which contains the edge e, but is otherwise disjoint from Q. Now think o f R 1 as being embedded in R 4 , and let L c R 4 be an affine three space whose intersection with R 3 is Lo. Let x be the midpoint of the edge e, and let 0 be a point in JR 4 _{ L v Rl } whose distance to x is some small (;> 0 and which is separated from Q - e by L. For any point ye Rl, if the line through y and 0 intersects L, then let f(y) be the point in the intersection of this line with L. The mapping f defined thusly is a projective map from R3 to L. (To be more precise, f extends to a projective map from real projective 3-space to the projective closure of L.) The image of Q, Q' = f(Q), is a realization of P in L. The points on the edge e remain fixed under f On the other hand, let h be the distance from L to the vertices of Q other than z"o' z"" let y be some vertex of Q other than z"o' z"" and let I be the length of the part of [y,o] which lies in the same side of L as y. As Fig. 5.1 illustrates, d(o, L)/ d(o,J(y)) ~ d(y, L) .

Using d(o, L)1=d(o, x)=e, that

I ~ d(y, o) ~ d(y,

x)+d(x, 0), and dey,

L) ~ h,

we see

d(o,j(y)) ~ , (d;amet:,(Q)+e) .

Thus, because the points x, t vo, t " , remain fixed under f, the polyhedron Q' = f(Q) satisfys the required conditions, provided e is sufficiently small. 0 100

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"

L f(y)

o

Fig. 5.1

Proof of Theorem 1.1 Let Vo• VI' f2 be as in Theorem 4.1. We start with a realization of P, P(zo), zO e Zp , which sa tisfies the conclusion of Lemma 5.2 with C=6. For each edge e of P(zo), let Q e be the midpoint of the edge. From the properties of P(zo) it follows that the diameter of the set of all such Q e is < t. For each edge e of P(zo) choose some plane me which supports P(ZO) at that edge; i.e.• P(ZO)nme= e. Now let KO be a C2 ·smooth strictly convex body of diameter < 1 with positively curved boundary and whose boundary conta ins each Q e and is tangent to me at Q e (for each edge e of P(zo)). Clearly, such a K O exists. (To get an explicit construction, one can start with the convex hull of small balls Bu with each Be tangent to me at a~ from the side of m~ which contains interior(P( zo» , and then minutely modify this convex body away from the Qe to make its boundary positively curved and e 2 ). By construction, the polyhedron P(ZO) midscribes K O. Denote by Xo and XI the two vertices zeo and of P(ZO), let eo be the edge of P(zo) joining them, and let Y2=ZJ l' Let KI be a copy of K , rescaled and translated so that it has diameter < t and is tangent to meo at a ~o' from the same side that KO is. It is clearly enough jf we show that KI can be P midscribed. Now choose a curve K J , se[O, I], of strictly convex bodies joining Ko and K I ' We require that each of the sets KJ, se(O, I), is tangent to meo at Q"o' that its boundary be C1-smooth and positively curved, and that its diameter be less than I (and the curve must be continuous, in the Hausdorff metric). Clearly, there exists such a curve. Let A be the set of se [O, I] such that there exists a realization reZI' which midscribes K Sand satisfys z:'o = Xo, r.;, = x!, zil = Y2' Clearly, OeA, because P(zo) midscribes K O. Our goal is to show that l eA, by demonstrating that A n [O, 1) is relatively open and A is closed in [0, I]. Showing that An [0, I) is open in [0, I] is easy after the preparations in the previous section: Consider an seA - {I}. The K · midscribing configuration

ze,

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55'

O. Schramm

z" is in Zp,K" By Theorem 4.1, applied to K ' , the intersection (4.1) is transverse al Z· , This implies that if the intersecting submanifolds in (4.1) are perturbed sligbtly, an intersection close to z' remains. For K' a smooth strictly convex body sufficiently close to K', the corresponding submanifolds are close to those of K', (For each w>eE' there is a homeomorphism of S ( w.... )(K·) to S{u.w) (K') which is arbitrarily close to the identity on compact subsets of Z +, if K ' is sufficiently close to K', For example, one can take as this homeomorphism

0, f is bi-Lipschitz (in the hyperbolic metric) on the set L£ of points in 0 having hyperbolic distance > € from J. Furthermore the restriction of f to L€ extends to a bi-Lipschitz homeomorphism from U onto U. LEMMA

118

FIXED POINTS AND KOEBE UNIFORMIZATION

383

Proof. We will first verify a Lipschitz condition near O. Assume that J does not separate 0 from O. Let a > 0 be some number less than the euclidean distance from 0 to J and set g(z) = z/a. Then g(J) does not intersect U. Let 'Y C g(O) be some Jordan curve separating g(J) from U and let ~ be the connected component of g(O) - 'Y containing O. The restriction of the map f 0 g-1 to ~ then satisfies the conditions of Lemma 4.1. We conclude from that lemma that f 0 g-1 is hyperbolic-length-decreasing on ~ n U. But the restriction of 9 to the disk D(O, a/2) of euclidean radius a/2 around 0 is Lipschitz in the hyperbolic metric, with the Lipschitz constant 1 = l(a) depending only on a. Since f 0 g-1 is contracting, it follows that f is Lipschitz with constant l(a) on D(O, a/2) nO. For every number t E (0,1) let h(t) denote the hyperbolic radius of a circle of euclidean radius t around O. Since we may precompose f by any hyperbolic isometry, our above Lipschitz condition near 0 translates to every point in U - J as follows: Let P be a point in U whose hyperbolic distance from J is greater than h(a); then f has Lipschitz constant l(a) on 0 n Dhyp(p, h(a/2)), where Dhyp(p, r) denotes the disk of hyperbolic radius r centered at p. By taking limits, we find that the continuous extension of f to 0 - au, which we continue to denote by f, is Lipschitz with constant l(a) on On Dhyp(p, h(a/2)). Let M ¢ {J, aU} be some circle boundary component of 0, which bounds a disk D(M), and let M* be the corresponding boundary component of 0*. As we have seen, for every point P in M, f satisfies a local Lipschitz condition at p. (The local Lipschitz constant of f at P is, by definition,

lim sup dhyp(f(Pl) , f(P2))/ dhyp(Pl,P2) as PI and P2 tend to p, while PI i- P2.) But one can further extend f to D(M) by mapping the hyperbolic center of D(M) to the center of the corresponding disk D(M*) and extending radially. Clearly the local Lipschitz constant of this extension at any point in D(M) is at most the supremum of the local Lipschitz constants of f in M. We extend f in this way over the interior of every such boundary circle M to obtain a map F. Now take any E' > 0 and let Rfl be the set of points in U having distance of at least E' from J and which are not separated from au by J. From the above, it follows that F satisfies a uniform local Lipschitz condition on Rf/; that is, there is some constant 1 such that all of the local Lipschitz constants of F in Rfl are bounded by l. Since the same arguments can be applied to the inverses of f and F, and since F(Rf/) is bounded away from J*, we conclude that F-l also satisfies a uniform local Lipschitz condition on F(Rf/). It is easy to see that one can modify F in the complement of Rf and then extend F to the whole of U so that the resulting homeomorphism, which we will continue to denote by F, as well as its inverse will satisfy a uniform local 119

z.-x.

384

HE AND O. SCHRAMM

Lipschitz condition on U. (To get an explicit construction take an analytic Jordan curve (3 c n, which separates J from R f • Let Dl be the topological disk bounded by (3, let D2 be the topological disk bounded by F((3) and let 91 and 92 be Riemann maps from a euclidean geometric disk D to Dl and D2, respectively. Note that 91 and 92 extend analytically to aD. Consider the map 9 : aD --t aD defined by 9(Z) = 92"I(F(91(Z))), This map 9 is analytic and therefore bi-Lipschitz in aD. Let G be the radial extension of 9 to D. Then G is bi-Lipschitz. Let F stay as it is outside Dl and, in Dl, redefine F by F(z) = 92(G(91 1(Z))). In other words, with 91 and 92 parametrize Dl and D2 as euclidean disks and then use the usual euclidean radial extension. Because 91,92 and G are bi-Lipschitz, so is the restriction of F to Dd Since the domain of F is now U, which is hyperbolically convex, clearly F is bi-Lipschitz (globally). (If p, q E U, take the hyperbolic line segment joining p and q; the image of that segment will be a path of length at most dhyp(p, q)l joining F(p) and F(q), where l is the uniform local Lipschitz constant of F. This gives dhyp(F(p) , F(q)) ~ l dhyp(P, q). A similar argument with F- 1 gives lf dhyp(F(p) , F(q)) ~ dhyp(p, q).) Since Lf C R f , this completes the proof of Lemma 5.1. D Remark. Though we will not use this, the above argument can be used to show that the Lipschitz constant of f on the set Ld tends to 1 as d --t 00. 5.2. Let f : n --t n* be a conformal homeomorphism between open connected subsets of C. Let J =1= K be boundary components of n and let J* and K* be the corresponding boundary components of n*. Suppose that n has at most countably many boundary components and all of them, with the possible exception of J, are circles and points. Similarly assume that all of the boundary components of n*, with the possible exception of J*, are circles and points. Also assume that f extends continuously to a homeomorphism of n - K onto n* - K*. Then K is a circle if and only if K* is a circle. LEMMA

Proof. We assume that K is a circle and K* is a point. Since the situation is symmetric, it is enough to reach a contradiction in this case. Since K* is a point, f extends continuously to K and maps K to K*. By replacing J with some Jordan curve in n separating J from K, we assume, without loss of generality, that J and J* are Jordan curves. Normalizing with Mobius transformations allows us to assume that the situation is as in Figure 5.1; that is, K* = {O}; K is a circle with center o which separates 0 from J; J separates K from J*; and J* separates J from 00. By further renormalization we want to replace f with a map which fixes some point in n, without losing the above properties. To achieve that let a = max{lzl : z E J}, let r be the (euclidean) radius of K and let d 120


385

-J*

FIGURE 5.1

be the (euclidean) distance from a to J*. Pick some point P E 0, so that If(P)1 < rdla, and consider the map g(z) = f(z)pI f(p). We have g(p) = p. Since Ipi > r, it follows that Ipi f(p) I > aid and, therefore, gB(J) is disjoint from J and separates J from 00. Let F : 0 - t g(O) be the continuous extension of g. By the Circle Index Lemma 2.2, the index of the restriction of F to J is 1 and the index of the restriction of F to K is -1, when K has the orientation induced by 0; the sum of these is O. But F has a fixed point at p, which contradicts Corollary 2.4 with Bo = {K, J}. This completes the proof of Lemma 5.2. D Proof of Theorem 3.2. We will prove by transfinite induction on a that, for each countable ordinal a, the map f extends continuously to a homeomorphism from 0 U (UKEW" K) to 0* U (UK*EW~ K*), where Wo: is the collection of boundary components in W having rank at most a and W~ is the collection of corresponding boundary components of 0* . Suppose that this holds for all ordinals f3 < a. Let K E W be a boundary component of rank a and let K* be the corresponding boundary component of 0*. Let J c 0 be a Jordan curve separating K from every other boundary component of 0 that has rank ~ a, and from every boundary component outside W. And let E be the connected component of 0 - J, which has K as a boundary component. Denote by E* the image of E under f, E* = f(E), and let F denote the restriction of f to E. We shall show that F extends to a homeomorphism from E onto E*. This will clearly suffice to complete the inductive step. By the inductive hypothesis we already know that F extends to a homeomorphism from E - K onto E* - K*. Lemma 5.2 shows that K and K* are either both points or both circles. If K and K* are points, then it is obvious 121

386

z.-x.

HE AND O. SC HRAMM

that F extends as needed, and we only consider the case where K and K* are circles. Since we are free to renormalize by Mobius transformations , we assume, without loss of generality, t hat K = K* = au and that E , E* c U. Now Lemma 5.1 applies and shows that the restriction of F to the set of points having hyperbolic distance at least 1, say, from J extends to a hi-Lipschitz homeomorphism 9 : U --+ U. We briefly reproduce here, for the convenience of the reader, a standard geometric argument (maybe due to Mostow), which shows that 9 extends to a self-homeomorphism of U. (Alternatively one can conclude that 9 extends t o a homeomorphism of U from the fact that 9 is quasiconformal.) Let 1 be t he hi-Lipschitz const ant of g. Consider some straight ray A of infinite hyperbolic length starting at O. We now show that g(A) has 1 limit point in au. Take some r > O. Since the diameter of a disk of radius r + 2 is 2(r + 2) , the preimage under 9 of the disk with center the origin and radius r + 2 has diameter at most 2(r + 2)1. Therefore the total length of the part of g(A) , whose distance from 0 is between rand r + 2, is at most 2(r + 2)l2. This implies that

L

9(E) " 2n 2(r + 2)/', EEH (, j p(r) where H (r) denotes the collection of connected components of the intersection of g(A) with the open annulus between the circles of radii rand r + 2 around 0, and where each 9(E ) denotes the angular diameter of E with respect to 0, O(E) = sUPx,yEE L(x , O,y) , and p(r) is the length of the perimeter of a hyperbolic circle with radius r. Since p(r) increases exponentially as r -+ 00, it follows t hat lim n....oo L~n LEe H(r) O(E) = O. This shows that g(A) , which clearly has some limit points in au, has in fact a unique limit point there. Now we extend 9 to au by letting g(p) be t he unique limit point of the ray g(A), where A is the ray [O,p) and p is any point in au. One easily uses the above inequalities to verify that 9 extended thusly is continuous. Checking t hat it is a homeomorphism is also straightforward. This clearly implies that f extends as needed, completing the inductive step. An appeal to the principle of transfinite induction now establishes the theorem . -(Not e t hat it is not necessary to verify the base of the induction, since the inductive hypothesis is empty when Q: = O. Of course the base of the induction is also standard.) 0 6. Maximum modulus, normality and angles

The results of this section, besides their independent interest, will prepare us for the proof of the existence part of Theorem 0.1. 122


387

MAXIMUM MODULUS THEOREM 6 .1. Let A and A · be Jordan domains in It; let 0 be a domain which is obtained from A by the deletion of a closed disjoint union of at most countably many closed (geometric) disks and points in A; and similarly let 0' be a domain obtained from A· by the deletion of a closed disjoint union of at most countably many closed disks and points. Suppose that f : 0 - 0 * is a conformal homeomorphism between 0 and 0' that extends continuously to oA, and that fB(oA ) = oA'. Then

I(z) - z E convex hull {f(w) - w, w E oA} for every point

Z

E O. In particular, sup I/(z) zEn

zl

= max I/(w) wE8A

wi.

Proof. Note first that Theorem 3.2 implies that f extends to a homeomorphism between the closures of 0 and 0*. Let ZO E 0 and define g(z) = f(z) - f(zo) + zoo Then ZO is a fixed point for g. Assume that f( zo) - zo is not in the convex hull of {few) - w : w E fJA} . It then follows that 0 is not in the convex hull of {g(w) - w : w E fJA}. Therefore the winding number around 0 of the restriction of w - g( w) - w to fJA is O. This means that the restriction of 9 to fJA has index O. However 9 has a fixed point at ZO, in contradiction to Corollary 2.4. This proves the first 0 assertion, and the second assertion clearly follows. By the same method as above, it is possible to get estimates analogous to Cauchy'S estimates for the first and second derivative. This will be done in a subsequent paper. COROLLARY 6.2 (Normality). Let 0 be as above and let fk : 0 - Ok be a sequence of conformal homeomorphisms such that each fk and Ok satisfy the conditions placed on f and 0* above. Suppose that the ik converge uniformly on compact subsets of 0 to a function g. Then 9 is either a constant map or a conformal homeomorphism, and gB (K) is a circle or a point for every boundary component K E B (n) - {fJA}. Moreover the convergence to g is uniform on any subset of 0 whose closure does not intersect fJA.

Proof. We start with the last assertion. Let E be some subset of n whose closure does not intersect fJA, and let f > O. There is some Jordan curve , c n separating E from fJA. The convergence on , is uniform . Therefore there is some integer N so t hat Ifm(z) - fk(z)1 < f for all k,m > N and all z E ,; equivalently, Ifm 0 f k- 1 (w) - wi < f for k,m > N, wE lkh). Now apply the Maximum Modulus Theorem 6.1 to the maps fm 0 f;l to conclude that 11m 0 I,l(w) - wi < , for all k, m > N and all w E ME). This gives 123

388

Z.-X. HE A ND O. SCHRAMM

Ifm(z) - ik(Z)1 < €, for all k, m > N and all Z E E, and implies the uniform convergence. That 9 is either a constant or a conformal homeomorphism is a well-known consequence of Rouche's theorem. Let K E 8(0) - {8A}, let W be an open set that contains K and whose closure is disjoint from 8A, and set Wo = W n n. By the above, the convergence ik -+ 9 is uniform on Woo Let € > 0 and let k be large enough that lik(z) - g(z) 1< € for Z E Woo Then clearly the distance from any point in gB(K} to ff!(K) is at most E, and the distance from any point in ff!(K) to gB(K) is at most L In other words, the Hausdorff distance from gB(K) to ff!(K) is at most (, Since ff!(K) is a circle or a point, and since € was arbitrary, we conclude that g8 (K) is a circle or a point, because the collection of circles and points is closed in the Hausdorff metric. This completes the proof of the normality corollary. 0 Definition 6.3 . Let 1} C t be a circular arc with endpoints p, q and let z be some point not on the circle containing 1}. We define the angle of 1} from z, denoted by ang(z,1}), to be the length of m(1}) , where m is any Mobius transformation taking z to 0 and ." into au, the unit circle. (This is the same as the angle at z between the two circular arcs that join z to p and q , respectively, and which are orthogonal to 1}.) The definition is clearly Mobius invariant.

Let 11 be a domain obtained from a Jordan domain A c t by the removal of a closed, disjoint, countable union of disks and points in A. Suppose that f is a conformal homeomorphism from 11 onto a circle domain that extends continuously to a homeomorphism from A to a circle. Further suppose that Zo E nand D is an open geometric disk containing Zo such that the boundary of the connected component of DnA that contains ZO , is the union of the arcs 0 c aA n D and (3 c aD n A, as in Figure 6.1. Then ang(J(zo),J(a)) "ang(ZQ, ~), ANGLE LEMMA 6.4.

a

where 1} is the arc of aD complementary to {3. In a slight abuse of notation we are using extension of f to aA .

f

to denote also the continuous

Proof. By normal izing with Mobius transformations , we assum e without loss of generality that f maps aA onto aD, respecting orientation, and that D = U , zo = f(zo) and 00 E 11. Striving for a contradiction , we assume that ang(f(zo),J(a)) < ang(ZQ , ~) , which is equivalent to ang(f(zo) , !(a)) + ang(zo, {3) < 27r. Then, by further normalizing with a hyperbolic isometry of U that fixes ZQ , we assume that {3 and f(o) are disjoint. 124


389

D

o

0

00

FleURE 6.1

Now let i C 0 - U be some simple curve with endpoints in aA - U, such that i U aA separates Zo from any other connected component of U n A and from 00. (See Figure 6.1.) It is easy to check that such a i exists. Let 0- be the connected component of n - i containing Zo . We now examine the index of the restriction of I to the boundary component KI of n- that contains i and a. Let h denote that restriction and let H : a x [0, 1J - u U {3 be an endpoints-fixing homotopy in U U f3 from the identity map on a to some homeomorphism hi from a to {3. Since h(a) = I(a) C au is disjoint from {3, it follows that H(z , t) =f:. fl(z) for each z E Q:, t E [0,1). Therefore the index of It is equal to the index of the map h : {3 U KI - Q; _ au, defined by J,(z) ~ h(z) for z E K! - " and J,(z) ~ h 0 h i! (z) for z E (3. But the index of h is clearly 0, because the Jordan domain in rt determined by {3 U Kl - a is disjoint from U . So the index of II is 0. However, every boundary component K E B(O - ) - {KI} is a circle or a point, and I maps these to circles and points (J extends continuously to the boundary, by Theorem 3.2). Since J(zo) = Zo and the index of the restriction h of I to KI is 0, this gives a contradiction to Corollary 2.4, as usual. This completes the proof of the lemma. 0 We will also need the following elementary geometric lemma: LEMMA 6.5. Let Tf be a circular arc with endpoints p, q, say. Suppose that z is a point not on the circle containing 1} and let 6: be the circular arc with endpoints p, q that passes through z. Then ang(z, 7]) + 29 = 211", where e is the angle between"., and 6: at p (or at q).

Proof. By normalizing with a Mobius transformation, we assume that z = 00.

Let

0

be the center of the circle containing 125

7].

Then ang(o,,,.,) = ang(z, 7]).

z.-x.

390

t

H E AND O. SCHRAMM

, I

'

t

- ---'-----k--------------;;,-, ;A- - ' - - - --

ang(o,T/)

FIG URE 6.2

Now consider the angle 1/J indicated in Figure 6.2. Clearly 1/J and ang(o, 1]) + 21jJ = 7[, From these the lemma follows.

+ 1f /2

=

(J

0

7. Uniforrnization

We will now restate and prove the existence part of Theorem 0.1. UNIFORMIZATION THEOREM 7.1. Every connected planar domain n with at most countably many boundary components is conformally equivalent to a

circle domain.

Proof. The proof will proceed by transfinite induction on the type of n. Recall that the type of 11, denoted by tp(fl), is defined as the pair (.\, n) such that). is a countable ordinal, n is a positive integer and B(D)A has n elements. The collection of such pairs is ordered lexicographically; that is, ().\,nt) < (A2,nz) if A\ < ).2 , or)q = >'2 and nl < nz . Since this is a well ordering , one can transfinitely induct with respect to it. Let ("\, n) be the type of n. If). = 0, that is, if n has finitely many boundary components, then the existence was proved by Koebe. Therefore we will assume that 0 has infinitely many boundary components and the theorem holds for all domains of lesser type. Let Ko be some boundary component of 0 of rank), (Ko E B(O)~). Let Jk, k = 1,2, .. ., be a sequence of Jordan domains satisfying 8Jk C 0, Jk C Jk+ 1 and U ~\ Jk :::) n - Ko. Define Ok = Jk n n. By the inductive hypothesis, since tp(Ok) < tp(O), eacil Ok is conformaJly homeomorphic to some circle domain. For each k let fk : Ok - Ok be such a homeomorphism. By normalizing with a Mobius transformation, we assume without loss of generality that ff(8Jk) = and A(zo) = 0, where Zo is some arbitrary point in n\. Since the sequence Uk} is a normal family, by choosing a subsequence if necessary, we also assume that the maps !k converge

au

126

FIXED POINTS AND KOEB E UNIfORM IZATIO N

391

uniformly on compact subsets of O. Let I be the limit of the sequence Uk}. Then I is either the constant 0 or a conformal homeomorphism I : 0 -> 0* , where 0' cU. We will consider these two cases separately. Hyperbolic Case (J f- const ant). Since the maps Ik converge to I uniformly on compacts of 0 , we conclude from Corollary 6.2 that IB(K) is a circle or a point whenever K E 8(0) - {Ko} . The application of t hat corollary is feasible here, because every such K can be separated from Ko by a J ordan curve in 0. It only remains to show t hat IB(Ko) = 8U . L EMMA 7.2. Let W be the connected component 01 U - IB(Ko) which contains 0* = 1(0 ). The n W is convex in the hyperbolic metric dhyp 01

u.

Proof. Let x, y be 2 distinct points in 0 and let their images under I be x",y" E n", x" = [(x) , y' = [(y). Let, > 0 and let m be such that x,y E nm, and dhyp(x"'/m(x)) < ,and dhyp(Y", [m(y)) < ,. Let e be the hyperbolic line segment joining Im(x) and Im(Y) . For k > m now consider t he map hk = !k 0 1;;;,1, whose domain is Im(Om ). By the Schwarz- P ick lemma, Theorem 0.6, it follows that hk is a contraction in the hyperbolic metric and extends to a contraction hk of U. Define ek = hk(e). Then t he length of each path €k is at most the length of e, which is less than dhyp(x ' , yO) + 2€. Taking a limit of the €k, we get some curve e, which joins x* and y., has length at most dhyp(X*, yO) + 2€ and obviously lies in W. This shows that any 2 points in 0* can be joined by a path in W , whose length is arbitrarily close to the distance between t he points. Since O' is open and W is simply connected, this implies that W contains t he convex hull of 0*; but because aw c aO" , we see that W is the convex hull of 0*. T his complet es the proof of the lemma. 0

Knowing now that W is convex, in order to reach a contradiction assume t hat W f- U. Then there is some hyperbolic line L , which contains a. point in aw, say p, and has W entirely on one side of it. (Through every point p E aw n U passes such a line L. ) Let L' be the hyperbolic line t hat is an arc of a euclidean circle with euclidean radius 1 and whose euclidean midpoint pi lies on the negative real ray. Let 9 be the hyperbolic orientation-preserving isometry that takes L to LI and p to p' and takes W into the region to the left of L'. (That is, Re(w) < Re(p') for W E g( W ).) Now let q be the euclidean translation q(z) = z + 1 - pl. Then q takes g(W ) into U. Furthermore q is clearly (strongly) expanding in t he hyperbolic metric at points in g(W) near pl. Therefore the ma.p q 0 9 t akes 0' into U and is expanding in the hyperbolic metric at some point of n* near p, at x· = f(x), say. Let 0: > 1 be the expansion factor of q 0 g at x· . 127

392

Z.-X. H E AND O. SCHRAMM

Since

n, the

Ik - f

and

Ik - !' as k

-

00,

uniformly on compact subsets of

expansion factor of the map J 0 1k- 1 at Ik{X) tends to 1. Let k be large enough that this expansion factor {3 is greater than 1/0:. Then the map q 0 9 0 10/;;1 , Ik(n k ) ~ q(g(W)) has an expansion f""tor /3a > 1 at Ik(X). But this contradicts the Schwarz- Pick lemma, Theorem 0.6, since the image

of this map is contained in U) and q and 9 preserve circles. This contradiction completes the proof of the uniformization theorem in the hyperbolic case.

Parabolic Case (f(n) = (o}). Define maps gk(Z) = !k(z)/ I~(zo), 9k : Ok - C. These maps clearly form a normal family, since g~(Z(l) = l. Therefore we assume without loss of generality that the 9k converge uniformly on compact subsets of n to some map , say, g. Since g'(zo ) = 1, 9 cannot be a constant and therefore is a conformal homeomorphism of n onto a domain, say, n·. As in the hyperbolic case, it follows from Corollary 6.2 that each boundary component of n*, with the possible exception of gB(Ko), is a circle or a point. We will show that = gB(Ko ) is a point (the point 00), and then the proof will be complete. Striving for a contradiction, assume that consists of more than a single point and let F be the connected component of t - fr containing We first consider the case where F has interior points. Then there is a Mobius transformation m, which maps fr into U, and with 00 E m(F). For each k the map

Ko

Ko

Ko.

mogo/;;l,

!kInk)

~

U

satisfies the hypotheses of the Schwarz- Pick lemma, Theorem 0.6, and is therefore a contraction in t he hyperbolic metric. This implies that the inverse map !k 0 g - l 0 m - 1 is an expansion, which clearly contradicts our assumption that fk --+ 0 as k --+ 00. The contradiction shows that int F = 0. The argument for the case where int F = 0 is more involved , but will still use the Schwarz- Pick lemma. Let D c t be some open geometric disk , whose closure D contains F such that there are at least two distinct points, say, p and q, in aD n F. One can take for D, for example, the closed disk with minimal radius, which contains F , in the spherical metric. (We allow 00 E D. ) We also assume that at least one of the two relatively open arcs of aD with endpoints p and q is disjoint from F, as we may, without loss of generality. Let 0 be such an arc. For each angle (J let 09 be thf! circul ar arc. whosf! endpoi nt.s ~re p and q, such that the oriented angle from 8 to 09 at P is O. We take 08 to be a {p,q} let O(z) be relatively open arc; that is, p,q ~ O(). For every Z E that angle 0 such that Z E 09(z)' This defines O( z) modulo 211' . However, since F is simply connected , there is a continuous function iJ : F --+ lR with B(z) = 8(z ) modulo 21r for Z E t - F.

t-

t-

t-

128

393

FI X ED POINTS AND KOEBE UN I FO RMI ZAT ION

D,

D,

- p,

-ih

FIGURE 7.1

rn

Now let 0, ~ iof{O(z) : z E ~ iof{O(z) : z E t - F) , aod let 0, ~ sup{O (z) : z E f1') ~ sup{O(z) : z E t - F) . From the fact that F C D it follows that 81 and 82 are finite. Let Zl be a point in 0 ° with 8(Zl } < 81+(-11"/ 4) and let Z2 E 0° with 8(Z2) > 82 - {11"/ 4}. Let 111 = 6(1l' 1J2 = 6~, PI = 6(11+71' and {h = 6~ -71" Let D I a.nd D2 be the open disks

D, ~

U

D, ~

6"

th 0 > 02} C F. However, since F has empty interior, this gives 0, + 211" :s;;;: 02 , which establishes HI n H2 = 0. Let k be some integer such that g- l(Zl), g- '(Z2 ) E Ok. Let A be a J ordan domain , which contains g(Ok} and whose boundary J A is contained in 0' . We also require that 8A intersect /3, in precisely 2 points P!,ql and intersect fh in precisely 2 points 1'2, q2. Thanks to Lemma 7.3, it is not difficult to see t hat such a Jordan domain exists. (One can just take some Jordan curve in 0* , which circles around F very close to F , and then modify it, if necessary, to avoid access intersections with /3, and ih The domain disjoint from F bounded by this curve is taken as A. ) By the inductive hypothesis, there is a conformal map fA : A n O· -+ U, which takes each boundary component of A n O' to a circle or a point, and with fA(I!A) = I!U. Let " I = I!A n H I, '" = I!A n H" {3 = ~I n A and ~ = aD l - {3. Now we can apply the Angle Lemma 6.4 , with fA , A n O*, z" D"a, in place of f , 0, Zo, D , a, respectively. We then conclude that

By similar reasoning we also have ang(!A(z,),fA("')) > ang(z,,'12) . Because 8, < 8( Zt) < lh + (11"/4), it follows from Lemma 6.5 that ang( 371"/ 2. Likewise ang( z2, 1"/2} > 371" / 2. From the above we conclude that (7.1 )

and (7.2 ) 130

z" 1"/1) >

395

FIXED POINTS AND KOEBE UN IFO RMIZATION

But 0'1 n 0'2 = 0, because HI n H2 = 0; and therefore, !A(O'I) n !A(O'2) = 0. It thus follows from (7.2) that

ang(JA(z2),!A(a!l) < 1 0 for the hyperbolic distance from /A(ZI) to /A(Z2) in the hyperbolic metric on U. The number C is an absolute constant, which can be described as the hyperbolic distance between two distinct circular arcs in U that have common endpoints in au, each having an angle of rr/ 4 with au (again by Lemma 6.5). The domain An f!* contains g(f!k ), and so we can consider the map fA 0 go/;;I : A(f!k) _ u. By the Schwarz- Pick lemma, Theorem 0.6, it follows that this map is a contraction in the hyperbolic metric. This tells us that

dhyp (Jk(g - I(Z I)),!k(g-I(Z2))) "dhyp(JA(ZI),!A(Z2)) > C, and dhyp(fk(g- l(z!l),!k(g - I(Z2))) > C > 0 holds fa; every sufficiently large k. That is a contradiction to our assumption that A - 0 as k - 00. And this D contradiction completes the proof of the theorem. 8. Domains in Riemann surfaces In this section we prove Theorem 0.2. We will make use of the space of ends £(f!) of a Riemann surface f!, which is defined exactly as in Section 1. The term closed sur/ace means a compact surface with no boundary and an open sur/ace means a surface without boundary. We use the term "surface" to mean "connected surface" . Pro%/ Theorem 0.2. We start with the proof of existence. If the genus of f! is 0, then f! can be conformally embedded in the Riemann sphere t, and existence follows from Theorem 0.1. Assume therefore that the genus is nonzero. Because f! has finite genus, there is some compact subset F c f! such t hat each connected component of f! - F is a O-genus (planar) surface, which has one boundary component in f!. It follows that there is a topological embedding i : f! _ S of f! into a closed topological surface such that every boundary component of i(f!) in S is contained in some t opological disk in S. (One can "fill in the holes" in each connected component of f! - F.) Let S be the universal cover of S , let p: S --- S be the covering map and let = p- l(i(O»). Then is a covering surface for 0, with covering map Pn = i - lop, and thus has the structure of a Riemann surface. Moreover is planar, since S is a topological disk. From this it follows that can be conformally embedded in the plane and, therefore , by Theorem 0.1 , is conformally homeomorphic to a circle domain c C.

n

n

n n

n*

131

n

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Let r be the group of deck transformations for the covering p ; § --+ S. The group r acts by conformal homeomorphisms on and, via the conformal homeomorphism of and fl*, it also acts by conformal homeomorphisms on Furthermore the uniqueness part in Theorem 0.1 shows that the action of r on O· is by Mobius transformations . Suppose that K is some boundary component of in 5. Since p(K) is contained in a topological disk in S, it follows that no nontrivial element of r stabilizes K. Let eoo be the only end of 8. Consequently no nontrivial element of r fixes an end of 0*, except for the end e~, which corresponds to e oo _ Let Eoo be the connected component of corresponding to e~ and let R = t - Eoo_ Then r acts freely, co-compactly and discretely on it The pair (Rlr, 0* If) is then the required pair. To prove uniqueness let RI and R2 be closed Riemann surfaces, let D 1 , D2 be circle domains in RI and R2 , respectively, and suppose that h : DI -- D2 is a conformal homeomorphism. Let RI , R2 be the universal covers of R 1 , R2 , with covering maps PI,P2, respectively. We think of RI c t and R2 C as being the unit disk, the plane or the sphere. Set 0 1 = p,ID J , 02 = p;-ID2i these are circle domains in From the fact that DI and D2 are circle domains in RJ and R2 , respectively, it follows that the homeomorph i s~ h lifts to a homeomorphism h : 0 1 __ 02. From Theorem 0.1 we know that h is a Mobius transformation. Therefore h extends to a conformal homeomorphism if : RI -- R2. From the fact that h conjugates the deck transformations of the cover PI : 0 1 - DI to the deck transformations of the cover P2 : 02 -- D2, it follows that if conjugates the deck transformations of the cover PI : Rl -- Rl to the deck transformations of the cover 1>2 : R2 -- R2 and therefore descends to a conformal homeomorphism H : RI __ R2 • The restriction of H to DI is obviously h. This shows uniqueness, completing the proof of the theorem. 0

n*.

n

n

n

t - n"

t.

t.

9. Uniforrnizations of circle packings Surely circle packings and circle domains are closely related. The following definitions give a common generalization of these two concepts.

t.

Definitions. Let D be any domain in (or, more generally, in a Riemann surface). A D-packing in D is an indexed collection P = {Pi: i E V} of compact topological disks in n with disjoint interiors. The nerve, or graph, of the packing P is the abstract graph G = (V, E), whose vertex set is V and where an edge (i, j) occurs in E precisely when the disks ~ and Pj intersect. An interstice of the packing is a connected component of D- U{Pv : v E V}, a finite interstice is an interstice whose boundary is contained in finitely many of the packed disks, and the carrier of the packing is the union of all the packed 132


397

sets and all the finite interstices. A decent packing is a D-packing in which the intersection of any two sets is at most a single point, and the intersection of any three sets is empty. The limit points of a packing are the set of all points p in t. with the property that every neighborhood of p intersects infi nitely many of the packed sets. A packing P in a domain n is said to be an acceptable packing in n if P is decent and has no limit points in n. If P is an acceptable packing in n, then np = n - U{int(p'-) : i E V} is a generalized domain; specifically it is the generalized domain associated with P and n. Note that, in general, a generalized domain is not a domain , since it is not open. However it is connected and is the closure of its interior. Given aD-packing P in t., one can get a planar embedding of its graph. To do that choose a point in the interior of each packed set to be the image of the associated vertex. Then the image of each edge (i, j) can be chosen as a simple path that lies in ~ U Fj and connects the images of the vertices. There is no problem in making the images of the edges disjoint, except at the vertices. Thus nerves of planar D-packings are planar. Of particular interest are nerves that are maximal with respect to being planar; that is , the introduction of one additional edge to the graph would make it nonplanar. These are the I-skeletons of a triangulation of an open planar surface. We will call them planar triangulations, for short, and when we use the term triangulation , it will be implicitly assumed that the triangulation is connected. Planar triangulations have another important property: their embedding in the sphere t is topologically unique (or unique up to reflection , if the orientation of the sphere is taken into account). This means that any two embeddings of a planar triangulation in t are related by a self-homeomorphism of t. Thus a decent packing in t, whose nerve is a triangulation, is topologically determined by its nerve. We will see below that when the packed sets are geometric disks and the nerve is a given planar triangulation, then under certain conditions the packing is also geometrically determined. One can study either a packing or the generalized domain associated to it. The difference is like the proverbial difference between looking at the halffull glass or at the half-empty glass . Of course, when there are few edges in the nerve of the packing, there is little to work with , and one must look at the domain. Historically both approaches to the subject are present. For example, Koebe looked at the domain and achieved the circle packing theorem as a consequence of his uniformization theorem. On the other hand , Thurston mostly looked at the circles, while the Rodin- Sullivan work [RoSu] and the paper [He2] adopt a mixture of the two views. In retrospect , the incompatibility theorem, which is the main tool in [Sch2] and [Sch3], can be seen as some kind of fixed-point theorem for packings. There, two finite topological 133

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packings satisfy some boundary conditions analogous to the condition that the associated map on the boundary of the domain have a fixed-point index - 1. The conclusion is that there exists a "fixed point of negative index" between the packings.

Definitions. A conformal homeomorphism between generalized domains h : n _ O· is a homeomorphism that is conformal in the interior of n, while an anticon/ormal homeomorphism is a homeomorphism that is anticonformal in the interior of n. If such an h exists, then n and 0* are said to be conformally or anticonformally homeomorphic, respectively. Example 9.1. Let P and P* be decent packings in t. having carriers !1 and fr and planar triangulations T and T* as nerves, respectively. Then P and p. are acceptable packings in S1 and S1*, respectively; and the associated generalized domains S1 p and S1j,. are conformally homeomorphic or anticonformally homeomorphic if and only if T and T* are combinat orially isomorphic. To see this, note that all of the interstices in the packings must be triangular interstices; that is, their boundary lies on three of the packed sets. Thus all one has to do to show that S1p and nj,. are conformally or anticonformally homeomorphic is to construct the conformal , or anticonformal, maps between combinatorically corresponding triangular interstices and glue them properly. Since there is a freedom of choice of the image of three points on the boundary for Riemann maps between Jordan domains, one can do this while maintaining the continuity at the points of contact between any two interstices. This shows that S1 and S1* are conformally or anticonformally homeomorphic. The other direction is obvious. When we speak of a circle packing, we will mean a D-packing of geometric, rather than topological , closed disks. If a circle packing has no limit points in a domain S1, then it follows that it is acceptable in S1. A circle packing, whose nerve is a triangulation, is always acceptable in its carrier.

Definition. Let S1 be a circle domain and let P he an acceptable circle packing in S1. The associated generalized domain S1p will be called a generalized circle domain. We can now state a generalization of Theorem 0.1 , which is applicable to circle packings.

Any generalized domain n in t that has at most countably many ends is conformally homeomorphic to a generalized circle domain S1* C t. Moreover S1* is unique up to Mobius tmnsformations, and every conformal automorphism of S1* is the restriction of a Mobius transformation. THEOREM 9.2.

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Theorem 0.3 follows immediately as a corollary: Proof of Theorem 0.3. We start with existence. One easily constructs a decent planar packing p. with nerve T. Since P* is acceptable in its carrier 0· , one can form the generalized domain Op.. From Theorem 9.2 we conclude that there is a generalized circle domain 0 c t that is conformally homeomorphic to Op. . The circle packing associated to 0 is then the required packing P . This proves existence. The uniqueness follows from Example 9.1 and Theorem 9.2. D

To prove Theorem 9.2 one must essentially adapt, for generalized domains, the proof of Theorem 0. 1 and the proofs of all the theorems that precede it. There are two minor difficulties, which require some changes in the proofs. The first has to do with the fact that the interior of a generalized domain is not connected. When working with domains, we used the fact that if a conformal function has nonisolated fixed points, then it is the identity. This is no longer true when the domain of the function is not connected and, therefore, in the proof of Theorem 9.2, one must take special care to avoid nonisolated fixed points. In the proofs above, quite often we have chosen a Jordan curve I in the domain 0 to cut and isolate a part of the domain we wanted to examine from other parts. This can still be done in generalized domains, but the resulting two pieces that n breaks into may no longer be generalized domains. An example of this phenomenon can be seen in the generalized domain OH obtained from the plane by the deletion of the interiors of disks that form an infinite hexagonal circle packing. The bounded part of nH determined by any Jordan curve I C nH will not be a generalized domain, unless it is contained in one interstice, because I would have to touch some boundary circles more than once. This forces us to further broaden the class of "domains" under discussion .

n

t

n.

Definitions. Let be some domain in and let P be a D-packing in Suppose that t he intersection of any three sets in the packing P is empty and the intersection of any two contains at most a finite number of points. Further suppose that at most one of the connected components of the complement of any pair of sets in P intersects with other sets in the packing. Then 0 = U{int(Pv) : v E V} will be called a degenerated generalized domain. A bi-gon in a degenerated generalized domain is a finite interstice whose boundary lies in two of the sets in the packing. Thus a degenerate generalized domain without bi-gons is a generalized domain. A morphism of degenerate generalized domains is a continuous map f n -+ n* between degenerate generalized domains that is conformal and

n-

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injective in int(f2) - B , where B is some union of bi-gons of stant in each bi-goo contained in B.

n and f

is con-

As explained above, the advantage of working with degenerated generalized domains over generalized domains is that it is easy to cut a degenerated generalized domain along a Jordan curve and get two degenerated generalized domains.

Proof of Theorem 9.2. Let nand P = (~ : i E V ) be the domain and t he packing, which define n; that is, n = U {int(p;) : i E V}. We need a construct for degenerated generalized domains analogous to

n-

t he sp ace of boundary components of domains. The elements of this space will be called the boundary elements. These come in two flavors: the elements at infinity are just the boundary components of fl, and t he border elements

a

are the sets of the form F1 . The topology on the set of boundary elements B e(o) is such that any neighborhood in B e( o) around any element at infinity K E B(s1) is t he set of all boundary elements that intersect a neighborhood of K in t., and {8Pd is a neighborhood of any border element aPi . (Thus the set of border elements is discrete in B e(f2), and the inclusion of B(s1) in B e(o) is a homeomorphic embedding.) As with B (s1) , the set B e(o) is compact, Hausdorff and countable. The type and rank are then defined for B e(o) as for B(n). In the following paragraphs we outline the modifications needed in the t heorems and lemmas leading to Theorem 0.1 to get corresponding statements for degenerate generalized domains. Note t hat Theorem 2.1 also holds if we allow the boundary of A to be a fi nite union of Jordan curves, which may intersect at finit ely many points . We will need a slightly more general version of Lemma 2.2. In the more general version we do not require f to be a homeomorphism, but we do require t he preimage of a ny point in K to be a point or an arc on J. The hypothesis t hat f is orientation preserving should t hen be weakened to the requirement that the image of a positively oriented arc from a point x to a point y in J be a positively oriented arc from f(x) to f(y) in J , or a single point. In this situation part (1) of Lemma 2.2 is dropped, the proof for the case J c Kin part (2) remains unchanged and the proof for the case K c j is done similarly as t he proof for J c K. The proofs of parts (3) and (4) remain unchanged. Some adjustments are needed in Corollary 2.4 as well. First f2 a nd 0· are permitted to be possibly degenerated generalized domains, with f a morphism between them. Obviously any mention of boundary components is replaced by boundary elements. (This same change is needed in all of the lemmas and theorems we discuss here a nd will not be mentioned , unless there is some 136

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special need.) Of course we no longer require that F, the continuous extension of I, be a homeomorphism, since I does not have to be a homeomorphism, but only t hat the restriction of F to each boundary element at infini ty be a homeomorphism. The concl usions of the corollary also need some revision. The new conclusions are t hat f has at most n isolated fixed points a nd the number of fixed points in any set S of isolated fixed points is at most n, counting multiplicities. This change is needed, since f may fix whole connected components of the interior of O. The proof remains almost unchanged . One only needs to note that z -+ F(z) + c has only isolated fixed points if it has no fixed points on the boundary. The description of 0 in the statement of the Schwarz- Pick lemma, Theorem 0.6, is modified to the following: 0 is a possibly degenerate generalized domain contained in A, and every boundary element of it is a circle or a poi nt, except possibly for 8A, which is also a boundary element of O. A similar change is done for 0*. The statement of Lemma 4.1 cha nges only in that the continuous extension of f to n is not required to be a homeomorphism, only its restriction to each boundary element at infinity needs to be a homeomorphism. A change is needed in t he proof of this lemma, since when one of the fixed points p, q is a nonisolated fixed point, one cannot immediately conclude that f is the identity. Suppose this to be the case. Then f must fix a connected component of the interior of 0 that is not a bi-gon. Let H be the union of t he connected components t hat f fixes. If H is the interior of 0, the lemma follows; if not, t hen t here is some border boundary element K t hat has nontrivial arcs in H and in t he closure of some connected component B of the interior of 0 which is not in H. Since f fixes an arc of K , then f(K) = K , because both are circles. If f is t he identity on K , t hen f must fix B , which contradicts our assumptions. If not, t hen there will be two points, x, y E K , such t hat dhyp(X,y) < dhyp(f(X),J(y)) . T he same would hold for some points x', y' in t he interior of 0 sufficiently close to x, y, respectively. Then one can postcompose f with a Mobius transformation m, which contracts distances in the hyperbolic metric on U and takes f(x ' ) and f(y') to x' and y', respectively. Since one has 2 dimensions of freedom in choosing m, one easily arranges that x' and y' will be isolated fixed points of m 0 f . Then the contradiction follows as in t he original proof of Lemma 4.l. The statement and proof of Lemma 5.1 for possibly degenerated generalized domains remain essentially unchanged. In t he formulation of Lemma 5.2, again the requirement that the continuous extension of f to 0 - K be a homeomorphism needs to be changed to the requirement that its restriction to each boundary element at infinity H E B~( O ) be a homeomorphism. In the proof, the only modification is that 137

402


one must make sure that p is an isolated fixed point for g. It is easy to see that p can be chosen so that this is the case. (Note that p is chosen before g , and a different choice of p may give a different choice of g.)

The statement and proof of the Boundary Extension Theorem remain essentially the same, and the generalized form of Theorem 0.6 follows from it and Lemma 4.1. The uniqueness part of Theorem 9.2 now clearly follows from Theorem 0.6. Except for the obvious modifications, similar to those in the Schwarz- Pick lemma (Theorem 0.6) , no change is needed in the formulation of the Maximum Modulus Theorem 6.1. In the proof, one must take care of the possibility that Z(} is a nonisolated fixed point of 9. In that case, let H be the union of the connected components of the interior of 11 that are fixed by g. Obviously H cannot contain all of the interior of 11 and, therefore, there is a connected component B of the interior of 11 that is disjoint from H , but whose closure intersects the closure of H. If p is in the intersection of the closures, then in B near p one can find a point q with g(q) - q very close to O. FUrthermore q can be chosen to also satisfy l (q) i- 1. Then q is an isolated fixed point for g(z) = g(z) - g(q) + q, and a contradiction follows. Corollary 6.2 requires substantial changes, and it will be replaced by a discussion below. The changes necessary in the formulation and proof of the Angle Lemma 6.4 are similar to those made in the previous lemmas and theorems above. These changes are left to the reader. [n the modified proof of existence, the inductive claim is that given a degenerate generalized domain 11 c t with tp(11) < ().,n) there exists a morphism of it onto a generalized circle domain. [t is not very well known, but Koebe [Ko4] also proved the existence statement in the case of degenerate generalized domains with finitely many boundary elements by taking limits of complements of packings, where the packed disks almost touch. This covers the base of the induction. The maps A are defined as in the original version, but one has to work a little harder to argue that their limits are either a constant or a morphism. First it is clear that a subsequence of the A converges uniformly on compact subsets of the interior of 11. Using the reflection principle, one easily concludes that a subsequence of {A} converges uniformly on compact subsets of 11. Then exactly the same argument as in Corollary 6.2 shows that fB(K) is a circle or a point whenever K E B e(11) - Ko , where f is the limit of the A. We will now show that f is ei ther a constant or a morphism. Restricted to each connected component of the interior of 11, f is either a constant or a conformal homeomorphism. Also f is clearly injective where it is not a constant. Suppose that f is constant on some interstice L, which is not a 138

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bi-gon. Say it takes the value c there. Let M be the connected component of f - l(c) containing L and let 80M be the relative boundary of M in f! . If K is some border boundary element contained in M , then clearly every connected component of the interior of n whose boundary has an arc on K , is also contained in M. This shows that 80M consists of contact points; that is, points in the intersection of the closures of two distinct interstices. The number of points in 80M cannot be 1; on t he other hand, if there are 2 or more contact points in 80M , then there are at least 3 border boundary elements intersecting 8nM . These elements will have the property that their images under f contain c, but also contain other points. However this is impossible, since at most two circles can touch at any given point. This implies that 80M is empty and thus shows t hat f is a constant if it is constant on some connected component of the interior of n that is not a bi-gon. This same argument is repeated for the hyperbolic and parabolic cases. Except for this, the proof remains intact. This completes the proof of Theorem 9.2. 0

Proof of Theorem 0.4. Theorem 0.4 follows from Theorem 9.2 exactly as Theorem 0.2 followed from Theorem 0.1. 0

Adden dum : A lmost circular d omain s with u ncountably many boundary components

We now state and outline the proof of a generalization to the existence part of our main result , Theorem 0.1 , which was obtained after this article was accepted. The details will appear in a forthcoming paper. THEOREM 10.1. Let n be a domain in C and let B.(O) c B(f!) be the collection of all boundary components that are not circles or points. If the closure of B.(f!) in B(n) is countable, then n is conformally homeomorphic to a circle domain.

The proof of Theorem 10.1 proceeds by induction in much the same way as the proof of existence presented earlier. The sticky point is , however, that the rigidity results used, primarily the Schwarz- Pick lemma, require some extra hypotheses when there are uncountably many boundary components. For this purpose, the theory of quasiconformal maps is useful. More specifically the Schwarz- Pick lemma, and most of the other results here, are applications of Corollary 2.4 , which in general fails when there are uncountably many boundary components. The first step in the proof of Theorem 10.1 is a variation of that corollary, as follows: 139

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LEMMA 10.2. Let f ; n _ n* be a conformal homeomorphism between bounded plane domains and let Bo C B(O) be a finite collection of boundary components 0/0,. Suppose that f extends to a quasicon/ormal homeomorphism F : C _ C and that the following conditions hold: is 0 a.e. (this is automatically (1) the conformal dilatation of F on satisfied if an has measure 0) ,(2) all of the boundary components in B(f!) - Bo and B(f!') - f8(Bo) are circles and points, and all of the boundary components in 80 are Jordan curves; (3) Eo contains the boundary component of 0 , which is contained in the unbounded component of C - n and, similarly, f8(8 0) contains the boundary component of fl·, which is contained in the unbounded component of C - 0*; (4) F has no fixed points in any of the boundary components in Bo. Let n be the index of the restriction of F to the boundary components in Bo. Then f has at most n fixed points in n. Furthermore, if S is a set of fixed points of f, then the total number of fixed points for fin S, counting multiplicity, is at most n .

an

The proof of t his lemma is based on approximations by fi nitely connected domains, on Corollary 2.4 for finitely connected domains and on a rigidity result of Sullivan [Su] to show that t he approximations converge to f. The rest of the proof of Theorem 10. 1 is like the proof of the existence part of Theorem 0.1, but with the extra burden that whenever Corollary 2.4 is directly or indirect ly applied, it can be arranged that the hypotheses of Lemma 10.2 are satisfied. PRINCETON UNIVERSITY, PRINCETON , NEW J ERSEY W EIZMANN INSTITUTE, REHOVOTH , ISRAEL

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[BStelJ A.F. BEARDON and K. STEPHENSON, The uniformization theorem for circle packings, Indiana Math. J . 39 (1990) , 1383- 1425. [BSte2] _ __ , Circle packings in different geometries, T6hoku Math. J. 43 (1991 ), 27- 36. [BSte3] _ _ _ , The Schwar"l- Pick lemma for circle packings, Illinois J. of Math. 35 (1991 ), 577-606. [Be] L. BERS, Uniformization by Beltrami equations, Comm. P ure Appl. Math. 14 (1961), 215- 228. [BoS] P.L. BOWERS and K. STEPHENSON, The set of circle packing points in the Teichmliller space of a surface of finite conformal type is dense, Math. Proc. Cambridge Phil. Soc. III (1992), 487- 513. [Brl ] R. BROOKS, On the deformation theory of classical Schottky groups, Duke Math. J . 52 (1985) , l Q09.-1024. [Br2] _ _ _ , Circle packings and co-compact extensions of Kleinian groups, Invent. Math. 86 (1986), 461-469. [CR] I. CARTER and B. RoDIN, An inverse problem for circle packing and conformal mapping, Preprint. [CdVl ] Y. COLIN DE VERDIERE, Empilements de cercles: Convergence d 'une methode de point fixe, Forum Math. 1 (1 989) , 395-402. [CdV2] _ _ _ , Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991 ), 655-669. [De] R. DENNEBERG, Konforme Abbildung einer Klasse unendlich vielfach zusammenhangender schlichter Bereiche auf Kreisbereiche, Diss. , Leipziger Berichte 84 (1932), 331- 352. [Gr] H. GROTZSCH, Eine Bemerkung zum Koebeschen Kreisnormierungsprinzip, Leipziger Berichte 87 (1935), 319- 324. [GPo] V. GUiLLEMIN and A. POLLACK, Differential Topology, Prentice-Hall, Englewood Cliffs, 1974, p. 222. [Haa] A. HAAS, Linearization and mappings onto pseudocircle domains, Trans. A.M.S. 28 2 (1984) , 415- 429. [Hau] F. HAUSDORff, Set Thoory, 2nd ed. , Chelsea Pub. Co., New York , 1962. [Hel] Z.X. HE, Solving Beltrami equations by circle packings, nans. A.M.S. 322 (1990), 657-670. [He2] _ _ _ , An estimate for hexagonal circle packings, J. Diff. Geom. 33 (1991) , 395-412. [HeSch] Z.X. HE and O. SCHRAMM, A hyperbolicity criterion for circle packings, Preprint. [Kol] P. KOEBE, Uber die Uniformisierung beliebiger anaiytischer Ku rven, III, Nachr. Ges W iss. Gott. (1908), 337- 358. [K02] _ _ _ , Abhandlungen zur Theorie der Konformen Abbildung: VI. Abbildung mehrfach zusammenhiingender Bereiche auf Kreisbereiche, etc., Math. Z. 7 (1920), 235-301. [K03] _ _ _ , Uber die konforme Abbildung endlich- und unendlich-vielfach zusammenhangender symmetrischer Bereiche auf Kreisbereiche, Acta Math. 43 (1922), 263- 287. [K04] _ _ _ , Kontaktprobleme der konformen Abbildung, Berichte Verhande. Sachs. Akad. Wiss. Leitylig, Math.-Phys. Klasse 88 (1936) , 141- 164. [LV] o . LEIITO and K.I. VIRTANEN, Quasioon/Qrmal Mappings "in the Plane, Springer- Verlag, Berlin, 1973, p. 258. [MRoJ A. MARDENand B. RoOIN , On Thurston's formulation and proof of Andreev's theorem, in Computational Methods and FUndion Theory, Ruscheweyh , Saff, Salinas, Varga, eds., Lecture Notes in Math. 1435 , Springer- Verlag, 1989, pp. 103-115. [Mel ] H. MESCHOWSKI, Uber die konforme Abbildung gewisser Bereiche von unendlich hohen Zusammenhang auf Vollkreisbereiche, I, Math. Ann. 123 (l951) , 392-405. (Me2] _ _ _ , Uber die konforme Abbildung gewisser Bereiche von unendlich hohen Zusammenhang auf Vollkreisbereiche, II, Math. Ann. 124 (1952), 178- 181. [Mi] J.W. MILNOR, Topology from the Differential V"iewpoint, Univ. Press of Virginia, 1965, p.64. 141

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[Ro1] B. RODIN, Schwarz's lemma for circle packings, Invent. Math. 89 (1987), 271-289. [Ro2] _ _ _ , Schwarz's lemma for circle packings, II, J. Diff. Geom. 30 (1989), 539-554. [Ro3] _ _ _ , On a problem of A. Beardon and K. Stephenson, Indiana Math. J. 40 (1991), 271-275. [RoSu] B. RODIN and D. SULLIVAN, The convergence of circle packings to the Riemann mapping, J. Diff. Geom. 26 (1987),349-360. [Sa] L. SARlO, Uber Riemannsche Flachen mit hebbarem Rand, Ann. Acad. Sci. Fenn., Ser. AI 50 (1948), 1-79. [Sch1] O. SCHRAMM, Packing Two-dimensional Bodies with Prescribed Combinatorics and Applications to the Construction of Conformal and Quasiconformal Mappings, Ph.D. Thesis, Princeton Univ., 1990. [Sch2] _ _ _ , Rigidity of infinite (circle) packings, J. A.M.S. 4 (1991), 127-149. [Sch3] _ _ _ , Existence and uniqueness of packings with specified combinatorics, Israel J. Math. 73 (1991), 321-341. [Sill R.J. SIBNER, Uniformizations of symmetric Riemann surfaces by Schottky groups, Trans. A.M.S. 116 (1965), 79-85. [Si2] _ _ _ , Remarks on the Koebe Kreisnormierungsproblem, Comm. Math. Helv. 43 (1968), 289-295. [Si3] _ _ _ , 'Uniformizations' of infinitely connected domains, in Advances in the Theory of Riemann Surfaces, Proc. 1969 Stony Brook Conf., Ann. of Math. Studies 66, Princeton Univ. Press, 1971, pp. 407-419. [Ste] K. STEPHENSON, Circle packings in the approximation of conformal mappings, Bull. A.M.S. 23 (1990), 407-415. [Str1] K.L. STREBEL, Uber das Kreisnormierungsproblem der konformen Abbildung, Ann. Acad. Sci. Fenn. 1 (1951), 1-22. [Str2] _ _ _ , Uber die konforme Abbildung von Gebieten unendlich hohen Zusammenhangs, Comm. Math. Helv. 27 (1953), 101-127. [Su] D. SULLIVAN, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Ann. of Math. Studies 97, Princeton Univ. Press, 1981, pp. 465-496. [Th1] W.P. THURSTON, The geometry and topology of 3-manifolds, Lecture notes, Princeton Univ., 1980, Chapter 13. [Th2] _ _ _ , The finite Riemann mapping theorem, invited address, International Symposium in Celebration of the Proof of the Bieberbach Conjecture, Purdue University, 1985. (Received July 8, 1991)

142

Discrete Compul Geom 14:123- 149 (1995)

Hyperbolic and Parabolic Packings. Zheng-Xu He! and O. Schramm2 1 University

of california at San DIego, La Jolla, CA 92093, USA

[email protected]

2Mathematics Department, The Weizmann Institute, RehOYOt 76100, Israel [email protected]

Abstract. The conlacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. Let G be the I-skeleton of a triangulation of an open disk. G is said to be CP parabolic (resp. CP hyperbolic) if there is a locally finite disk pack..ing P in the plane (resp. the unit disk) with contacts graph G. Several criteria for deciding whether G is CP parabolic or CP hyperbolic are given, including a necessary and sufficient combinatorial enterion. A criterion in tenns of the random walk says that if the random walk on G is recurre nt, then G is CP parabolic. Conversely, if G has bounded valence and the random walk on G is transient, the n G is CP hyperbolic. We also give a new proof that G is either CP parabolic or CP hyperbolic, but not both. The new proof has the advantage of being applicable to packings of more general shapes. Another new result is that if G is CP hyperbolic and D is any simply connected proper subdomain of the plane. then there is a disk packing P with contacts graph G such that P is contained and locally finite in D.

1.

Introduction

We consider packings of compact connected sets in the plane C ... R2 or in the Riemann sphere t _ 52. Given an indexed packing P - (Pu: v E V), its conractgraph, or nerve G - G(P), is defined as follows. The set of vertices of G is V. the indexing set for P, and an • Both aUlhors acknowledge support by NSF grants. The first aUlhor was also supported by Ihe A Sloan Research Fellowship.


124

Zheng-Xu He and O. Schramm

edge [v, u] appears in G precisely when the sets P" and P" intersect. Thus G encodes some of the combinatorics of P. If all the sets P" are smooth disks] in C, then it is easy to see that the contacts graph is planar. The circle-packing theorem (16] says that for any finite planar graph G there is some packing of (geometric) di~ks in the plane whose contacts graph is G. This fantastic theorem has received much attention since Thurston conjectured that the Riemann map from a simply connected domain to the unit disk can be approximated llsing circle packings with prescribed nerves. The conjecture was later proved by Rodin and Sullivan [20]. Some proofs of the circle-packing theorem appear in 11], 12], (28. Chapt"' 13]. (18]. (10]. (4], (13]. (6]. (7]. (21]. (24]. and (23]. Here, we are concerned with infinite packings. Suppose, for example, that G is a disk triangulation graph; that is, the I-skeleton of a triangulation of an open topological disk. By taking a Hausdorff limit of packings corresponding to finite subgraphs of G, an infinite packing P of disks in C whose contacts graph is G can be obtained. A few questions then naturally arise about the properties of P. Can P be bounded? Can P be locally finite in the plane? (This means that every compact subset of the plane intersects finitely many of the sets in the packing.) To what extent is P unique? It is not hard to see that (still assuming G to be a disk triangulation graph) there is a unique open topological disk D ee such that P is contained in D and is locally finite in D. The boundary of D is just the set of accumulation points of p.2 This D is called the carrier of P, and is denoted camP}. It was proved in [15] that P can be chosen such that camP) is the plane or the unit disk U = {z E C: Izi < 1}. Beardon and Stephenson [3] have obtained this result under the additional assumption that G has bounded valence. 3 There is a strong uniqueness statement valid when camP) = C: any other disk packing P' c C with nerve G is the image of P under a Mobius transformation [221. U5J. (The Mobius group is the group generated by inversions in circles. It is six dimensional.) In particular, it follows that there cannot be two disk packings p . P' with camP) = C, camP') - U, and G = G(P) "" G(P'). If carr(P) = U, then there is a weaker form of uniqueness: any disk packing P' with carr(P') "" U that has nerve G is the image of P under a Mobius transformation. All this parallels neatly with the analytic theory. The existence of a locally finite packing in U or C is a discrete analog of the uniformization theorem. which says that any simply connected noncompact Riemann surface is conformally equivalent to C or U. The parallels of the uniqueness statements are that any conformal map from the plane into the sphere or from U onto U is a Mobius transformation. Let us say that a disk triangulation graph G is CP parabolic (resp. CP hyperbolic)

1 The !erm disk means a geometric disk. a IOpologica/ disk means a set homeomorphic to a compact disk, and a SfTI()Qth disk is a topological disk with C I boundary. 1 A point z is an accumulation point of P if every neighborhood of z in!erseelS infinitely many sets in P. 3 The valence or rkgree of a vertex is the number of neighbors it has. "G has bounded valence" means that there is some C < oc such that every vertex has valence less than C.

144

Hyperbolic and Parabolic Packings

125

if there is a disk packing P with contacts graph G and carrier earnP) - C (resp.

camP) - U). We introduce the notion of a VEL parabolic graph. VEL parabolicity is a combinatorial property. which is defined using Cannon's vertex extremal length [8]. The precise definitions appear later. A graph which is not VEL parabolic is called VEL hyperbolic. We prove that a disk triangulation graph is CP parabolic iff it is VEL parabolic. This gives a complete combinatorial characterization of the "CP type" of any disk triangulation graph. Using this equivalence of CP parabolic and VEL parabolic. we prove: 1.1. Theorem. Let G be a disk trnmgulation graph. If the random walk on G is recurrent. then Gis CP parabolic. Conversely, if the degrees of the vertices in G are bounded and the random walk on G is transient , then Gis CP hyperbolic.

It will be shown that there arc CP parabolic disk triangulation graphs on which the random walk is transient. We also give new proofs to the above-quoted results that every disk triangulation graph is either CP parabolic, or CP hyperbolic, but not both. The results here actually generalize these theorems, since the proofs apply not only to packings by geometric disks, but to more general sets. In order to state some of our results, we introduce the notion of fat sets. Heuristically. a set is fat if its area is roughly proportional to the square of its diameter, and this property also holds locally. The precise definition is:

Definitions [26). The open disk with center x and radius r is denoted D(x, r). Let 7" > O. A measurable set X c t is T-fat if, for every x E X, .t -+ 00, and for every , > 0 such that D(x, r) does not contain X , the inequality arca(X n D(x,r» holds. A packing P "'" T-fal.

(P~:

~ T

area(D(x,r))

v E V) is fat if there is some T> 0 such that each Pu is

For example, any smooth disk is 7"-fat for some that K-quasi- 0 such that each Qv is 7"-fat. Let D e e be a simply connected domain, and suppose that D *" C (resp. D = C) if G is VEL hyperbolic (resp. VEL parabolic). Then there is a packing P '" (Pu : v E V) in D, which is locally finite in D, whose contacts graph is G,

and such that p~ is homothetic to Q~ for each v E V. 4 Conversely, suppose tho.t P = (Pv: v E V) is a fat packing in t of smooth disks whose nerve is G. Then G is VEL parabolic if and only if C-carr(P) consists of a single point.

145

126


From (24] we know that given a finite planar graph G* = (V* , £*) and a smooth disk Qt c C for each v E V*, there is a packing p. = (P.,* : U E V*). with G(P*) ,.. G* and p~* homothetic to for each v E V·. This constitutes the " finite case" for the existence part in Theorem 1.2. The basic innovation here is the control one gets on camP). The situation where the Q" are disks, G is VEL hyperbolic, and D

Q:

is an arbitrary simply connected proper subdomain of C seems interesting in itself. Although the equivalence of CP parabolicity to VEL parabolicity gives a com· plete characterization for disk triangulation graphs, it is quite natural to ask for other criteria. It has been shown by Beardon and Stephenson (5) that if every vertex in G has degree greater than 7, then G is CP hyperbolic. while if every vertex has degree at most 6, G is CP parabolic. We show that if finitely many vertices in G have valence greater than 6, then G is CP parabolic, while if the lower average valence (see Section 10 for the definition) in G is greater than 6, G is CP hyperbolic. From Rodin and Sullivan's proof of the length- area lemma [20], it follows that if 'YI, 'Y2" " is a sequence of nested simple closed paths in G and r:j 1/1 'Y;1 - 00, the n G is not CP hyperbolic. This can be seen as a criterion for CP parabolicity. In Section 9 we present a criterion of CP hyperbolicity based on a perimetric inequality in G. There will also be a somewhat restricted converse to this criterion, which is in the spirit of Rodin and Sullivan's length- area lemma. The interested reader may wish to consult Soardi's paper [27], which studies problems related to those discussed here.

2.

Discrete Extremal Length

In this section, we define discrete extremal length. Later, a brief discussion of the history of these definitions appears. We have chosen to start with an abstract notion, and then specialize to more geometric situations. Combinatorial Extremal Length . Let f be a nonempty collection of nonempty subsets of some set X. A (discrete) metric on X is a function m: X -) [0, (0). The area of m is just the square of the L 2 norm 'of m: area(m) = Jl m Jl2

=

E m(x)2. •eX

The collection of all metrics m on X with 0 < area(m) < 00 is denoted .L(X). Given a set A c X , we define the length of A in the metric m to be

This is also called the m-Iength of A. If f is a collection of subsets of X , we define its m-length to be the least m-Iength of a set in f: L.(f) -

in! L.(A).

Ae r 146


Finally, the extremal length of

127

r is defined as

EL(r) ~ sup

L.(r)' ) ():m E L(X) .

\ area m

This is a number in (0,00]' Note that the ratio L",(f)2jarea(m) does not change if we multiply m by a positive constant. Also note that EUf) does not depend on X ; that is, the value of EUf) does not change if we replace X with any other set that contains every A E r. The verification of the following simple monotonicity property of extremal length is left to the reader. 2.1. Monotonicity Property.

If each 'Y E r contains some y' E r', then EUr)

~

EL(r ' ).

At least when X is finite , a geometric interpretation can be given to EUf). Consider the Euclidean space IR x of all fun ctions f: X --+ R. For each subset 'Y of X , let XI' E n:x be defined by X/ x) = 1 for x E y and X/x ) = 0 otherwise. Now let f be. as before, a collection of subsets of X. 2.2. Theorem (Geometric Description of Extremal Length). least nonn in the convex hull of (X,),: y E f}. Then

LeI mo be the point of

We do not use this theorem. The simple proof is left to the reader. Extremal Length in Graphf. In the following. G = (V, E) is a locally fini te can· neeted graph. It will always be a simple graph; that is, each edge has two distinct vertices, and there is at most one edge joining any two vertices. A path y in G is a finite or infinite sequence ( vo , VI"") of vertices such that [Vi' Vi + I ] E E for every i = 0, 1, .... The edges and vertices of yare denoted by E(y) = {[ Vj,VIII ]: i - D.I .... } and V(y) - (vo,v\, ... }. respectively. Likewise, for r a set of paths in G, we set V(n = (V(y): yEn and E(f) = (E(y): yE n . A set A C V of vertices is said to be connected if, for every v , wE A, there is a path y in G from V to w with V( y) c A. (We allow trivial paths, paths that contain only one vertex.) Given subsets A, B c V , we let nA , B) = rG( A, B) denote the set of all paths in G with initial point in A and terminal point in B. We let r v(A, B) (resp. r E( A, denote the sets of vertices (resp. edges) of such paths:

B»

rv(A,B) - (V(,): yE r(A,B»), r.(A, B) - (E(,) : , 147

E

r(A, B»).

128

Zheng-Xu He and O . Schramm

A function m: V -+ [0, (0) is called a v-metric on G, and a function m: E --+ [0, 00) is called an e-metric. When m is a v-metric (resp. an e-metric) we use L",( y) as a (resp. L",(E( 'Y »). shorthand for L",(V( The vertex extrema/length VEL and edge extremal length EEL between A and B are defined by

y»

VEL - VEL G(A , B) - EL(rv(A , B» , EEL - EELG(A , B)

~

EL(l'E(A,B» .

To make the definition of VEL(A, B) more explicit, we have L.(y)' VEL(A , B) - supinf () ",yarcam

Here m runs over L(V) and 'Y runs over f G( A, B). These definitions give two discrete analogs for the classical notion of extremal length. (Reference [171 is a good introduction to continuous extremal length.) As we will see below, both arc useful. The edge extremal length was introduced by Duffin, who showed in [I2l that EEUA, B) is equal to the electrical resistance between A and B, if each edge in G is considered to be a resistor with unit resistance. The vertex extremal length was introduced by Cannon (8). Cannon's motivation was to obtain criteria for deciding when a group can be made to act conformally on the Riemann sphere C. Later it was discovered [9), [25] that extremal metrics of vertex extremal length (that is, metrics realizing the supremum in the definition of the extremal length) give square tilings of rectangles with prescribed contacts. An infinite path y in G is transient if it contains infinitely many distinct vertices. The set of transient paths in G that have an initial point in A is denoted by f(A ,oo). The edge and vertex extremal length from A to 00. are defined as EL({E(y), yE

r(A ,~)}) ,

- EL({V(y) , y E

r(A ,~)J).

EEL(A, ~ ) ~ VEL(A ,~)

Of course, this makes sense only for infinite G . For a v·metric or e·metric m, we let dm(A, B) (resp. dm(A ,oo» denote the distance from A to B (resp. to 00) in the metric m; that is, d.(A, B) - Lm(r(A , B» - inf{L.(y), y d.(A, ~)

-

L.(r(A ,~» ~

E

inf{L.(y), y E

r(A , B)}, r(A ,~».

An infinite graph G is VEL parabolic if VEL({v}, oo) - 00 for some v E V. Otherwise. G is VEL hyperbolic. Similarly, G is EEL parabolic if EEU{v}, (0) - 00 for some v E V, and is EEL hyperbolic, otherwise. 148


129

2.3. Remark. If VEL({v}, R12. Therefore, Observation 3.2 shows that area(D(z,3R) n Pu) ~ 1T'TR 2/ 4. In particular, we see that V(C(z, R» n V(C(z, R12)) is finite. This implies that there is an r, E (0, R1 2) such that V(D(z, r,» is disjoint from V(C(z, R» n V(C(z, RI2». Then it follows that D(z, r, ) is disjoint from U ~ e V( K) Pu ' We define inductively a sequence r, > r2 > ... of positive numbers. The first number in this sequence, r" has been defined already. Suppose that n > I, and that r1. ... ,r.. _ 1 have been defined. Let rn E (O,rn _ t /2) be sufficiently small so that V(C(z,r,,)) n V(C(z,r,, _1/2» = 0. The argument above shows that such an r" exists. 150


131

For each n let A" be the closed annulus bounded by C(z, r,,) and C{z. r,,/2). Define a v-metric m on G by setting m( v) ~

'" diameter(P n A,,) 1: ---'-"----"o

n,"

for each v E V. By the construction of the sequence r", at most one term in this sum is nonzero. Using this and Observation 3.2, we get an estimate for the area of m, as follows: area(m)

=

=

... ( ;. diameter(Po '(lEV ,, _ I nr" i.J

E E diameter;P~ n n,,,

,, _I VE V

;. 't'" S'-L.. ,, - I (lE V

's

nA,,»)'

;. 't'" L..L..

diameter( Po n D( z, '" 22

•

1:

,, _ 1

»2

n r" ..". - 17"- 1

,, - l ve Y

= 97"-1

A,,)2

1 "2
2r", and V(C{O,2r,,» n V(C(O,r,, + ,» "'" 0. The annulus All is defined as the annulus whose boundary is C(D, r,,) U C{O.2r,,). The rest of the proof remains essentially the same, A1tematively, using Lemma 3.3, the case z "" 00 can be reduced to the case

t

x c; O.

0

With this lemma, the proof of the first part of 3.1 is easy. Proof of 3.1(1), Suppose that P is locally finite in C - {pl. Pick some Applying Lemma 3.4 with K and let y be the Jordan curve "Y "'" U 7~J f([n i , n i ,;. I where we take n" :.. n o- The curve y is contained in U v e N P" and is disjoint from P"o' We say that two distinct triangles [ V I, V2,v 3 1 ,[w l ,w2 ,w3] in T neighbor if they share an edge. If [ v l , V 2 ,v3 1 is a triangle of T that does not contain Vo but neighbors with a triangle containing vo> say with [vo , n i , nj+ d, then D""v" II, and Duo.",.";,, lie on opposite sides of the arc f([ n" n ,·+d). Consequently, D"l ' v ~, " ) is not in the same connected component of " c - Y as P"o ' If [VI' v 2, v3 1 and (WI' w2, w]J are two neighboring triangles that do not contain vo ," then it is clear that D.,V '. .,V I. V) .. and D~Wh , ~. are in the same connected ... component of C - y. Hence it easily follows that for every triangle ( v), V2, U3 ] that docs not contain Vo the set D",. "I . v ) is disjoint from the connected component of " C - y that contains P"D ' This implies that y separates Pvo from U v e V - eN U{ QoJ) P" , and the lemma follows since y c U " e N Po - P"o ' D

n,

~, WJ

4.2. Corollary. Let G be a disk triangulation graph, and let P be a pacldng of $mooth disks in t with G(P) = G. LeI Z be the set of accumulation points of P . Then there is a connected component D of t - Z that contains P , P is locally finite in D, and D is a topological disk .

This D is called the carrier of P , D = carr(P). The verification of Corollary 4.2 is left to the reader. 4.3. Lemma. Let P - (P,, : U E V) and G - (V, E) be as in Lemma 4.1 , and let e V, C c V - {u}. Suppose that C is finite and u is contained in a finite component of G - C. Then U v e C Pu separates P" from the set of accumulation points of P .

U

Proof. Let Vo be the set of vertices that are contained in the same connected component of G - C as u is, and let K c t - U .' eC PQ be a connected set that intersects P" . For w E V, let N(w) c V - {w) denote the neighbors of w in G. From Lemma 4.1 we know that for each w e Vo there is a Jordan curve y... c U V E NI"') Pv - P,., that separates P,., from U v e V - ( N(w)U{ ... )) PV ' Let Q ... denote the component of t - y,., that contains P"" and let Q = U " e Vo Qu ' Suppose that p E K () J Q ... , where w E Yo' Then p E K n y... Since K is disjoint from U "eC P", and y... c U " E N( I 0 the set of accumulation points of P is covered by a finite collection of sets such that the sum of their diameters is less than B. This 0 clearly implies Theorem 6.1.

7. Uniformizations of Packings 7.1. Uniformization Theorem. LeI G => (V, E) be a disk triangulation graph, for U E V let Q u c C be a smooth disk, and let De e be a simply connected domain.

each

156


137

Assume tlUH there is aT > 0 such that Qv is T-fat for each v E V. Also suppose that D oF- C (resp. D = C) if G is VEL hyperbolic (resp. parabolic). Then there is a packing p - (Po: v E V) with camP) = D wlwse contacts graph is G, and such that PI,) is homothetic to QI,) for each v E V.

We note that the continuous analogue of this theorem appears in [26]. The proof is also similar. Proof. Let T be the triangulation of a disk that bas G as its ]-skeleton. Let T 2 c r 3 c··· be an exhaustion of T. By this we mean that T - U j T i, and each T i is a finite triangulation of a disk (with boundary). It is easy to see that such an exhaustion exists. We also require, without loss of generality, that TI has some interior vertex, say 0 0 , For each j - 1,2, . .. , let G i,", (Vi , Ei) denote the I-skeleton of r i. Suppose, without loss of generality, that 0 E D and 0 is in the interior of QI,) . Let D i be a sequence of smooth Jordan domains 6 in C such that 0 E Dl c n 2 ~ •.• and D .., U I D i. From the packing theorems of [24} we know that for each j ". 1, 2, ... there is a packing p i = (Pj: v E V i ) in the closure of D i. such that each pj is homothetic to QI,)' the sets p j are tangent to aDi when v is a boundary vertex of T i, and Pjo has the form tj Q vD for some tj > O. Let (j(k)} be a subsequence of {t , 2, ... } such that the Hausdorff limit Tl C

_

P =

I,)

1

lim

p iCk) k~CD diameter(Pj~k) ) I,)

(7.1)

exists for every 0 E V. The Hausdorff limit is taken in C; that is, a priori we must allow for the possibility that CIO is contained in some PI,) ' We show now that the sets PI,) do not degenerate to single points and do nOI contain 00. The set P~ certainly is OK, since it contains 0, has diameter 1, and is homothetic to QI,) , by construction. Let u be any neighbor of vo' Since Pw is a Hausdorff limit of sets homothetic to Q.. , which is smooth. Pw is either homothetic to Q.. , or is a single point. or a half-plane, or Pw = C. The last case is clearly impossible, since the interior of p.. does not intersect PI,) . It is also clear that Pu . mtersects PI,)O but does not intersect its interior. Let u\. u 2 , ••• , u" be the neighbors of vo, in circular order. For every j such that Vo and all its neighbors are in the interior of Ti, the set pj, U .. , U pj contains a Jordan curve that separates pj from 0;). (This follows from lemma 4.1.) Therefore. for at least two neighbors u of Vo the sets p.. contain more than a single point. Suppose, for example, that p.. , is a single point p, and that - Pu is not a single point. Let m be the largest number in {I , 2, ... , n} such that Pu "" {p} for each r < m in U. 2•...• n}._Since at least _two p.., do not Jegenerate to 'points, m < n. It is clear that each Pu!. intersects p.. ,., and that p.. , intersects p.. • . Therefore. the th ree smooth sets P,. ,Pu. , P.... contain the point p. This implies that the interiors of two

..

.

-

-

.

.

.

~o

o A smooth Jordan domain is the interior of a smooth disk. 157

138


.

of these sets must intersect, which is clearly impossible. Thus we conclude that none of the sets . Pu, consists of a single point, and that the ratios diameter(Pj )j diameter(PJ/) are bounded from above. (The reader may wish to compare the above argument with the Ring Lemma of [20].) Is it possible that P" I is .a half-plane? To see that it is not, consider a Hausdorff , . limit of the packings (hiPt ): v E VI ), where hj is the homothety that takes Pd, to Q" ,. The same argument as above, but with the roles of u l and Vo switched, shows . . then that the ratios diameter(Pd)/diameter(Pj ) are bounded from above. Similarly, . for every edge [u, w 1 the ratio diameter(Pd)/diameter(P~) is bounded independently of j. Since G is connected, this also holds when u, w E V are not neighbors. Therefore, each set Pv is not a half-plane, nor a point, and thus is homothetic to Q~ . If G is VEL parabolic, then part (2) of 3.1 implies that P is locally finite in t - {p} for some PEt. It is easy to see that p -= 00, and thus P is locally finite in C. This completes the proof in the case that G is VEL parabolic. Now suppose that G is VEL hyperbolic. The set Pjo is contained in D ~ C and has the form tj Q~~ , ti > O. Since 0 is an interior point of Q~o ' this implies that the sequence tj is bounded from above, and hence diameter(Pjo ) is bounded from above. By passing to a subsequence of j(k), if necessary, assume that t = Iimk_ ... diameter(PJo) E (O,oo) exists. We have established above that for any v, w E V the ratios diameter(Pj )jdiameter(P!> remain bounded as j -> 00. Consider the Hausdorff limits

..

P

"

_ _ pj(t) = klim

" .

(7.2)

If t - 0, then, because G is connected, it follows that Pu ... {O} for each u, and in particular the limits (7.2) exist. If t > 0, then comparing with (7.1), we conclude again that these limits exist, and that each Pv is homothetic to Qv ' We now prove that each Pv is contained in D. Consider some vertex U E V, and let N( u) denote the neighbors of u. By Lemma 4.1 , for each j sufficiently large (so that N(u) is contained in the interior of T j ) there is a Jordan curve in U " E N (v ) pj - pj that separates pj from aDj. Assuming that I > 0, since for any fixed u the sets pj vary within a compact collection of homotheties of Q" , the above implies that the distance from pj to aDj is bounded from below independently of j. Therefore, Pv cD. The same conclusion is true, of course, if t = 0, because then Pv = (D). So we have established that the packing P is contained in D. Clearly, the interiors of the sets Pv are disjoint. and Pv () P", ". 0 whenever [v, w 1E E. Therefore, the proof will be complete once we show that the packing P "" (Pv : u E V) is locally finite in D. (This will also rule out the possibility t "" 0, Pv = (ot.) That is actually the most significant part of the proof. It turns out that the packing P is useful to proving this property of P. Let F be some compact connected subset of D that contains O. We prove that F intersects finitely many sets in P, and this shows that Pis locaUy finite in D. Let F' be any compact connected subset of D that contains F in its interior. Let z be some accumulation point of P. From Lemma 4.3 we know that P is disjoint from its accumulation points. By Theorem 3.1, z is not the only accumulation point of P. 158

"9


Therefore, there is a compact connected set K that intersects Pu , contains an accumulation point of P, and is disjoint from z. In the following, for a °set X c let V(X) denote the set of v E V such that Pu intersects X. Since K is connected and oontains an accumulation point of P, it is clear that each component of V( K) is infinite. (This follows from Lemma 4.3.) We let e be a small positive number whose value is determined below. By Lemma 3.4, there is some open set W -= W(z , K , e) containing z such that

t.

I VELG(V( K) , V(W)) > -. e

Without loss of generality, we assume that W is connected. Then every component of V(W) is infinite. Assume for the moment that D has finite area. We show that if e is chosen sufficie ntly small, then Pu is disjoint from F for every v E V(W). Let C be some component of V(W). Let j be sufficiently large so that C intersects Vi, and let C i be any component of C n Vi. Since every component of V(W) is infinite, C is infinite, and therefore C i must contain boundary vertices of r i. Let H i be the component of V(K) n Vi that contains vo. The above argument tells us that H i i denote the family of all subsets of V i that contains boundary vertices of Ti. Let intersect every path in r o,(Hi,Ci). Proposition 5.2 now implies that

r-

Consider the v-metric m = mj that assigns to each v By the r-fatness of the sets Qu, we have .

area(PJ )

(

E

Vi the diameter of Pj.

,

~ T7Tm v) .

This implies area(m) ::s; r - I1T-

1

area(D)

147

On the other hand, the only vertices in K" +I that neighbor with D" +I are in

Dn U Cn' the vertices in D" have at most two neighbors in D" ... I ' and the vertices in Cn have at most three neighbors in Dn +I . Therefore,

which gives (10.2)

Using induction and inequalities (to.l) and (to.2), we see that

3nleol ID"I" IDol + - 2- · Therefore,

laK"I - Ie"

U

D"I " IDol + (2n + Oleo I.

(10.3)

Let m be the v-metric on G defined by m( u) = l / (n log n) for u E aK" , n > 1, and m( u ) - 0 for u ~ U" > 1 aK". Since aK" intersects every transient path meeting K o' we see that dm(KO ' ~) ~ E" > 1 l / (n log n) ,. 00. On the other hand, (10.3) implies that area(m) < 00, Hence G is VEL parabolic. From Theorems 8.1 and 2.6 it 0 follows that G is EEL parabolic and recurrent. Let G be a disk triangulation graph. For u

E

V, let deg(u) denote the degree of

u in G. The average valence of a finite nonempty set of vertices W is just 1 av(W) - -I -I [ deg( v ). W !l e W

The lower average valence of G is defined to be lav(G)

=

sup

inf av(W) ;

Wo W~ Wo

where Wand Wo are nonempty finite connected sets of vertices. (The authors do not know if this notion appears in the literature.) 10.2. Theorem. Let G be a locally finite connected planar graph, and suppose that laV(G) > 6. Then G is VEL hyperbolic, and therefore EEL hyperbolic and transient. Note that the lower average valence of the hexagonal grid is 6. Beardon and Stephenson [5] have shown that if every vertex of G has degree at least 7, then G is not CP parabolic. The above theorem is a generalization of this result. 167

148


Proof. In any finite planar 'graph G· with vertex set satisfies

V.,

the average valence (10.4)

This is a well·known fact but for the convenience of the nonexpert readers, we give the proof here. Let n, e, f be the number of vertices, edges, and faces of the graph (which is embedded in the plane). The Euler fonnula gives n + f => e + 2, and the inequality 3f s 2e holds if f > I, since every face must have at least three edges on its boundary, and each edge is on the boundary of at most two faces. From these it follows that n > el3 (actually it is this inequality which we need later). However, av(V·) "* 2eln, since every edge is counted exactly twice in the sum Lue V' deg( ul This establishes av(V·) < 6. We now return to the infinite graph G. Let Wo be a finite connected nonempty set of vertices such that av(W) > C > 6 for some constant C and every finite connected set of vertices W ::) Wo. Consider such a W, and let G· be the restriction of G to W u aW; that is, the vertices of G· are W u aW, and an edge of G appears in G· iff both its endpoints are in W ua W. Denote by nand e the number of vertices and edges in G·, respectively. Then, clearly, 2e ~ IW Iav(W), and therefore, by the previous paragraph,

IWI+lawl-lwuawl-n>

e

3""

I W ~v(W)

6

>

(C) 6" IWI.

This gives

laWI > g(IWIl with g(x) - (C - 6)xI6. Now, since L.:. t g(n) - 2 < 00, part (1) of Theorem 9.1 shows that G must be VEL hyperbolic, and the proof is complete. 0

It would be interesting to narrow the wide gap between Theorems 10.1 and 10.2. Suppose, for example, that G is a bounded valence disk triangulation graph and that Vo is some vertex in G. Let k~ "" L v (6 - deg(u», where the sum extends over all vertiCes v at distance at most n from vo' Can criteria for the type of G based on the sequence (kll) be given? For example, if k" is bounded, does it follow that G is VEL parabolic? Aclmowledgments

We thankfully acknowledge fruitful discussions with Peter Doyle and Burt Rodin. We also thank the anonymous referees for helpful comments, and especially for the improvement of the statement and proof of Theorem 9.1. References 1. E. M. Aodreev, On convex polybedra in Loba~l spaces, Mat. Sb. (N.S.> 81 (123) (1970), 445- 418; English transl. Math . USSR-Sb. 10 (1970),413- 440.

168

Hyperbolic and Parabolic Padtings

149

2. E. M. Andreev, On convex polyhedra of finite volume in Lobacevskii space, Mat. Sb. (N.S.) 83 (125) (1970), 256- 260; English transl. Math. USSR-Sb. 12 (1970), 255- 259. 3. A. F. Beardon and K. Stephenson, The uniformization theorem for circle packings, Indiana Uniu. Math. J. 39 (1990), 1383- 1425. 4. A. F. Beardon and K Stephenson, The Schwarz-Pick lemma for circle packings, Illinois J. Marh. 141 (1991), 577- fi:J6. 5. A. F. Beardon and K. Stephenson, Circle packings in diffe rent geometries, TOhoIru Math. J.43 (991),27- 36. 6. P. L Bowers, The upper Perron method for labeled complexes with applications to circle packings, Math. hoc. Cambridge Philos. Soc . 114(993),321 - 345. 7. G. R. Brightwell and E. R. Scheinerman, Representations of planar graphs, SIAM J. Discrete Math. 6 (993), 214- 229. 8. J. W . Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155- 234. 9. J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, The Mathematical Heritage of Wuhelm Magnus- Groups, -Geometry and Special Funcrirms, Contemporary Mathematics Series, American Mathematical Society, Providence, RI, 1994. lO. Y. Colin De Ve rdiere, Un principe variationnel pour les empilements de cercles, In vent. Ma/h. 104 (1991), 655- 669. 11. P. G. Doyle and J. L Snell, Random Walks and Electric Networks, The Carns Mathematical Monographs, Vol. 22, Mathematical Association of America, Washington, DC, 1984. 12. R. J. Duffin, The extremal length of a network, J. Math. Anal. Appl. 5 (1962), 200-215. 13. B. T. Garrett, Circle packings and polyhedral surfaces, Discrete Comput. Geom. 8 (1992), 429-440. 14. A. A. Grigor'yan, On the e: O, let H~ denote the hexagonal grid with mesh E. The vertices of H f form the hexagonal lattice:

The fin;l named a uthor was su pported by NSF Grant OMS 96-22068, and the second named author by NSF' Grant OMS 94-03548.


220

Z .• X. HE AND O. SCHRAMM

. .

. .

• • • . C . • • • •

. . . . . . a

Fig. 1.1 . The packing:;

.....

ne and

~,

and the ma p

r . Several pointti have been marked to aid

in grasping the correspondence.

where w is the cube root of -1 ,

w =exp A1I" i=!(J3i+l), and an edge connects any two vertices of He at distance e. Let n be a simply-connected domain in C wit h Ole. Then there is a subgrid B tl of HE, equal to the I-skeleton of a triangulation of a closed topological disk contained in n that approximates n. Let V~ denote t he set of vertices in H n. From the Disk Packing Theorem it follows t hat there is a disk packing pC=(P::vEV6) in iJ whose nerve is H fi and with the property t hat boundary disks are tangent to au. For each vE V(i, let JE(v) denote the center of the disk ~ (see Figure 1. 1). Then the Rodin- Sullivan Theorem (15] tells us t hat, assuming that the packings p~ are suitably normalized (by a VJ-U Mobius transformation, possibly ori entation~ reversing), the discrete functions converge "locally uniformly" in n to the (similarly normalized) Riemann map f: O-U, when E-O. There is a natural definition for the d iscrete partial derivatives of fun ctions defined on the lattice VJ. Given discrete fu nctions g~ : VJ - C, where EE(O, 1), we say that g ~ converge in GOO(O) to a smooth function g: Q-C as E-O, if g~ converge locally uniformly in 0 to g, and the discrete partial derivatives of gft: of any order converge locally uniformly to the corresponding part ial derivatives of g. The precise definitions will be given in §2. For each VE VJ, let r~(v) denote the radius of p;. Our main t heorem is:

r:

COO-CONVERGENCE THEOREM

1.1. The discrete junctions

po: Vrl'-U converge in

COO (!l) to the Riemann mapping j :!l - U. The discrete junctions 2r£(v) jE converge in COO (rl) to 11'1216

221

COO_CONVERGENCE OF' DISK PACKINGS

As a consequence, the derivatives of the conformal map up to order n can be approximated by quantities which depend on the positions of the centers of some n+ 1 conse0 there are some 6), 62 >0 such that If(z) - r(v) 1O, consider the subset of vertices of Ven{ ZEC: Izi ~ l /e} 8uch that their distance to C - 0 is bigger than E; and let Vd be the set of vertices which are either an interior vertex of this subset, or shares an edge with an interior vertex. Let V& be the connected component of V& which contains some vertex with distance at most E to Zo, and let H~ be the subgraph of H~ spanned by the vertices in Vd - Then for small E, H~ is equal to the I-skeleton of a geometric triangulation (with equilateral triangles of side length E) of a closed topological disk contained in 0 that approximates O. As we stated in the introduction, it follows from the Disk Packing Theorem that there is a disk packing f>€ = (P';:vEV&) in V whose nerve is H~ and with the property that boundary disks are tangent to au. We normalize pE by a (possibly orientationreversing) Mobius transformation, so that (1) for a vertex VoEV& that is closest to Zo the center of the disk p~o is 0, (2) for a vertex voEVn that is closest to Zo the center of the disk p:~ is on the positive real ray, and (3) the six disks Pf.tv, k=O, 1, ... , 5, surround P; in positive circular order. Let r:V&-U be the function that maps every vEV& to the converges center of the disk P;. Then the Rodin- Sullivan Theorem [15) tells us that locally uniformly in 0 to the Riemann map f:O - U satisfying f(zo) = O and f(zo»O. Let R! denote the regular hexagonal disk packing whose disks are centered at the vertices in the hexagonal grid H E. The disks of R£ all have radius ~E. Let Rh be the subpacking corresponding to the subset of vertices V&. For any triplet of mutually tangent disks Pl> P2 , P3 in p£ or R£, there is Il. disk D wbose boundary aD pas5C8 through the inlersection points P1 n P2 , P2 nP3 and P3nP1 •

r

221

226

Z._ X. HE AND O. SCHRAMM

The disk D is called a dual disk of the packing. Note that aD is orthogonal to each of the circles uPj • j=1,2,3.

4. The discrete Schwarzians Let f be a complex analytic function defined on a domain in the plane C, such that I'(z) never vanishes. The Schwarzian derivative of

Sf(z)

=

f is defined by

(I"(Z))' _~ (I"(Z))' _I"'(z) _ 3(f"(z))' I'(z) 2 f'(z) - f'( z) 2(f'(z))' .

(4.1)

The Schwarzian derivative of f is itself a complex analytic function. It is elementary

to check that for a Mobius transformation T (z}={az+b)/(cz+d) we have S(T of)(z) = S/(z). Moreover, 5/=0 if and only if f is equal to the restriction of a Mobius transformation. These and some further properties of the Schwarzian can be found in [11, §1], for example.

In the same spirit, we will define the Mobius invariants of hexagonal disk pa.ckings, and derive their immediate equations. Analogous invariants and equations were worked out in [16] in a similar way for circle patterns based on the square grid , where applications were found to the study of global properties of immersed patterns in C. Here, we will use Mobius invariants as an intermediate means in the study of t he convergence problem; and the Schwarzians will be defined as some suitably scaled measure of deformation of the Mobius invariants from their regular values. As an important step, we will derive a formula (or the Laplacian of the Schwarzians in the next section . Our method will yield the same result if Rodin's equation (1. 1), which the radii satisfy, is used instead. Thus, the use of Mobius invariants is not essential. We have chosen to work with the Mobius invariants as they yield relatively easier equations. For any edge e=[v, ul in Htl' we let Pe denote the point of tangency of the two disks p~, p;. Let e= [v, ul be some edge in H~, let WI, W2 be the two vertices of V t that neighbor with both v and u, and suppose that WI, w2EVfj. Let T be a Mobius transformation that sends the tangency point P e to infinity. Then the two circles 8Pv ,8Pu are mapped to lines. It follows that t he four points T(p[v ,Ul d), T(P[v,""'J), T(p(u ,Uld), TCPtu .UI~J) are at the corners of a rectangle, see Figure 4.1. Since T is well-determined up to post-composing by a similarity, the aspect ratio of the rectangle, IT(PI.,wd) -T(PI•. w,])1 IT(PI.,wd ) - T(Plu.wd)I ' 222

Coo·CONVE RGENCE OF DISK PAC KINGS

TP.

227

TP.

Fig. 4.1. The configuration after applying T .

is independent of the choice of T. It also does not change if we modify

transformation. Set

Se

p E by a Mobius

to be this aspect ratio divided by ..j3, (4. 2)

and let (4.3)

be called the (discrete) Schwarzian derivative, or Schwarzian of PE at e. The factor 1/..j3 in (4.2) is justified by the fact that when the disks P,., PlI , PWI' pw~ are all the same size, we get s( e) = 1 and h( e) = 0. The factor E- 2 is reasonable, because of t he behavior of the SchwaI"7.ian derivative under rescaling, namely, (Sg)(z)=e- 2(SJ)(EZ) when g(Z) = J( EZ). (This is also justified by the estimate of §6.) For a vertex v in V E, denote ek(v)= [v, Lk(v)J. Sec Figure 4.2. Let Sk , hk: (VJ)(I ) - fR be defined by Sk(v)=8(ek(v» hk(V)=hk+3(LkV) .

and hk(v)=h(e/c(v».

Clearly, Sk(V) = S/C+3(L/ev) and

L EMMA 4.1. Let v be an interior vertex in Vct. Then,

(4.4)

is valid for any kEZ6. Although we will not prove it here, the equations (4.4) are sufficient to guarantee that a positive function s on the edges of HE corresponds to an (immersed) hexagonal circle pattern (d. [16]). 223

228

Z ._ X. HE

A~O

0. SCHRAMM

L 2v

\

e2(V)

Lav -

e3(V) -

/

€1(V)

\/ V/\

e4(V)

/

Llv

eO(v) -

Lov

e5(V)

\

L,v

Fig. 4.2. The edges around

II.

vertex.

Fig. 4.3. The spedal points or a flower .

Proof. For any kE Z6, let VA: be the point of tangency ~ n pi. . ", and let qk be the • point in Pf.."nPf. _ 1 to' See Figure 4.3. There is no loss of generality, because t he packing p~ may be reRected about a line. Let mk = mk(v) be the orientation-preserving Mobius transformation that takes Pk, Pk -I, qk-l to 00,0,1, respectively. T hen, by the definition of the Sk'S,

--/3 ski, m,(v)(q,) ~ 1- J3 s,i,

mk(V}(Pk+l) =

m,(v)(p,) ~oo.

M,(-J3s,i) ~ oo,

M,(I-J3"i) ~ I, M,(oo)~O.

224

COC·CONVERGE"'CE OF DISK PACKINGS

T herefore, M _ J; -

(0 i

i ) -J3SJ; ,

229

(4.5)

where the usual matrix notation for Mobius transformations is used. Note t hat the com· position Ms oM4 oM3 oMl oMI oMe is the identity. Hence, Ms oM4 0M3 = Mo-I OMj- l OMil, which evaluates to 3is3 s 4 - i ) ~ (v'3(SO+Sl -3S08182) i-3iS eSl ) . (4.6) V3S4 ( 3is4SS -i v'3 (53 + SS -3535455) i - 3i5152 v'381 This is an equality of Mobius transformations, and is therefore valid up to a scalar factor. However, both sides have determinant 1, because the matrix in (4.5) has determinant 1,

and therefore (4 .6) is valid up to sign. From the upper left entries, we get

(4.7) Because 54 is never zero, and since the set of configurations of 6 disks in a 'flower' around a given disk is connected, the sign in (4.7) does not depend on the configuration . When all circles have the same radius, sJ;=1, so the correct sign is minus. This proves (4.4) for k=O. The equations for the other values of k are valid by symmetry. 0

5. The Laplacian of the discrete Schwarzians In this section, we will use the equations of the 5k'S of §4 to obtain t he equations for the hk's, and these will be used to show that t::lhk(v) is equal to a polynomial in e,hjo(v),Li~hh(v}; jo,jl ,hE Z6' We consider only e in the range (0,1), and therefore, if all hJ;'s are uniformly bounded in a vertex subset W , then so are the !::l;hk's in the set W (I). We substitute Sk(V)=1+e 2hJ;(v) in equation (4.4), simplify, and get h, (v) +h'+2 (v) +h'+4 (v) ~ 3h,(v) +3h'+1 (v) +3h,+, (v)

+ 3 l /e} - 0 is greater than 2e. Then

(6.1) for some constant C = C(O), which depends only on

o.

The proof is quite similar to the proof of Lemma 1.5 in [5]. The boundedness of the Schwarzians is equivalent to the boundedness of the third order derivatives of JE. It is an open problem whether the above lemma can be proved directly using Rodin's equation (1.1 ) for the radii , or from the formu la for the Laplacian of the Schwarzians.

Proof. By [6], there is a homeomorphism rf from the carrier of Rh onto the carrier of pc with the following properties: (1) For each vE V&, the image of the disk R~ under rf is the corresponding disk P! . (2) The restriction of ff to a dual disk of the packing Rh is equal to a Mobius transformation. (3) There is a universal constant C 1 > 1 such that for each vE (V,V(2 ), the map g£ restricted to Rt is C1·quasiconformal. (4) For each vE(V&) (2), there is a constant C2= C2(0(v»>O, which depends only on the distance 6(v ) from v to {zEC: Izl > l/e} -0 , such that the area of the subset of ~ where g£ fails to be conformal is bounded by C2 e4 . (Note that the area of ~ is l1l"e 2 . ) Consider the restriction of g£ to R!o. Let Dk be the dual disk bounded. by the circle which passes through the tangency points of pairs of the disks £"Vo R £ , Ri.~ v 0 and Rt.~ +I v. 0 See Figure 6.1. Let Zl, Z2, Z3 , Z4 be a quadruple of points on the circle which are: (1) sufficiently spaced out, say, r ZiI-zhr~ 160€, for any j1 =F j2 ; (2) away from the points of tangency

amo

227

232

Z .• X . HE AND O . SCHRAMM

Fig. 6.1. The six dual d isks.

{ wk } = R~on Ri.\: 1Io' say, IZj -wr.:1 ~ Ike; and (3) cyclic1y ordered in the counter-clockwise direction. We claim that for any such quadruple,

(6.2)

where [ . • '; . ,.J denotes t he cross ratio, and C3 depends only on 6. In fact, there are positive numbers m and m" such that the quadrilaterals (~o; Zit %2, %3, %4) and (gt(R!o); gt(z d, 9'"(Z2),9' (Z3), 9'(%4» are conformally homeomorphic to t he standard rectangles

(Qm = [0, m[ x [0 , 1[; (0, 0) , (m, 0), (m, 1), (0, 1))

and (Qm' = [0, mO[ x [0, 1[; (0, 0), (mO , 0), (mO, 1), (0, 1)),

respectively. T he map gt will then be translated to a CI-quasiconformal map F tc between the standard rectangles (Qm ; (0, 0) , (m, 0) , (m, 1), (0, 1))

a nd

(Qm' ; (0, 0), (mO, 0), (m", I ), (0, 1) ).

The relations I Zil - Zhl~n\oc, jl#-12. imply that m and m"E[m j C1 ,mC1 ] are bounded from above and below by some universal positive constants. On the other hand, since IZj-Wk l): l~E, by property (2) above we deduce that gE is conformal in the ~e neighborhood of the points Zj. 1:(j:( 4. Outside the 2~e- neighborhoods of the Z;'8, the conformal homeomorphism from R( onto Qm is clearly Lipschitz, wit h Lipschitz constant bounded by C4 e - 1, where C4 is a universal constant. Thus, using property (4) , it follows that f'! is conformal except on a subset of area bounded by c'fC2e2, It then 228

233

COO _CONVERGENCE OF DISK PACKINGS

follows by a standard Grotsch argument (compare [6, §2]) that Im*jm - 11:;;;:Cs €2 , where Cs depends on Cl and C2 • Then (6.2) follows as m and m- are related to the cross ratios [Zt, Z2;Z3,Z.,.l and [gl: (Zl),gl: (Z2);gl: (Z3),gl:(Z4)] ' respectively, by the same smooth function. Let Tk be the Mobius transformation which agrees with gl: on D k . Since a Mobius t ransformation is uniquely determined by its values at three points, it follows from (6.2) that for any kE Z6, (6.3)

where C6 depends only on C3 , and consequently, on D. It is then elementary to check that (6.3) implies that ISk(Vo) -1 1:;;;:,2C7, with C7 depending only on 6, which gives (6.1). 0

7. Regularity of solutions of discrete e lliptic equations

Let W be a subset oj V I:, let voEW(l ), and let Euclidean distance from Va to V I: - W. Let 1'/: W _ R be any function. Then

REGULARITY LEMMA 7.1.

6

be the

(7. 1)

holds for any kE Z6' This lemma is known in the continuous setting, and may certainly also be known in the discrete setting. The estimate is not sharp. For lack of a good reference, we include a proof.

Proof. Note that both sides of (7.1) are scale invariant; i.e., if we define ij:a- lW_ R by i1(v)=1}(av), where a>O, then (7.1) for 1'/ is equivalent to (7.1) for i1 at a-IVa. Hence, we assume with no loss of generality that D=1. Also assume for convenience , k=3, Va = '. Again, there is no loss of generality. Let R(x+iy)=(c-x)+iy be the reflection in the line x=!c, and set

g(v) ~ ~(v) - ~(Rv). As qVa=va+( -l)c=O=Rvo, what we need to prove is that

(7.2) This is obvious if 2:;;;:7c , so assume 7€I/E)-n is at least 8. 8.1. Let 6>0, and let n be an integer. Then there are constants C=C(n, 6), a = a(n,6»O such that (8.4) 118k.. 2 be the center, and £{h he the radius of C2 · Then 1>2 and (!z are constants. the center of P;, in terms of Zd(v)? The How can we find a formula for inversion of rev) in 8P; is 00, obviously. The preimage of that inversion under ZMv) is the pole of Z6(v). Let q be t he inversion in C2 of the pole of Zd(v). Note that when a point Z2 is the image of a point Zl under an inversion in a circle c, then any Mobius transformtion m will take Z2 to the inversion of m(zd in the circle m(c). Consequently, Z~(v)(q)=r(v). The pole of Zd(v) is just the point - d"(v )/c(v). This gives Let

Cl

rev),

It is therefore dear from Lemma 2.2 that

r(v)=Z~(v)(q)

238

converges Coo to

f.

243

C""'·CONVERGENCE OF DISK PACKINGS

Note that r' (w)+r'(L~w)

= If'(w) - f'(L~w)1 = 0 when z i=- Zl are in X. Now (4.2) is used to verify one case in the triangle inequality for d.

d(x, m)

+ d(m, y)

~

Ix - al V Ix - bl + sup

wEX

(Iy -

wi - la - wi V Ib - wi)

Ix - al V Ix - bl + (Iy - xl - la - xl V Ib - xl) = Ix - YI = d(x, y) . ~

Another case of the triangle inequality is proved as follows,

d(m,x)

+ d(x,y)

=

D/2 + sup

(Ix - wl-Ia - wi V Ib - wi) + Ix -

+

wi - la - wi V Ib - wi)

wEX

~ D /2 =

sup (Iy -

wEX

yl

d(m,y).

Since the triangle inequality for d holds for a~ three points jn X, it now follows that it holds for any three points in X, and hence X is a metric space. We now verify that (X, d) is 8-hyperbolic. For this, it is enough to prove the inequality (3.3) for m and points x, y, z E X. For an arbitrary w EX, we have

Ix - wi + Iy - zl

~ (Iy -

wi + Ix - zl) V (Iz - wi + Ix -

yl)

+ 28 .

Therefore,

d(m, x) +d(y,z)

=

D/2+ sup

wEX

(Ix -wl-Ia -wi V Ib-wl) + Iy- zl 251

EMBEDDINGS OF GROMOV HYPERBOLIC SPACES

Vol. 10, 2000

~

D/2 + 28 + sup ((Iy-wl + Ix-zl) wEX

=

(Iz-wl + Ix-YI) - la-wi

(D/2 + Ix - zl + sup (Iy - wl-Ia - wi wEX

V =

V

V

275

V

Ib-wl)

Ib - wi))

(D/2+ Ix-yl + SUp(lz-wl-la-wl V Ib-wl)) +28

(d(m, y)

which verifies that Observe that

wEX

+ d(x, z))

V

(d(m, z)

+ d(x, y)) + 28,

(X, d) is 8-hyperbolic.

d(m, a) = =

D/2 + sup (Ia - wl-Ia - wi V Ib - wi) wEX

D /2 + sup 0 A wEX

(Ia - wi - Ib - wi) .

As b may be chosen for w, this shows that d(m, a) d( m, b) = D /2, and the proof is complete.

=

D /2. Similarly, D

Proof of 4·1. Suppose Z is a metric space and x, y E Z. We say that m is a midpoint of x and y if Ix - ml = Iy - ml = Ix - yl/2. The space Z has the midpoint property, if for all x, y E Y there exists a midpoint m E Z. In the following, when a metric space Zl admits an isometric embedding into another metric space Z2, we can think of Zl as being contained in Z2 and write Zl C Z2. The first step in the proof is to construct a metric space X, ::) X, which has the midpoint property and is 8-hyperbolic. For a 8-hyperbolic metric space Z and a, b E Z, denote by Z[a, b] the space constructed according to the previous lemma. Let ¢ : w ----+ X x X be a bijective mapping from some ordinal w onto the set of pairs in X. By transfinite induction we define a metric space X (a) for each ordinal a ~ w + 1 with the property that X(a) ::) X((3) if a> (3. Set X(O) = X. When a = (3 + 1 is a successor ordinal, let X(a) be X((3)[a,b], where (a,b) = ¢((3). For limit ordinals a, set X(a) = U,6

(1) The map

f

f :X

---+

281

Y be a bijection

is an (a, A)-snowflake map if for all x, y E X A-llx - yla :::; If(x) - f(y)1 :::; Alx _ yla.

(2) The map f is an (a, A)-quasisymmetry iffor all distinct points x, y, zEX

(IxIx --zl) yl . °< t < 1 ,

If(x) - f(z)1 If(x) - f(y)1 :::; 7]a,A Here

7]a,A(t)

=

{

Atl/a for Ata for

1:::; t.

It seems that the class of maps in (1) has not been given a name in the literature. We call them snowflake maps, because these maps behave similar as the map giving the parameterization of the well-known von Koch snowflake curve. In contrast to this, quasisymmetries have appeared in the literature before (cf. [TuVI], [V]). Our usage of this term slightly differs from the common one. In general, a map is called a quasi symmetry if the above condition holds for some increasing homeomorphism 7] : (0, (0) ---+ (0, (0). So our notion of quasisymmetry is stronger. To emphasize this distinction we call a map a power quasisymmetry, if it is an (a, A)-quasisymmetries for some a > and A ?:: 1. For some spaces, e.g., connected spaces, quasi symmetries are always power quasisymmetries (cf. [TuVl, Cor. 3.12]). Every snowflake map is a power quasisymmetry. It is straightforward to check that inverse maps and compositions of snowflake maps are again snowflake maps. The same statement is true for power quasisymmetries. In fact, the inverse of an (a, A)-quasisymmetry is an (a, C(A, a))-quasisymmetry. Quasisymmetries are homeomorphisms. Two metrics d l and d 2 on a space X are called bilipschitz equivalent, snowflake equivalent or quasisymmetricaUy equivalent if the identity map idx : (X, dI) ---+ (X, d2 ) is bilipschitz, a snowflake map or a quasisymmetry, respectively. This defines equivalence relations for the metrics on a space X. Following is the standard construction of the metrics on ax, where X is a Gromov hyperbolic space. If x, y E ax, W EX, E > 0, let

°

dax,w,Jx,

y) = dw"(x, y) = inf {

°

t,

e-,(x,-,Ix;)w } ,

where the infimum extends over all finite sequences x = y in ax. Here the convention e- oo = is understood. 258

(6.1)

XO, Xl, X2, ... ,X n =

282

M. BONK AND O. SCHRAMM

GAFA

LEMMA 6.1. There is some constant EO > 0 with the following property. If X is 8-hyperbolic and E8 ~ EO, then l2 e - E(xly)w "~ dW,E (x ,y), , ~ e-E(xly)w , v W (6.2) x, Y E ax . For the proof, see, for example, [GH, Ch.7]. This lemma leads to the following definitions. DEFINITIONS. The canonical gauge Q(X) on ax is the set of all metrics of the form d = dW,E. We say that (El, dd is B-equivalent to (E2' d2 ) if there is a constant c > 0 such that El ~ d E2 ~ C dEl c- 1d2 " 1" 2· A B-structure on Z is an equivalence class of this equivalence relation. The lemma tells us that (E, dW,E) and (E', dW' ,E/) are B-equivalent when w, w' E X and E, E' > 0 satisfy E8 ~ EO and E'8 ~ EO. Therefore, when X is Gromov hyperbolic, there is an associated B-structure on ax. It is called the canonical B-structure on ax. Mostly, it is convenient to work with a fixed metric in the canonical gauge and to suppress the ambiguity in choosing this metric. We often abbreviate such a metric by d8X. PROPOSITION 6.2. The boundary ax of any Gromov hyperbolic space X is bounded and complete. Note that boundedness and completeness are properties that do not depend on the particular choice of the metric d8X in Q(X). There are situations where ax is not compact. For example, when X is infinite dimensional hyperbolic space, ax is homeomorphic to the unit sphere in Hilbert space. Proof. The boundedness of ax follows from the fact that the Gromov product is nonnegative. Fix a basepoint a E X. Let {Yi} c ax be a d8X-Cauchy sequence for some metric in the canonical gauge. Then limi,j---+oo(YiIYj) = 00. There exists a sequence {xd C X with limi---+oo(XiIYi) = 00. Then {Xi} converges at infinity. If Y E ax is the equivalence class of {xd, then limi--->oo Yi = y. D We will now discuss the functorial properties of X f---+ ax. PROPOSITION 6.3. Suppose X, Y, Z are Gromov hyperbolic almost geodesic metric spaces, and! : X ---+ Y, 9 : Y ---+ Z are roughly quasi-isometric embeddings. (1) If {Xi} c X converges at infinity, then {!(Xi)} c Y converges at infinity. If {Xi} and {yd are equivalent sequences in X converging at infinity, then {!(Xi)} and {!(Yi)} are also equivalent. 259


Vol. 10, 2000

283

(2) If a E ax and {Xi} E a, let bE ay be the equivalence class of {1(Xi)} and define al(a) = b. Then al : ax -----+ ay is well-defined. Moreover, a(g 0 f) = ag 0 aj. (3) If il, 12 : X -----+ Yare rough quasi-isometric embeddings and il ~ 12, then ail = ah. (4) The map al is injective. If the image of I is cobounded in Y, i.e., if I is a rough quasi-isometry, then I is a bijection.

a

Proof The statements in (1)-(3) immediately follow from the definitions

and (1) of Proposition 5.5. For the first part of (4) note that if {xd and {yd are sequences in X converging at infinity such that {1(Xi)} and {1(Yi)} are equivalent, then limi-+oo(f(xi)II(Yi)) = 00. This implies limi-+oo(xiIYi) = 00 by 5.5.(2). Thus {xd and {Yd are equivalent. If I is a rough quasi-isometry, then it has a rough inverse h : Y -----+ X which is a rough quasi-isometry. Since a(id x ) = idax, the statements (2), (3) and the relations hoi ~ idx , I 0 h ~ idy imply ah 0 a I = idax and a I 0 ah = iday. From this it follows that a I is bijective. D 6.4. The inequalities in Proposition 5.5 are also valid if X, y, Z E ax, if x', y', z' are the images of these points under ai, and if we assume in part (2) that x, y, z are distinct points. COROLLARY

Proof This follows from the definition of ai, Proposition 5.5, and (3.4).

Note that for the proof of the second part we need inequality 5.5.(2) under the slightly stronger assumption (xlz)w - (xIY)w ;:: -C8, x, y, z, W EX (cf. the remark following Prop. 5.5). D Theorem 6.5. Suppose X and Yare Gromov hyperbolic almost geodesic metric spaces, and let I : X -----+ Y be a map.

(1) If I is a rough similarity, then al is a snowflake map. (2) If I is a rough quasi-isometry, then aI is a power quasisymmetry. Proof Let x' denote al(x) for x E ax. Note that al is bijective by Proposition 6.3 (4). The properties of a I in question do not depend on which metrics dax E Q(X) and day E Q(Y) in the canonical gauges on ax and ay we choose. We may assume that the image w' of the basepoint w of X is the basepoint in Y. Then there exists constant E, E' > a such that for all x, Y E ax

dax(x, y) ~

e-E(xly)

and

day(x', y') ~

e-E'(x'ly') .

(6.3)

Here ~ means equality up to a multiplicative constant independent of x and y. 260


284

°

GAFA

°

From this we see that statement (1) is equivalent to the existence of constants ..\ > and K ~ such that ..\(xIY) - K ~ (x'ly') ~ ..\(xIY) + K (6.4) for all x, y E ax. If I is a rough similarity, an inequality like this holds for x, y E X (the prime indicating the image under I in this case). The definition of ai, and (3.4) imply that (6.4) also holds for x, y E ax. Statement (2) is a straightforward consequence of (6.3), 5.5.(2), and Corollary 6.4. D We can summarize Proposition 6.2, Proposition 6.3, and Theorem 6.5 as follows. Let CI and C2 be categories, where the objects are Gromov hyperbolic almost geodesic metric spaces. The morphisms of CI are rough similarities, while the morphisms of C2 are rough quasi-isometries. Let VI and V 2 be categories, where the objects are bounded and complete Bstructures. The morphisms of VI are snowflake maps, and the morphisms of V 2 are power quasisymmetries. Then X I---t ax is a functor from Ci to Vi for i E {1,2}. Actually, as the results in section 7 will indicate, it is more appropriate to consider rough mapping classes of rough similarities and rough quasi-isometries as the morphisms of CI and C2, respectively.

7

The Metric Space Con(Z)

Given a bounded metric space (Z, d), we now construct a Gromov hyperbolic space Con(Z). The space Con(Z) has properties analogous to the hyperbolic convex hull of a set in the boundary of a real hyperbolic space. We refer to section 10 for some further discussion. Our construction is similar to one given by Gromov [Gr, 1.8.A.(b)] and to a construction of Trotsenko and Viiisiilii [TV]. Set

Z x (0, D(Z)] . Here and in the following we let D(Z) = diam(Z) if diam(Z) > 0, and D(Z) = 1 if diam(Z) = 0, i.e., if Z consists of a single point. It is convenient Con(Z)

=

to include this trivial case in the definition, for otherwise we would have to exclude it explicitly in some of the following statements. Define p: Con(Z) x Con(Z) -+ [0, (0) by

p( (z, h), (z', h'))

= 2 log (

d(z, zj;; V h')

(7.1)

The motivation of the factor 2 comes from the fact that it corresponds to curvature -1 for real hyperbolic spaces. 261

EMBED DINGS OF GROMOV HYPERBOLIC SPACES

Vol. 10, 2000

LEMMA

7.1.

285

P is a metric on Con( Z).

Proof The triangle inequality, p((z, h), (z", h")) ~ p((z, h), (z', h'))

+ p((z', h'), (z", h")) , (7.2)

reduces to

d(z, z") + h V h" ~ d(z, z') + h V h' . d(z', z") + h' V h" Jhh" "" Jhh' Jh'h'"

which is equivalent to

h' (d(z, z") + h V h") ~ (d(z, z') + h V h')( d(z', z") + h' V h"). (7.3) Since d(z, z") ~ d(z, z') +d(z', z"), inequality (7.3) holds, which verifies the triangle inequality. The other properties of a metric are immediate.

D

Henceforth, Con( Z) will always be equipped with this metric p. We will use the same letter p for metrics on different spaces Con(Z). Theorem 7.2. There are constants 8 ;? 0, k ;? 0 with the following property. If (Z, d) is a bounded metric space, then Con( Z) is 8-hyperbolic, k-visual, and k-roughly geodesic.

Proof First we prove the Gromov hyperbolicity of Con(Z). Suppose we are given numbers rij ;? 0 such that rij = rji and rij ~ rik + rkj for i, j, k E {1, 2, 3, 4}. Then r12r34 ~ 4( (r13r24) V (r14r23)). To see this, we may assume that r13 is the smallest of the quantities rij appearing on the right hand side of this inequality. Then r12 ~ r13 + r32 ~ 2r23 and r34 ~ r3l + r14 ~ 2r14. The inequality follows. Now let Xi = (Zi' hi), i E {1, 2, 3, 4}, be four arbitrary points in Con(Z). Set dij = d( Zi, Zj) and rij = dij + hi V hj . The numbers rij satisfy the above requirements. Hence

(dl ,2 + hI V h2)(d3,4 + h3 V h4) ~ 4((dl ,3 + hI V h3)(d2,4 + h2 V h4)) V ((dl ,4 + hI V h4)(d2,3 + h2 V h3)). This translates to

p(Xl' X2)

+ p(X3, X4)

~ (p(Xl' X3)

+ p(X2' X4))

V

(p(Xl' X4)

+ p(X2' X3)) + C ,

which shows that Con(Z) is C-hyperbolic (cf. (3.3)). Choose a point Zo E Z, and let 0 = (zo, D(Z)) E Con(Z). If p = (z, h) E Con(Z) is an arbitrary point, a C-roughly geodesic ray emanating from 0 and passing through p can be obtained from an appropriate parameterization of the set {o}U{z} x (O,D(Z)]. More precisely, define the ray , : [0,(0) --t Con(Z) by ,(0) = 0 and ,(t) = (z, D(Z)e- t ) for t > o. It is easy to check that , has the required properties. 262

286


GAFA

Finally, Con(Z) is C-roughly geodesic as follows from Proposition 5.6 and the first two parts of the proof. D We want to show that the assignment Z f---+ Con(Z) is a functor of appropriately defined categories. The basic problem is to define an induced map i: Con(X) --+ Con(Y) for a map f : X --+ Y, X f---+ x'. This is possible if f is a power quasisymmetry. As we mentioned in the introduction, the idea is similar to the fuzzy extension of a quasiconformal map on JRn to upper half-space in JRn+1 by Tukia and ViiisiWi [TuV2]. One would like to define i(x, h) ~ (x', lx' - y'I), where y is a point in X with h = Ix - YI. The fact that f may not be well-defined is not serious problem. For if z is any other point with h = Ix - zl, then lx' - y'l and lx' - z'l are comparable if f is a power quasisymmetry. In this case, the p-distance of (x', lx' - y'l) and (x', lx' - z'l) in Con(Y) is bounded by a constant only depending on the parameters of f. So f is "roughly" well-defined. More serious is the problem that there might exist no y with h = Ix-yl, or even worse, no y such that Ix-yl is comparable to h. In this case, one has to "interpolate" between those heights h which are attained as distances of points. The following lemma gives the basic step in the construction of f. For z E Z, we denote by Rz the ray in Con(Z) that ends at z E Z: Rz = {z} x (O,D(Z)] c Con(Z). The maps fx of the lemma will be used in the definition of and will be the restrictions of to the rays Rx (cf. the definition following Lemma 7.3). ~

i

1,

7.3. Suppose that f : X --+ Y, X f---+ x' = f(x), is an (a, c)quasisymmetry of bounded metric spaces X, Y. For all x E X we can define a map fx : Rx --+ R x' with the following properties:

LEMMA

(1) There exist A

°

A(a, c) ~ 1, k = k(a, c) ~ such that all maps fx, x E X, are (A, k)-rough quasi-isometries. (2) If x, y E X, x -::J y, h = Ix - yl, and fx(x, h) = (x', h'), then =

C(a, c)-llx' - Y'l ~ h' ~ lx' - y'IC(a, c). (3) Let x E X, tl, t2 E (0, D(X)], fx(x, tl) = (x', ti), fx(x, t2) = (x', t~). Then tl ~ t2 =? ti ~ t~ , i.e., fx is non-decreasing on Rx· (4) For all x, y E X, x -::J y, and h E [Ix - yl, D(X)] p(Jx(x, h), fy(y, h)) = Oa,c(1) . 263

Vol. 10, 2000

287


(5) If Z is a metric space with Enite diameter and 9 : Y quasisymmetry, then for all x E X and p E Rx

---+

Z is a ((3, c')-

p((g 0 J)x(p),gx,(fx(p))) = Oa,/3,e,c'(l). In particular, (g 0 J)x ~ gf(x) 0 Ix. (6) If I is a bilipschitz or snowflake map, then the maps Ix, x E X, can be deEned such that they are rough isometries or rough similarities with the same parameters, respectively. The parameters only depend on the parameters of I. These maps satisfy statements similar to (2)-(5) (cf. proof).

Proof We may assume diam(X) > 0. For x E X let Sx c N be the set of all lEN for which the annulus Ax(x, l) = {z EX: 2- l - 1 diam(X) < Iz - xl ::;; 2- l diam(X)} is nonempty. The set Sx can be regarded as the "scale spectrum" of X at x. Note that m = min Sx E {O, l}. Similarly, let Sx' be the scale spectrum of Y at the point x'. We define a map ¢x : Sx ---+ Sx' by

¢x(l) = sup {l' I :3yEX : 2- l - 1 diam(X) < Iy-xl , y' EAy(x', l')}, VlESx ' Obviously, ¢x is non-decreasing on Sx' If l E Sx there exists a point y E Ax(x, l). If we take any such point y, then for a unique l' E Sx' we have y' E Ay(x', l'). From the fact that I is an (a, c)-quasisymmetry it follows that Ii' - ¢x(l)1 = Oa,e(l). For h, l2 E Sx choose points Y1 E Ax(x, h), and Y2 E Ax(x, l2)' Since

I

is an (a, c)-quasisymmetry,

C(a c)-In

(2h -l2)-1

~ Iyi - x'i ~ C(a c)n

"" I' ",a,e Y2 -x'I "" This and the remark following the definition of ¢x show ",a,e

(2l2-h)

.

(7.4)

A- 11l2 - hl- C(a, c) ::;; l¢x(l2) - ¢x(h)1 ::;; AIl2 - hi + C(a,c) , (7.5) where A = A(a) ~ 1. The set ¢x(Sx) is C(a, c)-cobounded in Sx" For if l' E Sx' is arbitrary,

then, since I is bijective, there exists a point y E X, y -=I=- x, such that y' E Ay(x', l'). For some l E Sx we have y E Ax(x, l). By the above remark I¢x(l) - l'l = Oa,e(1). Since {O, l} n Sx' -=I=and ¢x is non-decreasing, this coboundedness implies

°

¢x(m) ::;; C(a, c). We now extend the map ¢x : Sx ---+ Sx' to a map ¢x : [0, 00)

---+

(7.6) [0,00).

We use the same notation for the new and the old map. The extension is essentially obtained by linear interpolation. 264

288


GAFA

More precisely, let M = sup Sx E N U {oo}. Define ¢x (t) = ¢x (m) for t E [0, m). If t E (m, M)\Sx, there a smallest interval (h, h) with endpoints h, l2 E Sx containing t. Then t = fJh + (1 - fJ)l2 for a unique fJ E (0,1). Define ¢x(t) = fJ¢x(h) + (1 - fJ)¢x(l2). Finally, if M < 00 and t > M put

¢x(t)

=

¢x(M)

+ (t -

M).

Obviously, ¢x is continuous and non-decreasing. Furthermore, limt-+oo ¢x(t) = 00. This follows from the definition of ¢x if M < 00 and from (7.5) if M = 00. Now (7.6) shows that ¢x([O, (0)) is C(a, c)-cobounded in [0,(0). Since ¢x : Sx ---+ Sx' is non-decreasing and satisfies (7.5), it follows that ¢x : [0,(0) ---+ [0,(0) satisfies (7.5) for all h,h E [0,(0). This statement is straightforward to prove. We leave the details to the reader. (Note that (7.5) is true, when h, l2 are in some minimal non-degenerate interval with endpoints in Sx U {a, oo}. Introducing appropriately chosen points in Sx n [h, h], the general case of (7.5) can be reduced to this special case.) These considerations show that ¢x : [0, (0) ---+ [0, (0) is a non-decreasing rough quasi-isometry with parameters only depending on a and c. Now let D = diam(X), D' = diam(Y), and define for (x, h) E Con(X)

Ix(x, h)

=

(x', D'2-¢x(IOg2(D/h))) .

(7.7)

The statements about the maps Ix follow directly from the properties of the maps ¢x. This is clear for (1) and (3). For (2) note that ¢x(log2(D/lxyl)) = log2(D' Ilx' -y'l) +Oa,c(1) for x, y E X, x =I- y. Statement (4) follows from (2) and (3). To prove (5), define 'ljJy : Sy ---+ Sg(y) , y E Y, and Ox : Sx ---+ Sg(f(x)) , x E X, in the same way for g and go I, respectively, as the maps ¢x, x E X, were defined for I. Let x EX. Making the parameters of the maps I and g larger if necessary, we may assume that both are power quasisymmetries with the same parameters, a and c, say. This implies that for t E Sx

(7.8) To establish this equation for general t E [0, (0) assume t E (m, M) \Sx. The cases t E [0, m] or t E [M, (0) are easier and can be treated in a similar way as below. Let (h, l2) be the smallest interval with endpoints in Sx containing x and write t = fJh + (1- fJ)l2 with fJ E (0,1). Then ¢x(t) = fJli + (1- fJ)l~, where li = ¢x(h) and l~ = ¢x(h). In particular, ¢x(t) E [minSx', sup Sx']' and so there exists some minimal interval [{1, {2] with endpoints in Sx' containing ¢x(t). The interval [{1,[2] is degenerate if ¢x(t) E Sx'. 265

Vol. 10, 2000


289

Since ¢x is non-decreasing and ¢(Sx) is C(a, c)-cobounded in Sx" we have II = l~ + Oa,c(l) and I2 = l~ + Oa,c(l). From the definition of Ox and ¢x, equation (7.8) for t E Sx, and the fact that 'l/Jx is a rough quasi-isometry with parameters only depending on a and c, we obtain

Ox(t)

=

= =

+ (1 - f-t)'l/Jx,(l~) + Oa,c(1) f-t'l/Jx,(ll) + (1 - f-t)'l/Jx,(12) + Oa,c(l) 'l/Jx'(¢x(t)) + Oa,c(l). f-t'l/Jx,(lD

This shows that (7.8) holds for all t E [0,(0). The maps (g 0 J)x, x E X, and gy, y E Y, are defined similarly as in (7.7) using the maps Ox and'l/Jy, respectively. Statement (5) then follows from (7.8). To see that (6) is true, define Ix(x,h) = (x', (D'jD)h) for (x,h) E Con(X) if I is a bilipschitz map. Statements (1)-(5) are immediate in this case with all constants only depending on the bilipschitz constant of I (and the bilipschitz constant of 9 in (5)). Suppose that I is an (a, c)-snowflake map. If S(X) := UXEX Sx is an infinite set, then a is uniquely determined. In this case let aU) = a. If S(X) is a finite set, let aU) = l. Note that the property of a space X that S (X) is finite or infinite is invariant under snowflake maps. This implies that if 9 : Y ---+ Z is a snowflake map, then a(g 0 I) = a(g)aU). The set S(X) is finite, if and only if d(X) := infx,YEx,xyfy Ix - yl > O. In this case, I is a A-bilipschitz map with A = A(a, c, diam(X), d(X)). Now define Ix(x,h) = (x',D'(hjD)a(f)). The stated properties of the maps Ix are immediate to check. The constants in (1), (2), (4) will only depend on a and c. If S(X) is finite, the constant in (2) will also depend on d(X) and diam(X). The snowflake D version of statement (5) is true with constant O. If X and Yare bounded metric spaces, and I : X ---+ Y is a power quasisymmetry (which includes snowflake or bilipschitz maps), we define a map Con(X) ---+ Con(Y) by

1:

l(x, h)

=

Ix(x, h) , \i(x, h)

E Con(X) .

Here Ix, x E X, are the maps defined in the last lemma. Theorem 7.4. Suppose that I : X ---+ Y is a power quasisymmetry of bounded metric spaces X, Y. Then Con(X) ---+ Con(Yl is a rough quasi-isometry. If I is a snowflake or bilipschitz map, then I is a rough similarity or a rough isometry, respectively.

1:

266

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GAFA

If Z is a bounded metric space, and 9 : Y ---+ Z is a map of tbe same type as I, i.e., a power quasisymmetry, a snowflake map, or a bilipscbitz ~ map, tben 9 0 I ~ go I·

--

Proof Assume I: X ---+ Y, X t---+ x' = I(x) is an (a, c)-quasisymmetry. We have f(Con(X)) = UXEX Ix(Rx), and UXEX R x' = Con(Y). Since Ix(Rx) is C(a, c)-cobounded in R x' for all x E X by Lemma 7.3.(1), the set f(Con(X)) is C(a, c)-cobounded in Con(Y). To show that Con(X) ---+ Coney) is a rough quasi-isometry, we have to establish inequality (2.1) for I. For two arbitrary points ql = (Xl, hI), q2 = (X2' h 2) E Con (X) we express the distance of ql and q2 and the distance of their image points q~ = (x~, h~) and q~ = (x~, h~) under by distances only involving pairs of points lying on the same ray R z . Then (2.1) will follow from the definition of and the fact that the maps lx, x E X, are rough quasi-isometries with the same parameters. To that purpose define h = IXI -x21 V hI V h2, PI = (Xl, h), P2 = (X2' h), p~ = f(pI) = (x~, h'), p~ = f(P2) = (x~, h'). The definition of p shows

f :

f

f

p(ql' q2) = p(ql,Pl) + p(p2' q2) + 0(1) . If h = IXI - x21, then from Lemma 7.3.(2) and (3) it follows that

C(a, c)-llx~ - x~1 ~ h', h' ~ C(a, c)lx~ - x~l,

and

(7.9)

h~ V h~ ~ h', h'.

This implies p(q~, q~) = p(q~,pD

+ p(p~, q~) + Oa,c(1).

(7.10)

If h =I- IXI -x21, i.e., if h = hI V h2, we may without loss of generality assume that h = hI. Then q~ = pi and Lemma 7.3.(4) shows that p(pi,p~) = Oa,c(1). Thus, (7.10) also holds in this case. Consequently, Lemma 7.3.(1) now gives

+ p(p~, q~) + Oa,c(l) ~ C(a, c) (p(ql,Pl) + P(P2' q2)) + Oa,c(1) = C(a, C)P(ql' q2) + Oa,c(1).

p(q~, q~) = p(q~,p~)

An inequality in .the opposite direction can be obtained in the same way. This shows that I is a rough quasi-isometry. If I is a snowflake or a bilipschitz map, the same argument based on (7.9), (7.10) and Lemma 7.3.(6) shows that is a rough similarity or a rough isometry, respectively. Finally, the last statement follows from Lemma 7.3.(5). D Again we may summarize the results of this section in the language of categories. Let Cl , C2 , C3 be categories, where the objects are bounded

f

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291

metric spaces, and the morphisms are power quasisymmetries, snowflake maps, and bilipschitz maps, respectively. Let V 1 , V 2 , V3 be categories, where the objects are visual Gromov hyperbolic spaces, and the morphisms are rough mapping classes of rough quasi-isometries, rough similarities, and rough isometries, respectively. Then Z 1-----+ Con(Z) is a functor from Ci to Vi for i E {l, 2, 3}. In particular, we get the following correspondence for the types of maps power quasisymmetry

rough quasi-isometry,

snowflake map

rough similarity,

bilipschitz map

rough isometry.

Note that the first two correspondences (going from right to left) are exactly what we found in the last section for the functor X 1-----+ ax.

The Relation of

8

ax

and Con(Z)

In this section we will investigate the relation of the functors X 1-----+ ax and Z 1-----+ Con(Z). We content ourselves with proving two statements about the objects of the categories involved. Similar questions can also be studied for the morphisms.

Theorem 8.1. Suppose (Z, d) is a complete bounded metric space, and let X = Con( Z). Then the spaces ax and Z can be identified as sets, and dis biJipschitz to a metric in Q(X), the canonical gauge on ax.

Proof. Take as the basepoint a E X some point of the form (zo, D), where D = D(Z). Let x = (z, h), x' = (z', h') EX. Then (xix')

=

10 (d(Zo, Z)+D) g ..)Dh

+ 10 (d(Zo, Z')+D) -10 (d(Z, z')+h V h')

..)Dh' g ..)hh' = -log (d(z, z') + h V h') + OD(1). From this follows that a sequence {(Zi' hi)} in Con(Z) converges at infinity if and only if {zd is a Cauchy sequence in Z and limi--+oo hi = O. Since Z is complete, {Zi} has a limit y in Z. It is immediate to check that if a sequence {(z~, h~)} is equivalent to {(Zi' hi)}, then limi--+oo Zi = limi--+oo z~ = y. So by assigning y to the equivalence class of {(Zi' hi)} we get a well-defined map from ax to Z. It is straightforward to verify that this map is bijective. g

Thus, ax and Z can be identified as sets. Moreover, from the above expression and (3.4) we conclude

(zlz')

=

-log(d(z, z'))

+ OD(l), 268

\;/z, z' E ax

=

Z.

292


GAFA

This shows that d is bilipschitz to a metric in the canonical gauge on ax.

D

Theorem 8.2. Suppose X is a visual Gromov byperbolic metric space. Tben X and Con( aX) are rougbly similar.

The proof will actually show more. If d is a metric in the canonical gauge on ax for which there exists a point W E X, and a constant a ~ 1 such that

z') ~ ae-(zlz')w ,z,z, E ax , a -Ie-(zlz')w -.~. ;: d(z ,-...;:: then X and Con( (aX, d)) are roughly isometric.

Proof Assume X is 8-hyperbolic and k-visual with respect to 0 E X. If we take different metrics d l , d2 in the canonical gauge on ax, then the spaces (aX, d l ) and (aX, d 2 ) are snowflake equivalent. Theorem 7.4 implies that the spaces Con((aX, d l )) and Con((aX, d2 )) are roughly similar. Therefore, the assertion does not depend on which metric in the canonical gauge on ax we choose. Fix such a metric d on ax. Then we may assume that

a-I exp ( - E(zlz')) :s:; d(z, z') :s:; a exp ( - E(zlz')),

\:jz, z'

E ax.

(8.1)

Making k larger if necessary, by Proposition 5.6 and Proposition 5.2.(2) for each z E az we can choose a k-roughly geodesic ray '"Yz : [0,00) ---+ X with '"Y( 0) = 0 and lim '"Yz = z. Let D = D((aX, d)). For (z, h) E Con(aX) , we define f(z, h) = '"Yz(cIlog(D/h)). We show that f : Con(aX) ---+ X is a rough similarity. First, note that the image of f is cobounded in X. For each x E X lies on some k- roughly geodesic ray emanating from o. If z = lim '"Y, then from Lemma 5.1 it follows that b(t) - '"YAt) I = 08,k(1) for t ~ o. So x has distance at most 08,k(1) to '"Yz which is contained in the image of f. Let (z, h) and (z', h') be two arbitrary points in Con(aX). Observe that (zlf(z, h)) = cIlog(D/h) + 08,k(1), and (z'lf(z', h')) = cIlog(D/h') + 08,k(1). Therefore, from Lemma 5.1 we get

Elf(z, h) - f(z', h')1 = 10g(D /h)+ 10g(D /h')-2( E(zlz') 1\ 10g(D /h) 1\ 10g(D /h') )+08,k,E(1) = -log h -log h' + 2( - E(zlz') V log h V log h') + 08,k,E,D(1) e-E(zlz') V h V h') = 2 log ( v'hJ1! +08kED(1). hh' , ,, 269


Vol. 10, 2000

+ v)/2 ~ u V v ~ u + v to obtain + V 2 log (d(Z, z') v'hh! + 08,k,E,D,a 1

Now apply (8.1) and the inequalities (u EI

f ( z, h) - f ( Z",h) I = =

293

p((z, h), (z', h'))

h h')

+ 08,k,E,D,a(1).

()

(8.2)

Since the image of f is cobounded in X, it follows from (8.2) that f is a rough quasi-similarity. Note that if E = 1, then f is a rough isometry. D The last two statements may be summarized as follows. If Z is a complete bounded metric space, then Z

sf

a(Con Z),

where sf means snowflake equivalence of the spaces. Note that for a space to be snowflake equivalent to the boundary of a Gromov hyperbolic space it is necessary that the space is complete and bounded by Proposition 6.2. On the other hand, if X is a visual Gromov hyperbolic space, then X rv Con(aX) , where rv means rough similarity of the spaces. It is not hard to see that the property of being visual and Gromov hyperbolicity are preserved under rough similarities. So the assumptions on X are necessary by Theorem 7.2. These two statements show that in some sense X f---+ ax and Z f---+ Con(Z) are inverse to each other.

9

Growth and Assouad Dimension

In this section we prove Theorem 9.2 which will be used in sections 10 and 1l. DEFINITION (cf. [AJ). Let X be a metric space. For a,{3 > 0, let 8(a,{3) be the maximal cardinality of a set V C X such that a ~ Ix - yl ~ (3 for all x, y E V, x =I y. Define t to be the infimum of all numbers s ~ 0 such that for some constant K ~ 0 the inequality

8(a, (3)

~

K({3/a)S

holds for all 0 < a ~ {3. (It is understood that t = 00, if no such sexists.) Then dimA(X) = t is the Assouad dimension of X. See [L] for a further discussion of the Assouad dimension. We will need the following theorem (cf. [AJ). Theorem 9.1 (Assouad's Theorem). Let (Z, d) be a metric space with finite Assouad dimension, and let p E (0,1). Then there is some integer n such that the metric space (Z, dP ) admits a biJipschitz embedding into ]Rn. 270


294

GAFA

Here, dP is the metric dP(x, y) = d(x, y)P, x, Y E Z. A metric space X is called doubling, if for all r, R with R > r > 0 there exists N E N only depending on R/r such that every open ball of radius R in X can be covered by N open balls of radius r. It can be shown that a metric space has finite Assouad dimension if and only if it is doubling. The next theorem gives a sufficient condition for a Gromov hyperbolic geodesic metric space X to have a boundary with finite Assouad dimension. To state the theorem we make the following definition. DEFINITION. A metric space X has bounded growth at some scale, if there are constants r, R with R > r > 0, and N E N such that every open ball of radius R in X can be covered by N open balls of radius r. Theorem 9.2. Let X be a Gromov hyperbolic geodesic metric space with bounded growth at some scale. Then the Assouad dimension of ax is nnite. As we remarked above, the assertion is equivalent to ax being doubling. So a weak doubling condition on X implies that ax is doubling. Note that the properties of a metric space to have finite Assouad dimension and to be doubling are preserved by quasisymmetric maps. In particular, the statement about ax in the theorem is independent of which metric in the canonical gauge we choose. Examples for Gromov hyperbolic geodesic metric spaces having bounded growth at some scale are all complete simply-connected Riemannian nmanifolds X with sectional curvature '" satisfying -b2 :::;; '" :::;; -a 2 < o. In this case open balls of radius 1 in X are bilipschitzly equivalent to the unit ball in ]Rn with bilipschitz constant only depending on a and b. This statement follows from Topogonov's comparison theorem, and implies that X has bounded growth at some scale. In particular, complex hyperbolic spaces are Gromov hyperbolic geodesic metric spaces with bounded growth at some scale. Note that any bounded degree graph has bounded growth at some scale. Hence, every finitely generated hyperbolic group has bounded growth at some scale. The proof of 9.2 is similar to the proof of Proposition 11 from [GH,

Ch.7]. Proof Assume X is 8-hyperbolic, and fix some metric dax = do,E in the canonical gauge on ax. Rescaling the metric on X by the factor E if necessary, we may assume E = 1. Then

(zlz')

=

-log (dax(z, z'))

+ 08(1),

271

't/z, z' E ax.

(9.1)

Vol. 10, 2000


295

Denote by B(q, s) the open ball in X with center q E X and radius s > O. By assumption, there exist constants R > r > 0, N E N such that every open ball of radius R in X can be covered by N open balls of radius r. Suppose that B(p, 2R - r) is some ball of radius 2R - rand center p E X. Then B(p, R) can be covered by N open balls of radius r with centers PI, ... ,pN EX, say. Since X is a geodesic metric space, it follows that B(p, 2R - r) is covered by the balls B(Pl' R), ... , B(PN, R). Therefore, B(p, 2R - r) can be covered by N 2 open balls of radius r. By induction it follows that any open ball of radius mR - (m - l)r, mEN, can be covered by N m open balls of radius r. Given any Z E ax, for every t > 0 there is an x E X such that Ixi = t and (xlz) - t = 0 8 (1). To verify this, take w E X with (wlz) ? t and let x be the point satisfying Ixl = t on the geodesic segment [0, w]. Fix some a, f3 with 0 < a ::::;; f3 ::::;; 1. Suppose Zl, Z2, ... ,Zn are points in ax such that

(9.2) Then (9.1) implies - log f3 - C (8) ::::;; (Zi IZj) ::::;; - log a

+ C (8) ,

i:lj.

(9.3)

Let Xl, ... ,Xn be points satisfying Ixj I = - log f3, I - log f3 - (x j IZj ) I ::::;; C(8), and let Yl, ... ,Yn be points satisfying IYjl = -loga, I-Iogf3(YjIZj) I ::::;; C(8). Then it is easy to see from Lemma 5.1 and (9.3) that

IXi - xjl IYi - Yjl IXi - Yil

=

08(1),

= 2( -loga

+ 10g(3) + 08(1),

(9.4)

10gf3 -loga + 08(1), for all i :I j. It follows that the open ball of radius =

R*

=

10g(f3/a)

+ C(8)

centered at Xl contains all the points Yi. Let m be the least integer such that m(R- r) ? R*. We have seen that an open ball of radius m(R - r) can be covered by N m balls of radius r. So the set {Yl, ... , Yn} can be covered by Nm balls of radius r. We now assume that 10g(f3/a) is much larger than r. It then follows from the middle equation in (9.4) that IYi - Yj I > 2r for i :I j. Consequently, any ball of radius r can contain at most one of the points Yj. This gives n::::;; N m ::::;; N1+R*/(R-r)::::;; C(R,r,N,8)(f3/a)C(R,r,N,8). By changing the constants if necessary, the same is true even without the assumption that 10g(f3/a) is much larger than r. Since the diameter of X 272

296


GAFA

is finite, the assumption j3 :::; 1 is also inconsequential. We conclude that the Assouad dimension of 8X is finite. D

10

Embeddings into Real Hyperbolic Spaces

The definition of the metric p on Con(X) was motivated by the upper half-space model of real hyperbolic n-space IHIn. In this model IHIn

= lR~ =

{(Xl, ... ,Xn )

E lRn : Xn

> o}

is equipped with the Riemannian metric given by the length element

ds 2

= ~(dxi Xn

+ ... + dx~).

The space IHIn is a Riemannian n-manifold with constant sectional curvature -l. We identify lRn- l with the hyperplane in lR n given by the equation Xn = O. For X E lRn we write X = (z, h), where z E lRn- l , hER Then IHIn = {(z, h) : z E lRn-l, h > O}. It is not hard to find explicit expressions for the hyperbolic distance dh((Z, h), (Z', h')) of two points (z, h), (Z', h') E IHIn. For our purpose it is sufficient to know that for some universal constant C > 0 I I ( Iz Idh((Z, h), (z, h)) - 2 log

+ h V hI) I :::; C. Jhhi

Zl I

(10.1)

Here Iz - Zl I is the euclidean distance of z and Z'. The spaces IHI n , n E N, are geodesic, and from (10.1) and the first part of the proof of Theorem 7.2 it follows that they are C-hyperbolic. The Gromov boundary 8IHIn can be identified with lRn - l U {oo}. If M c lR~ is bounded with respect to the euclidean metric, then (M, dh) is C-hyperbolic, and 8M can be identified with the intersection of the euclidean closure of M and lR n - l . Moreover, if w E IHIn , then (ZIZ')w = -log Iz - zll + OM,w(1) , \/z, z' E 8M c lRn- l . (10.2) In particular, the euclidean metric is bilipschitz to a metric in the canonical gauge of 8M. These statements can be verified using (10.1), and considerations as in the proof of Theorem 8.l. If A c IHI n U 8IHIn , let hull(A) c IHIn be the intersection of all closed half-spaces H C IHIn such that A C H U 8H. Recall that in the upper halfspace model of IHIn, half-spaces in IHIn are determined by hyperplanes in lRn perpendicular to lR n- l or spheres with center in lRn- l . The set hull(A) is the smallest set M C IHIn that is hyperbolically convex and closed, and 273

Vol.IO, 2000


297

satisfies A eMu 8M. Here we have to assume that A consists of more than one point if A c 8JH[n, for otherwise hull(A) = 0. We need the following facts about hull(A). PROPOSITION 10.1. (1) Let A c JH[n. For each P E hull(A), there exists a point q E JH[n that lies on a geodesic segment with endpoints in A and satisfies dh(P, q) ~ 0(1). (2) Let A c 8JH[n be a set with more than one point, and fix 0 E hull(A). For each P E hull (A) , there exists a point q E JH[n that lies on a geodesic ray from 0 to some point in A and satisfies dh(P, q) ~ 0(1). (3) Let Z c IRn - 1 c 8JH[n be a compact set containing more than one point. Then hull ( Z) rv Con( Z).

Remember the notation X rv Y, if two metric spaces X, Yare roughly quasi-isometric, X ~ Y if they are roughly similar, and X rv Y, if there are roughly isometric. It is worthwhile to note that the constant 0(1) in the statement is absolute. In particular, it does not depend on the dimension n. Proof. Statement (1) and (2) can easily be proved, for example, by using the Euclidean unit ball in IRn as the Klein model for JH[n (where the hyperbolic geodesics are line segments) with P corresponding to O. We leave the details to the reader. (3) Since Z is compact, 8(hull(Z)) = Z. To see this note that Z c 8(hull(Z)). On the other hand, assume z E IRn-l\z c 8JH[n. Since Z is compact, there exists a small closed euclidean ball B centered at z which is disjoint from Z. Then H = B n IR~ is a closed half-space in JH[n disjoint from hull(Z). Hence, z tt 8(hull(Z)). By the second part of (1), the union of the geodesic rays in hull(Z) emanating from some fixed basepoint 0 E hull(Z) is cobounded. Therefore, hull(Z) is visual. Since hull(Z) is also Gromov hyperbolic, Theorem 8.2 shows that hull(Z) rv Con( Z). To get the stronger statement hull ( Z) rv Con( Z) note that the euclidean metric on Z = 8(hull(Z)) is bilipschitz to a metric in the canonical gauge on 8(hull(Z)) by formula (10.2). The assertion now follows from the remark after Theorem 8.2. D

Theorem 10.2. Let X be a Gromov hyperbolic geodesic metric space with bounded growth at some scale. Then there exists an integer n such that X is roughly similar to a convex subset of hyperbolic n-space JH[n.

Note that the theorem has an easy converse. If X is geodesic and 274

298


GAFA

roughly similar to a subset of lHI n , then X is Gromov hyperbolic and has bounded growth at some scale. Let d be the metric on X. The theorem says that for some c > 0 the metric space (X, cd) is roughly isometric to a convex subset of lHIn. The proof shows that there is a C = C (8 (X)) > 0 such that c may be chosen as any number in (0, C). However, one cannot always take c = 1, as demonstrated by Proposition 10.3 below. Proof First suppose that the union of the geodesic rays starting from some basepoint 0 E X is cobounded in X, and that ax contains more than one point. Then X is visual, and so X ~ Con( (aX, d)) by Theorem 8.2. Here, d is some fixed metric in the canonical gauge on ax. By Theorem 7.4 applied to snowflake maps, Con((aX, d)) ~ Con((aX, d1/ 2 )). Theorem 9.2 shows that ax has finite Assouad dimension. By Theorem 9.1, (aX, d 1/ 2 ) admits a bilipschitz embedding into IRn- 1 for sufficiently large n. Let Z be the bilipschitz image of (aX, d 1 / 2 ) in IRn-l. By Theorem 7.4 for bilipschitz maps, Con((aX, d 1 / 2 )) rv Con(Z). Since ax is complete, bounded, and contains more than one point, Z also has these properties. In particular, Z is compact. Proposition 10.1 shows that Con(Z) rv hull(Z). Therefore, X rv hull ( Z) c lHIn , proving the assertion in this case. Now drop the additional assumptions on X. Since X has bounded growth at some scale, there exist constants R > r > 0, N E N such that every open ball of radius R in X can be covered by N open balls of radius r. Let Xo be a maximal set of points in X with the property that the distance between any two points in Xo is at least 5R. At each point Xo EX, we glue an isometric copy of the ray [0, (0) to ~ by identifying Xo with 0, the initial point of the ray. The new space X carries a unique metric which agrees~with the metric on X, the metrics on the rays glued to X, and makes X a geodesic space. If a geodesic segment connects two points lying on different rays glued to X, then this segment contains the initial points of the rays. Using this aEd the thin triangle definition of hyperbolicity, it is not hard to check that X is Gromov hyperbolic. If we assign to each point Xo E Xo the limit point in ax of the ray glued to xo, then we get an injective embedding of Xo into ax. From this we see that we may assume that ax contains more than one point. For otherwise, Xo is a one point set. Then X has bounded diameter. The theorem is true in this case, X being roughly similar to a point. 275

Vol. 10, 2000


299

Fix some basepoint 0 E X. For each Xo E X o, the union of a geodesic segment [0, xo], and the ray glued to X at Xo is a geodesic ray emanating from o. By definition of Xo this implies that the union of the geodesic rays from 0 is 5R-cobounded. To verify that X ~has bounded growth at some scale, observe}hat a ball of radius R in X can intersect at most ~ne component of X - X. Consequently, any open ball of radius R in X can be covered by N + IR/rl open balls of radius r, where IR/rl denotes the least integer greater than R/r. ~ It follows that X satisfies the additional assumptions of the special case ~ ------treated above. So we know that X '" W for some convex set W c lHI n , where n is sufficiently large. Let W denote the image of X under a rough ~ ----similarity f : X ---+ W. Then X '" W. We claim that W '" hull(W). To see this let w, w' E W. There exist points x, x, E X such that w = f (x) and w' = f (x'). Since X is geodesic, there exists a geodesic segment [x, x'] joining x and x'. The image f([x, x']) is a roughly quasi-isometric path joining wand w'. By geodesic stability in lHIn , dHaus (J ([x, x']), [w, w']) ~ c, where c depends on the parameters of f only. This shows that W = f(X) is cobounded in the set consisting of the union of all geodesic segments with endpoints in W. Since the latter set is cobounded in hull(W) by Proposition 10.1, the set W is also cobounded in hull(W). Therefore, the inclusion map i : W ---+ hull(W) is a rough isometry. We conclude X ~ hull(W) c lHI n , completing the proof of the theorem. D The following theorem shows that in 10.2 "roughly similar" cannot be replaced by "roughly isometric" .

10.3. Let dh denote the standard metric on 1HI2 , let c > 1 and n EN. Then the metric space X = (1HI 2 , c dh) is not roughly isometric to a subset of lHIn . PROPOSITION

Proof Recall that alHIn the (n - l)-dimensional unit sphere is sn-1, in the following sense. For every 0 E lHIn the metric dalHIn 01 is bilipschitz to dsn-l, the spherical metric on sn-1. We identify alHIn ~ith sn-1. Suppose that X is roughly isometric to a subset W of lHIn. Then it must be roughly isometric to hull(W). Fix points 0 E X, 0' E W, and fix a small E > O. It follows that the metric da1HI2 ,°,CE is bilipschitz to the metric dahull(W),o',o which is just the restriction of dalHIn,o',E to aw. It therefore follows that (dSl)C is bilipschitz to the restriction of dsn-l to aw, in the 276


300

sense that there is a homeomorphism 9 : 8 1 C such that

---+

oW

GAFA

c 8 n - 1 and a constant

C- 1ds n-l(g(X),g(y)) ~ dSl(X,y)C ~ Cdsn-l(g(X),g(y)) ,

°

for every x, y E 8 1 . However, for every x, y E 8 1 and every s > there is a sequence x = XO, Xl, ... , Xk = Y such that 2.:;=1 dSl (Xj-1, Xj)C < s. It follows that dSl (x, y) = 0, a contradiction. D It is interesting to ask what can remain of the conclusion of Theorem 10.2 if the assumption that X has finite growth at some scale is removed. For any metric space Z let

Conh(Z)

=

Z x (0, (0),

and define a metric dh on Conh(Z) as follows. Given two points p = (z, h),p' = (z', h') E Conh(Z), let dh(P,P') be the distance between the two points in the hyperbolic plane JHI2 whose coordinates in the upper half-plane model are (0, h) and (Iz - z'l, h'). Note that Conh(IRn-1) = JHIn. The spaces Conh(Z) are 8-hyperbolic for some univeral 8 ?:: 0. This is seen by checking the four point condition (3.3) using (10.1), and the argument of the first part of the proof of Theorem 7.2. If Y is any bounded subset of Z, then (10.1) implies that the set inclusion i : Con(Y) ---+ Conh(Z) is a roughly isometric embedding of Con(Y) into Conh (Z) . On first thought it seems tempting to expect that perhaps any Gromov hyperbolic space admits a rough similarity into Conh(H), where H is a sufficiently large Hilbert space. This seems to be natural, since Conh(H) of a Hilbert space H is the infinite dimensional analogue of JHIn. However, it follows from results of Enfio [E] (see also [Raj) that there is a bounded metric spaces (Z, d) such that none of the spaces (Z, dP ), p E (0,1]' has a bilipschitz embedding into any Hilbert space. This can be used to show that Con(Z) does not admit a rough similarity into Conh(H) for any Hilbert space H. On the other hand, the following is easy.

For every Gromov hyperbolic space X there is an l(X)space l(X)(A) and a rough similarity of X into Conh(l(X)(A)).

Theorem 10.4.

Recall that if A is any set, then l(X)(A) is the space of all bounded realvalued functions on A equipped with the metric coming from the supremum norm. 277


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Proof Using Theorem 4.1, we assume with no loss of generality that X is a geodesic metric space. Choose a basepoint 0 EX. To every point x E X glue a new copy of the ray [0, 00 ), wi0 the initialpoint of the ray identified with x, and call the resulting space X. Endow l! with the obvious metric, and note that X is Gromov hyperbolic. Then X has the property that for every point x E X there is a geodesic ray from 0 passing through x (cf. the proof of Theorem 10.2). Let Z = Then X rv Con(Z) by Theorem 8.2. Fix a metric d in the canonical gauge on Z. Then the map f : (Z, d) -----+ loo(Z), z 1-----+ d(z, .), is an isometric embedding of (Z, d) into loo(Z). Thus Con(Z) rv Con(J(Z)). Since the inclusion Con(J(Z)) C Conh(loo(Z)) is a rough isometric embedding by the remark following the definition of Conh, X admits an embedding into Conh(loo(Z)) by a rough similarity. The same D is then true for X.

ax.

11

Some Two-dimensional Applications

In this section a characterization of the hyperbolic plane among geodesic metric spaces up to rough quasi-isometry is given. It would be interesting to have similar statements for higher dimensional hyperbolic spaces as well. We also give a sufficient condition for a Gromov hyperbolic planar graph to be roughly quasi-isometric to a convex subset of llJI2 . DEFINITION. A metric space Z is called a >..-quasi-circle for >.. ;? 1, if it is homeomorphic to the circle 8 1 = {x E "Ix - YI. We need the following result (cf. [ThV1, Thm.4.9]). Theorem 11.1. A metric space Z is power quasisymmetric to 8 1 if and only if it is a quasi-circle and doubling.

The theorem was not exactly stated like this in [TuV1], but our statement is equivalent. To see this note that a metric space is doubling if and only if it has finite Assouad dimension, and this is equivalent to the condition that the space is homogeneously totally bounded as defined in [ThV1] (cf. Def. 2.7 and Rem. 3.20). Moreover, every quasi symmetry between connected metric spaces is a power quasisymmetry (cf. [ThV1, Cor. 3.12]). There are quasi-circles which are not doubling. The unit circle 8 1 with the metric d(x, y) = 1/ 10g(100/lx - yl), x -I- y, is an example. Theorem 11.2.

A geodesic metric space X is roughly quasi-isometric to 278

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GAFA

the hyperbolic plane 1HI2 if and only if X is Gromov hyperbolic, visual, has bounded growth at some scale, and aX is a quasi-circle. The property of a metric space being a quasi-circle is preserved by quasisymmetric maps. In particular, for the condition that aX is a quasicircle it does not matter which metric in the canonical gauge on aX we choose. There are Gromov hyperbolic geodesic metric spaces which are visual, have bounded growth at some scale, and for which aX is a topological circle, but which are not roughly quasi-isometric to 1HI2. For example, one can take X to be the hyperbolic convex hull of a topological circle Z in a1HI3 which is not a quasi-circle. If X and 1HI2 were roughly quasi-isometric, then Z = aX and a1HI2 would be power quasisymmetric. This is impossible, since a1HI2 is a quasi-circle (cf. the proof below). Proof First note that 1HI2 is a Gromov hyperbolic geodesic metric space which is visual and has bounded growth at some scale. If we use the unit disc model of the hyperbolic plane, then a1HI2 can be identified with the unit circle 8 1 , and it can be shown that the euclidean metric on 8 1 is bilipschitz to a metric in the canonical gauge. Therefore, a1HI2 is a quasi-circle. Conversely, if a metric space has all the properties stated in the theorem, then X rv Con(aX) by Theorem 8.2. By Theorem 9.2 the Assouad dimension of aX is finite, and so aX is doubling. Hence aX is power quasisymmetric to 8 1 by Theorem 11.1. By the first part of the proof, we can apply these considerations to 1HI2. In particular, 1HI2 rv Con(a1HI2). Moreover, both a1HI2 and aX are power quasisymmetric to 8 1 , and so there are power quasisymmetric to each other. This implies Con(aX) rv Con(a1HI2) by Theorem 7.4. We conclude that X rv 1HI2. D Following is an application of Theorem 11.2 to planar graphs. We consider only connected graphs G and think of G as being equipped with a path metric in the usual way, where the edges are intervals of length 1.

Theorem 11.3. Let G be a Gromov hyperbolic planar graph with bounded vertex degree. Suppose that the union of all geodesics is cobounded in G, and that every compact subset of the plane contains only finitely many vertices of G. Then G is roughly quasi-isometric to a convex subset of1HI2. The following lemma is probably known, but we have not found a reference. LEMMA

Z

=

11.4.

Suppose that a geodesic metric space Z is the union,

Xu Y, of two closed subsets X, Y c Z. Assume that X n Y is 279

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303

geodesic, and both X and Yare 8-hyperbolic. Then Z is 8' -hyperbolic, where 8' only depends on 8. A proof can be based on Rips' thin triangles definition. The details are left to the reader. We shall also need a recent theorem of P. Bowers. Theorem 11.5 [B]. Let G be the 1-skeleton of a triangulation of the plane, which is Gromov hyperbolic. Then the Gromov boundary 8G is either a single point or a topological circle. In the latter case, suppose that "( is a geodesic in G, and let G l , G 2 be the two halves of G "cut" by "(. Then Gl and G2 are Gromov hyperbolic and the closed arcs on the circle 8G determined by the endpoints of"( are the Gromov boundaries of Gl and G2.

Proof of 11.3. Assume that G is 8-hyperbolic. Our first goal is to obtain a better understanding of the planar embedding of G. A face is a component of ffi.2 - G. Let F be a bounded face. Then there are finitely many vertices of Gin 8F. We shall now show that the number of vertices in 8 F is bounded independently of F. Given any simple closed path a in G, let D(a) denote the closure of the bounded component of ffi.2 - a. Let n be the least number of vertices in a simple closed path a in G satisfying D(a) ~ F. Let a be a simple closed path in G with n vertices such that D(a) ~ F and with D(a) as small as possible. That is, there is no a' =F a with n vertices and F C D( a') C D( a). If we choose three almost equally spaced vertices in a, then a becomes a geodesic triangle. Therefore, from the Rips thin triangles condition it follows that there is an upper bound of 0 8 (1) on n, the number of vertices in a. No geodesic "( in G intersects the interior of D(a), because otherwise we could contradict the minimality of the length of a or of D( a) by replacing an arc of a by an arc of "( n D(a). Since the union of all geodesics is cobounded G, it follows that there is an upper bound on the distance from any vertex on 8F to a. Since the degrees of the vertices in G are bounded, and the length n, of a is bounded, it follows that the number of vertices in 8F is bounded. Consequently, we assume, with no loss of generality, that each bounded face is a triangle (that is, has three vertices on its boundary), because triangulating the bounded faces gives a planar graph roughly isometric to G. We now deal with the unbounded faces, and prove that with no loss of generality it may be assumed that the boundary of an unbounded face 280

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GAFA

is a geodesic. Let F be an unbounded face. Then there are infinitely many vertices in 8F. Fix two vertices a, b in 8F, and let (3 be a simple curve in the interior of F with endpoints a, b. Then there is a geodesic arc Cta,b joining a and b such that D( Cta,b U (3) is minimal with respect to inclusion. It follows that no geodesic enters D(Cta,b U (3). Now let a and b tend to infinity along 8F, "in opposite directions". Since the union of the geodesics is cobounded in G, there is an upper bound on the lower limit of the combinatorial distance of an arbitrary vertex v E 8F from Cta,b. One consequence is that we may extract a limiting geodesic CtF. It also follows that there are no geodesics that meet the component of JR2 - CtF which contains F. Let GF be the set of vertices and edges of G that are contained in the component of JR2 - CtF that contains F, and let G' be the graph obtained from G by deleting GF for every unbounded face F. Then G' is roughly isometric to G. Consequently, assume with no loss of generality that G = G' and for every unbounded F, its boundary 8F is a geodesic in G. Now let Q be a bounded degree triangulation of the upper half-plane that is roughly quasi-isometric to a hyperbolic half-plane and such that 8Q, the part of Q on the real axis, is a geodesic. For each unbounded face F of G, glue a graph QF combinatorially isomorphic to Q, with 8QF identified with the geodesic 8F, and with orientations agreeing. Let 8 be the resulting graph. ~

Suppose that Xl, X2, X3, X4 are four vertices in G. Then there are at most four unbounded faces F such that {Xl, X2, X3, X4} is contained in G union with the corresponding QF'S. Consequently, it follows by applying Lemma 11.4 three times that the distances determined by these four points satisfy (3.3) with some constant {)' independent of the choice of the points XI,X2,X3,X4. Therefore, 8 is Gromov hyperbolic. Moreover, it is (combinatorially isomorphic to) a triangulation of the plane where the union of the geodesics is cobounded, and G is isometric with a subset of 8. From Theorem 11.5 it follows that the Gromov boundary logical circle. We will show that it is a quasi-circle.

88 is a topo-

!ix a basepoint ° E 88. If x, y E 88, X =I- y, there exists a geodesic '"Yxy in G joining X and y, i.e., one end of '"Yxy converges to X and one to y. It can be shown that

(Xly)

=

dist(o,'"Yxy)

+ 08(1).

It follows that if d is some fixed metric in the canonical gauge on 281

88, then


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there exists

E

305

> 0 such that d(x, y)

::=:::

e-Edist(o,I'XY) .

(11.1)

Here ::=::: means equality up to multiplicative constants independent of x and y. Let Zl, Z2 be}wo distinct points in BO, let aI, a2 C BO be the two closed arcs on BG determined by Zl, Z2, and let WI E aI, W2 E a2 be points chosen to maximize d(Zl' WI) and d(Z2' W2). Note that Zl =I=- WI, Z2 =I=- W2· Let "Y C 0 be a geodesic joining Zl and Z2. Then 0 - "Y has two components, one, Al say, that contains a geodesic ray with limit point WI, and one, A2 say, that contains a geodesic ray with limit point W2. Changing notation if necessary, we may assume that 0 E "Y U AI. The geodesic j3 C X joining W2 and Z2 lies entirely in "Y U A 2 . In order to reach j3 from 0, one must cross "Y. Therefore, dist(o,"Y) ~ dist(o,j3). By (11.1) this implies d-

diam(a2) ~

2d(Z2' W2)

~ cle-Edist(o,,B)

~ cle-Edist(o,l') ~ C2 d (Zl, Z2) .

Here the constants CI, C2 > 0 can be chosen independently of the points ZI, Z2, WI, W2· This shows that BO is a quasi-circle. The graph G is visual, as easiq follows !rom the fact that the union of the geodesics is cobounded in G. Since G has bounded degree, it has bounded growth at some scale, and BO has finite Assouad dimension. Theorem 11.2 then tells us that 0 is roughly quasi-isometric to JH[2. Consequently, G is quasi-isometric to a subset, say W of JH[2. As in the proof of Theorem 10.2, it can be shown that W is roughly isometric to hull(W). The theorem follows. D ~

11.6. Assume additionally in Theorem 11.3 that BG is connected. Then G is roughly quasi-isometric to the hyperbolic plane, or to a hyperbolic half-plane.

COROLLARY

The proof easily follows from Theorem 11.3, and is left to the reader.

References [A] [B]

P. AssouAD, Plongements lipschitziens dans jRn, Bull. Soc. Math. France 111 (1983), 429-448. P.L. BOWERS, Negatively curved graph and planar metrics with applications to type, Michigan Math. J. 45 (1998), 31-53. 282

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[CDP] M. COORNAERT, T. DELZANT, A. PAPADOPOULOS, Geometrie et theorie des groupes. Les groupes hyperboliques de Gromov, Springer Lecture Notes in Mathematics 1441, Berlin, 1990. P. ENFLO, On a problem of Smirnov, Ark. Mat. 8 (1969), 107-109. [E] [GH] E. GHYS, P. DE LA HARPE, EDS., Sur les Groupes Hyperboliques d'apres Mikhael Gromov, Progress in Mathematics 83, Birkhauser, Boston, 1990. M. GROMOV, Hyperbolic groups, in "Essays in Group Theory" (G. Ger[Gr] sten, ed.), Math. Sci. Res. Inst. Publ. Springer (1987), 75-263. J. L UUKKAINEN, Assouad dimension, antifractal metrization, porous sets, [L] and homogeneous measures, J. Korean Math. Soc. 35 (1998), 23-76. F. PAULIN, Un groupe hyperbolique est determine par son bord, J. London [P] Math. Soc. 54 (1996), 50-74. J .G. RATCLIFFE, Foundations of Hyperbolic Manifolds, Graduate Texts [R] in Mathematics 149, Springer, New York, 1994. [Ra] Y. RAYNAUD, Espaces de Banach superstables, distances stables et homeomorphismes uniformes, Israel J. Math. 44 (1983), 33-52. [TV] D.A. TROTSENKO, J. VAISALA, Upper sets and quasisymmetric maps, preprint. [ThV1] P. TUKIA, J. VAISALA, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980),97-114. [ThV2] P. TUKIA, J. VAISALA, Quasiconformal extension from dimension n to n + 1, Ann. Math. 115 (1982), 331-348. [V] J. VAISALA, Quasisymmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), 191-204.

MARIO BONK, Institut fur Analysis, Technische Universitat Braunschweig, D-38106 Braunschweig, Germany [email protected] ODED SCHRAMM, Mathematics Department, The Weizmann Institute, Rehovot 76100, Israel [email protected] Submitted: October 1998 Revised version: January 1999

283

Correction to "Embeddings of Gromov hyperbolic spaces". The parts of the proof of Theorem 9.2 related to the relations (9.4) are somewhat garbled, but it is easy to correct the argument. The idea is to find for each of the points Zj suitable associated points Xj and Yj in X. The points Xl, ... , xn should be close together, the distance IXj - Yj I should be controlled, but we want the definite separation IYi - Yjl > 2r for i =I- j. As in the paper, one chooses Xj for j = 1, ... , n such that IXjl = -log,6 and l-log,6 - (xjlzj)1 ::::; C(O). Then IXi - xjl = 08(1) as follows from Lemma 5.l. The point Yj is chosen such that IYj I = t and I(Yj IZj) - tl ::::; C( 0) for appropriate t 2: O. If t 2: -log 0:, then again Lemma 5.1 shows that IXj -Yjl =t+log,6+08(1)

and IYi - Yjl 2: 2t + 2 logo: + 0 8(1) for i =I- j. If we pick t = - 2 log 0: + log,6 + C (0), then IXj - Yjl = 2( -logo: + log,6) + 0 8(1)

and

IYj - Yil > 2log(,6/0:). Here we may assume that log(,6/o:) 2: r, because if this inequality is not true, then we replace 0: by a smaller quantity that is comparable to 0: up to a factor only depending on r. The rest of the argument is as in the paper.

Mario Bonk

284

Part II

Noise Sensitivity

Oded Schramm's contributions to Noise Sensitivity Christophe Garban *

Abstract

We survey in this paper the main contributions of Oded Schramm related to Noise Sensitivity. We will describe in particular his various works which focused on the "Spectral analysis" of critical percolation (and more generally of Boolean functions), his work on the shape-fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation.

This paper is dedicated to the memory of Oded Schramm. I feel very fortunate to have known him. A sentence which summarizes well in what consisted of Oded's work on Noise Sensitivity is the following quote from Jean Bourgain. There is a general philosophy which claims that if f defines a property of 'High Complexity', then Sup], the support of the Fourier Transform, has to be 'spread out'. Through his work on models coming from Statistical physics (in particular percolation), Oded Schramm was often confronted with such functions of 'High complexity'. For example, in percolation, any large scale connectivity property can be encoded by a Boolean function of the 'inputs' (edges or sites). At criticality, these large scale connectivity functions turn out to be of 'High Frequency' which gives deep information on the underlying model. As we will see along this survey, Oded Schramm developed over the last decade highly original and deep ideas to understand the 'complexity' of Boolean functions. We will essentially follow the chronology of his contributions in the field; it is quite striking that three distinct periods emerge from Oded's work and they will constitute sections 3, 4 and 5 of this survey, corresponding respectively to the papers [BKS99, SSlOb, GPSIOj. We have chosen to present numerous sketches of proof, since we believe that the elegance of Oded's mathematics is best illustrated through his proofs, which usually combined imagination, simplicity and powerfulness. This choice is the reason for the length of the present survey. *UMPA, CNRS UMR 5669, ENS Lyon. Research supported in part by ANR-06-BLAN-00058 and the Ostrowski foundation

1 I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_11, 287

Contents 1 Introduction 1.1 Historical context . . . . . . . . . . 1.2 Concept of Noise Sensitivity . . . . 1.3 Motivations from statistical physics 1.4 Precise definition of Noise Sensitivity 1.5 Structure of the paper . . . . . . . .

2 3

2

9

4

6 8 9

Background 2.1 Fourier analysis of Boolean functions 2.2 Notion of Influence . . . . 2.3 Fast review on Percolation . . . . .

12 14

3

The 3.1 3.2 3.3

"hypercontractivity" approach About Hypercontractivity . . . . Applications to Noise sensitivity. Anomalous fluctuations, or Chaos

18 19 21 23

4

The 4.1 4.2 4.3 4.4

Randomized Algorithm approach First appearance of randomized algorithm ideas The Schramm/Steif approach . . . . . . . . . . Is there any better algorithm? . . . . . . . . . Related works of Oded an randomized algorithms

28

The 5.1 5.2 5.3 5.4 5.5

"geometric" approach Rough description of the approach and "spectral measures" . First and second moments on the size of the spectral sample. "Scale invariance" properties of Sl'fn . . . . . . . . . . . . . . . Strategy assuming a good control on the dependency structure The weak dependency control we achieved and how to modify the strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 40 41 43

5

6

1

Applications to Dynamical Percolation 6.1 The model of dynamical percolation. . . . . . . . . . . . . . . . .. 6.2 Second moment analysis and the "exceptional planes of percolation" 6.3 Dynamical percolation, later contributions . . . . . . . . . . . . ..

10

29 29 35 36

47 49 49 51 53

Introduction

In this introduction, we will start by describing the scientific context in the late nineties which lead Oded Schramm to work on the sensitivity of Boolean functions. We will then motivate and define the notion of Noise Sensitivity and finally we will review the main contributions of Oded that will concern us throughout this survey.

2 288

1.1

Historical context

When Oded started working on Boolean functions (with the idea to use them in statistical mechanics), there was already important literature in computer science dedicated to the properties of Boolean functions. Here is an example of a related problem which was solved before Oded came into the field: in [BOL87] it was conjectured that if f is any Boolean function on n variables (i.e. f: {O, l}n --t {O, l}), taking the value 1 for halfofthe configurations of the hypercube {O, 1 }n; then there exists some deterministic set S = Sf C {l, ... ,n} of less than clo~n variables, such that f remains undetermined as long as the variables in S are not assigned (the constant c being a universal constant). This means that for any such Boolean function f, there should always exist a set of small size which is "pivotal" for the outcome. This conjecture was proved in [KKL88]. Beside the proof of the conjecture, what is most striking about this paper (and which will concern us throughout the rest of this survey) is that, for the first time, techniques brought from Harmonic Analysis were used in [KKL88] to study properties of Boolean functions. The authors wrote at the time: These new connections with harmonic analysis are very promising. Indeed, as they anticipated, the main technique they borrowed from harmonic analysis, namely hypercontractivity, was later used in many subsequent works. In particular, as we will see later (section 3), hypercontractivity was one of the main ingredients in the landmark paper on noise sensitivity written by Benjamini, Kalai and Schramm: [BKS99]. Before going into [BKS99] (which introduced the concept of Noise sensitivity), let us mention some of the related works in this field which appeared in the period from [KKL88] to [BKS99] and which made good use of hypercontractivity. We distinguish several directions of research . • First of all, the results of [KKL88] have been extended to more general cases: non-balanced Boolean functions, other measures than the uniform measure on the hypercube {O, l}n and finally, generalizations to "Boolean" functions of continuous dependence f : [0, l]n --t {O, l}. See [BKK+92] as well as [TaI94]. Note that both of these papers rely on hypercontractive estimates . • Based on these generalizations, Friedgut and Kalai studied in [FK96] the phenomenon of "Sharp Thresholds". Roughly speaking, they proved that any monotone "graph property" for the so-called random graphs G( n, p), p ::; 1 satisfies a sharp threshold as the number of vertices n goes to infinity; see [FK96] for a more precise statement. In other words, they show that any monotone graph event A appears "suddenly" while increasing the edge intensity p (whence a "cut-off" like shape for the function p f---+ JPln,p(A)). In some sense their work is thus intimately related to the subject of this survey, since many examples of such "graph properties" concern connectedness, size of clusters and so on ..

°: ;

The study of Sharp Thresholds began in Percolation theory with the seminal 3 289

work of Russo [Rus78, Rus82], where he introduced the idea of sharp thresholds in order to give an alternate proof of Kesten's result Pc(Z}) = ~. See also the paper by Margulis [Mar74]. • Finally, Talagrand made several remarkable contributions over this period which highly influenced Oded and his coauthors (as we will see in particular through section 3). To name a few of these: an important result of [BKS99] (theorem 3.3 in this survey) was inspired by [TaI96]; the study of fluctuations in First Passage Percolation in [BKS03] (see section 3) was influenced by a result from [TaI94] (this paper by Talagrand was already mentioned above since it somewhat overlapped with [BKK+92, FK96] ). More generally the questions addressed along this survey are related to the concentration of measure phenomenon which has been deeply understood by Talagrand, see [TaI95].

1.2

Concept of Noise Sensitivity

It is now time to motivate and then define what is Noise Sensitivity. This concept was introduced in the seminal work [BKS99] whose content will be described in section 3. As one can see from its title: Noise sensitivity of Boolean functions and applications to Percolation, the authors introduced this notion having in mind applications in percolation. Before explaining these motivations in the next subsection, let us consider the following simple situation. Assume we have some initial data w = (Xl, ... , Xn) E {a, l}n and we are interested in some functional of this data that we may represent by a (real-valued) function f : {a, l}n ----; ffi.. Often, the functional f will be the indicator function of an event A C {a, 1 }n; in other words f will be Boolean. Now imagine that there are some errors in the transmission of this data wand that one receives only the sightly perturbed data WE; one would like that the quantity we are interested in, i.e. f(w), is not "too far" from what we actually receive, i.e. f(w E ). (A similar situation would be: we are interested in some real data w but there are some inevitable errors in collecting this data and one ends up with WE). To take a typical example, if f : {a, l}n ----; {a, I} represents a voting scheme, for example Majority, then it is natural to wonder how robust f is with respect to errors?

At that point, one would like to know how to properly model the perturbed data WE? The correct answer depends on the probability law which governs our "random" data w. In the rest of this survey, our data will always be sampled according to the uniform measure, hence it will be sufficient for us to assume that w '" lP' = lP'n = (~oo + ~OI)<SIn, the uniform measure on {O,l}n. Other natural measures may be considered instead, but we will stick to this simpler case. Therefore a natural way to model the perturbed data WE is to assume that each variable in w = (Xl, ... ,xn ) is resampled independently with small probability E. Equivalently, if w = (Xl, ... ,xn), then WE will correspond to the random configuration (YI, ... , Yn), where independently for each i E {I, ... , n}, with probability 1 - E, Yi := Xi and with probability E, Yi is sampled according to a Bernoulli(1/2). It is clear that such a "noising" procedure preserves the uniform measure on {O, l}n. 4 290

In computer science, one is naturally interested in the Noise stability of f which, is Boolean {i .e. with values in {O, I }), can be measured by the quantity p[f(w) f- f(w t )], where as above P = P" denotes the uniform measure on {O, I}" (in fact there is a slight abuse of notation here, since P samples t.he coupling (w,w when n ----; 00 the entire information about the largest cluster is lost from Wn to w~ (even though most of the "microscopic" information is preserved). For nice applications of the concept of Noise sensitivity in the context of computer science, see [KS06] and [O'D03].

°

1.3

Motivations from statistical physics

Beside the obvious motivations in computer science, there were several motivations coming from Statistical physics (mostly from percolation) which lead Benjamini, Kalai and Schramm to introduce and study noise sensitivity of Boolean functions. We wish to list some of these in this subsection .

• Dynamical percolation In "real life" , models issued from statistical physics undergo thermal agitation; consequently, their state evolve in time. For example, in the case of the Ising model, the natural dynamics associated to thermal agitation is the so-called Glauber Dynamics. In the context of percolation, a natural dynamics modeling this thermal agitation has been introduced in [HPS97] under the name of dynamical percolation (it was also invented independently by Itai Benjamini). This model is defined in a very simple way and we will describe it in detail in section 6; see [Ste09] for a very nice survey. For percolation in dimension two, it is known that at criticality, there is almost surely no infinite cluster. Nevertheless, the following open question was asked back in [HPS97]: if one lets the critical percolation configuration evolve according to this dynamical percolation process, is it the case that there will exist exceptional times where an infinite cluster will appear? As we will later see, such exceptional times indeed exist. This reflects the fact that the dynamical behavior of percolation is very different from its static behavior. Dynamical percolation is intimately related to our concept of Noise sensitivity since if (Wt) denotes the trajectory of a dynamical percolation process, then the configuration at time t is exactly a "noised" configuration of the initial configuration Wo (Wt == Wo with an explicit correspondence between t and E, see section 6). As is suggested by the picture 1.1, large clusters of (Wt)t>o "move" (or change) very quickly as the time t goes on. This rapid mixing of the large scale properties of Wt is the reason for the appearance of "exceptional" infinite clusters along the dynamics. Hence, as we will see in section 6, the above question addressed in [HPS97] needs an analysis of the noise sensitivity of large clusters. In cite [BKS99], the first results in this direction are shown (they prove that in some sense any large scale connectivity property is "noise sensitive"). However, their control on the sensitivity of large scale properties (later we will rather say: their control on the "frequency" 6 292

of large scale properties) was not sharp enough to imply that exceptional times do exist. The question was finally answered in the case of the triangular lattice in [SSlOb] thanks to a better understanding of the sensitivity of percolation. The case of 71} was carried out in [GPSlO]. This conjecture from [HPS97] on the existence of exceptional times will be one of our driving motivations throughout this survey.

• Conformal Invariance of percolation In the nineties, an important conjecture popularized by Langlands et al in [LPSA94] stated that critical percolation should satisfy some (asymptotic) conformal invariance (in the same way as Brownian motion does). Using Conformal Field Theory, John Cardy made several important predictions based on this conformal invariance principle, see [Car92]. Conformal invariance of percolation was probably the main conjecture on planar percolation at that time and people working in this field, including Oded, were working actively on this question until Stanislav Smirnov solved it on the triangular lattice in [SmiOl] (note that a major motivation of the introduction of the SLE", processes by Oded in [SchOO] was this conjecture). At the time of [BKS99], conformal invariance of percolation was not yet proved (it still remains open on Z?), and the SLEs from [SchOO] were "in construction", so the route towards conformal invariance was still vague. One nice idea from [BKS99] in this direction was to randomly perturb the lattice itself and then claim that the crossing probabilities are almost not affected by the random pertubation of the lattice. This setup is easily seen to be equivalent to proving Noise sensitivity. What was difficult and remained to be done was to show that one could use such random perturbations to go from one "quad" to another conformally equivalent "quad"; see [BKS99] for more details. Note that at the same period, a different direction to attack conformal invariance was developed by Benjamini and Schramm in [BS98a] where they proved a certain kind of conformal invariance for Voronol percolation.

• Tsirelson's "Black noise" In [TV98], Tsirelson and Vershik constructed sigma-fields which are not "produced" by White-noise (they called these sigma-fields "nonlinearizable" at that time). Tsirelson then realized that a good way to characterize a "Brownian filtration" was to test its stability versus perturbations. Systems intrinsically unstable led him to the notion of Black noise; see [Tsi04] for more details. This instability corresponds exactly to our notion of being "noise sensitive", and after [BKS99] appeared, Tsirelson used the concept of noise sensitivity to describe his theory of black noises. Finally, black noises are related to percolation, since according to Tsirelson himself, percolation would provide the most important example of a (two-dimensional) Black noise. In some sense [BKS99] shows that if percolation can be seen asymptotically as a noise (i.e. a continuous sigma-field which "factorizes", see [Tsi04]), then this noise has to be black i.e. all its observables or functionals are noise sensitive. The remaining step (proving that percolation asymptotically factorizes as a noise) was proved recently by Schramm and Smirnov ([SSlOa]).

• Anomalous fluctuations In a different direction, we will see that Noise Sensitivity is related to the study of

7 293

random shapes whose fluctuations are much smaller than "gaussian" fluctuations (this is what we mean by "anomalous fluctuations"). Very roughly speaking, if a random metric space is highly sensitive to noise (in the sense that its geodesics are "noise sensitive"), then it induces a lot of independence within the system itself and the metric properties ofthe system decorrelate fast (in space or under perturbations). This instability implies in general very small fluctuations for the macroscopic metric properties (like the shape of large balls and so on). In section 3, we will give an example of a random metric space, First Passage Percolation, whose fluctuations can be analyzed with the same techniques as the ones used in [BKS99].

1.4

Precise definition of Noise Sensitivity

Let us now fix some notations and definitions that will be used throughout the rest of the article, especially the definition of noise sensitivity which was only informal so far. First of all, from now on, it will be more convenient to work with the hypercube {-1, l}n rather than with {O, l}n. Let us then call Dn := {-1, l}n. The advantage of this choice is that the characters of { -1, l}n have a more simple form. In the remainder of the text a Boolean function will denote a function from Dn into {O, 1} (except in section 5, where when made precise it could also be into { -1, 1}) and as argued above, Dn will be endowed with the uniform probability measure JPl = JPln on Dn. Through this survey, we will sometimes extend the study to the larger class of real-valued functions from Dn into R. Some of the results will hold for this larger class, but the Boolean hypothesis will be crucial at several different steps. In the above informal discussion, "noise sensitivity" of a function f : Dn ---> {O, 1} corresponded to Var [lE [f(w) I WE]] or Cov(f(w),f(w E)) being "small". To be more specific, noise sensitivity is defined in [BKS99] as an asymptotic property.

Definition 1.2 ([BKS99]). Let (mn)n~O be an increasing sequence in N. A sequence of Boolean functions fn : {-1, l}mn ---> {0,1} is said to be (asymptotically) noise sensitive if for any level of noise E > 0, (1.1) In [BKS99], the asymptotic condition was rather that Var[lE[fn(w) I WE]]

----7 n-->oo

°

but as we mentioned above, the definitions are easily seen to be equivalent (using the Fourier expansions of fn). Remark 1.3. One can extend in the same fashion this definition to the class of (realvalued) functions fn : Dmn ---> R of bounded variance (bounded variance is needed to guarantee the equivalence of the two above criteria). Remark 1.4. Note that ifVar(fn) goes to zero as n ---> 00, then (fn) will automatically satisfy the condition (1.1), hence our definition of noise sensitivity is meaningful only for non-degenerate asymptotic events.

8 294

The opposite notion of noise stability is defined in [BKS99] as follows Definition 1.5 ([BKS99]). Let (mn)n>o be an increasing sequence in N. A sequence of Boolean functions fn {-I, l}n ---; {O, I} is said to be (asymptotically) Noise stable if

1.5

Structure of the paper

In section 2, we will review some necessary background: Fourier analysis of Boolean functions, the notion of influence and some facts about percolation. Then the three sections 3, 4 and 5 form the core of this survey. They present three different approaches, each of them enabling to localize with more or less accuracy the "Frequency domain" of percolation. The approach presented in section 3 is based on a technique, hypercontractivity, brought from harmonic analysis. Following [BKS99] and [BKS03] we apply this technique to the sensitivity of percolation as well as to the study of shape fluctuations in First Passage Percolation. In section 4, we describe an approach developed by Schramm and Steif in [SS10b] based on the analysis of randomized algorithms. Section 5, following [GPS10], presents an approach which considers the "Frequencies" of percolation as random sets in the plane; the purpose is then to study the Law of these "Frequencies" and to prove that they behave in some ways like random Cantor sets. Finally, in section 6 we present applications of the detailed information provided by the last two approaches ([SS10b] and [GPS10]) to the model of dynamical percolation. The contributions that we have chosen to present reveal personal tastes of the author. Also, the focus here is mainly on the applications in statistical mechanics and particularly percolation. Nevertheless we will try as much as possible, along this survey, to point towards other contributions Oded made close to this theme (such as [BS98b, PSSW07, BSW05, OSSS05]). See also the very nice survey by Oded [Sch07]. Acknowledgments I wish to warmly thank Jeff Steif who helped tremendously in improving the readability of this survey. The week he invited me to spend in Goteborg was a great source of inspiration for writing this paper. I would also like to thank Gabor Pete: since we started our collaboration three years ago with Oded, it has always been a pleasure to work with him. Finally, I would like to thank Itai Benjamini who kindly invited me to write the present survey.

2

Background

In this section, we will give some preliminaries on Boolean functions which will be used throughout the rest of the present survey. We will start with an analog of 9 295

Fourier Series for Boolean functions; then we will define the Influence of a variable, a notion which will be particularly relevant in the remainder of the text; we will end the preliminaries section with a brief review on percolation since most of Oded's work in Noise Sensitivity was motivated by applications in percolation theory.

2.1

Fourier analysis of Boolean functions

In this survey, recall that we consider our Boolean functions as functions from the hypercube Dn := {-I,I}n into {O,I}, and Dn will be endowed with the uniform measure lP' = lP'n = (~8-1 + ~81)0n.

Remark 2.1. In greater generality, one could consider other natural measures like lP'p = rr»; = ((1 - p)8_ 1 + p81)0n; these measures are relevant for example in the study of sharp thresholds (where one increases the "level" p). In the remainder of the text, it will be sufficient for us to restrict to the case of the uniform measure lP' = lP'1/2 on Dn. In order to apply Fourier analysis, the natural setup is to enlarge our discrete space of Boolean functions and to consider instead the larger space £2 ({ -1, I}n) of real-valued functions on Dn endowed with the inner product:

Xl,···,Xn

where IE denotes the expectation with respect to the uniform measure lP' on Dn. For any subset 8 C {I, 2 ... ,n}, let xs be the function on { -I,I}n defined for any x = (Xl,'" ,Xn ) by xS(X) :=

II

Xi·

(2.1)

iES

It is straightforward to see that this family of 2n functions forms an orthonormal basis of £2( {-I, 1 }n). Thus, any function f on Dn (and a fortiori any Boolean function J) can be decomposed as

f

=

L

}(8)Xs,

SC{I, ... ,n}

where }(8) are the Fourier coefficients of f. They are sometimes called the Fourier-

Walsh coefficients of f and they satisfy

}(8):= (J,Xs) = IE[fxs]. Note that }(0) corresponds to the average IE [f]. As in classical Fourier analysis, if is some Boolean function, its Fourier(-Walsh) coefficients provide information on the "regularity" of f.

f

Of course one may find many other orthonormal bases for £2( {-I, I}n), but there are many situations for which this particular set of functions (XS)SC{I, ... ,n} arises

10 296

naturally. First of all there is a well-known theory of Fourier analysis on groups, a theory which is particularly simple and elegant on Abelian groups (thus including our special case of {-I, l}n, but also ffi.jZ, ffi. and so on). For Abelian groups, what turns out to be relevant for doing harmonic analysis is the set G of characters of G (i.e. the group homomorphisms from G to C*). In our case of G = {-I, l}n, the characters are precisely our functions X8 indexed by 5 C {I, ... ,n} since they satisfy X8(X . y) = X8(X)x8(Y)· These functions also arise naturally if one performs simple random walk on the hypercube (equipped with the Hamming graph structure), since they are the eigenfunctions of the heat kernel on { -1, I} n. Last but not least, the basis (X8) turns out to be particularly adapted to our study of noise sensitivity. Indeed if f : On --t ffi. is any real-valued function, then the correlation between f(w) and f(w E ) is nicely expressed in terms of the Fourier coefficients of f as follows: IE [f(w)f(w E )]

IE

[(2: j(51)X81 (w)) (2: j(52)X82 (WE) )] 81

82

8

2: j(5)2(1 -

£)18 1.

(2.2)

8

Therefore, the "level of sensitivity" of a Boolean function is naturally encoded in its Fourier coefficients. More precisely, for any real-valued function f : On --t ffi., one can consider its Energy Spectrum E f defined by

Ef(m):= L..J f(5) 2 , VmE {l, ... ,n}. 181=m '"'

A

Since Cov(f(w) , f(w E )) = 2::=1 E f (m)(l- £)m, all the information we need is contained in the Energy Spectrum of f. As argued in the introduction, a function of "high frequency" will be sensitive to noise while a function of "low frequency" will be stable. This allows us to give an equivalent definition of noise sensitivity (recall definition 1.2): Proposition 2.2. A sequence of Boolean functions fn : {-I, 1 }mn (asymptotically) noise sensitive if and only if, for any k ::::: 1 k

2: 2: m=1

181=m

--t

{O, I} is

k

jn(5)2

=

2: Efn(m) ~ O. m=1

Before introducing the notion of influence, let us give a simple example: Example: let n be the Majority function on n variables (a function which is of obvious interest in computer science). n(X1, ... ,xn) .- sign (2: Xi), where n is 11 297

an odd integer here. It is possible in this case to explicitly compute its Fourier coefficients and when n goes to infinity, one ends up with the following asymptotic formula (see [0 '0 031 for a nice overview and references therein):

-(5)' _ { .,~,m (::: 1) E4>.. (111. ) -_ 'L" ~ ,,2 lSI""", 0

+ O(m/n)

if m is odd,

••

If m IS even.

n

1 3 5 Figure 2.1:

Picture 2.1 represents the shape of the Energy Spectrum of ill,,: its Spectrum is concentrated on low frequencies which is typical of stable functions. From now on , most of this paper will be concerned with t he description of the Fourier expansion of Boolean functions (and more specifically of their Energy Spect rum).

2.2

Notion of Influence

If f 0" --+ IR is a (real-valued) function , the influence of a variable k E [nJ is a quantity which measures by how much (on average) the funct ion f depends on the fixed variable k. For this purpose, we introduce the functions for all k E In] , where a~. acts on 0" by flipping the derivative along the klh bit). The Influence I,,(f) of the

kth

kth

bit (thus 'Vld corresponds to a discrete

variable is defined in terms of 'V d as follows

Remark 2.3. If f On --+ {O, 1} is a Boolean function corresponding to an event A C Or! (i.e. f = lA ), then I,,(J) = I ,,( A ) is the probability t hat the k t h bit is pivotal £0' A (Le. 1,(1) ~ lP[ f(w ) "f(.,· w)]) . 12 298

Remark 2.4. If f : On ----+ R is a monotone function (i.e. f(Wl) :::; f(W2) when WI :::; W2), then notice that

by monotonicity of f. This gives a first hint that influences are closely related to the Fourier expansion of f. We define the Total influence of a (real-valued) function f to be I(J) := This notion is very relevant in the study of sharp thresholds (see [RusS2, FK96]). Indeed if f is a monotone function, then by Margulis-Russo's formula (see for example [GriOS])

L: Ik(J).

ddt [1/2 lPp(J)

=

I(J)

=

L Ik(J) . k

This formula easily extends to all p E [0,1]. In particular, for a monotone event A, a "large" total influence implies a "sharp" threshold for p f---+ lPp(A). As it has been recognized for quite some time already (since Ben OrjLinial), the set of all influences Ik(J), k E [n] carries important information about the function f. Let us then call Inf(J) := (Ik(J))kE[n] the Influence Vector of f. We have already seen that the £1 norm of the influence vector encodes properties of the type "Sharp Threshold" for f since by definition I(J) = IIInf(J) IiI. The £2 norm of this influence vector will turn out to be a key quantity in the study of noise sensitivity of Boolean functions f : On ----+ {O, I}. We thus define (following the notations of [BKS99])

H(J) :=

L Ik(J)2 = IIInf(J)II~· k

For Boolean functions f (i.e. with values in {O, I}), these notions (I(J) and H(J)) are intimately related with the above Fourier expansion of f. Indeed we will see in the next section that if f : On ----+ {O, I}, then

I(J)

=

4

L 151}(5)2 . 8

If one assumes furthermore that

~H(J)

=

f

is monotone, then from remark 2.4, one has

L}({k})2

=

k

L

}(5)2

=

(2.3)

E j (1),

181=1

which corresponds to the "weight" of the level-one Fourier coefficients (this property also holds for real-valued functions, but we will use the quantity H(J) only in the Boolean case). We will conclude by the following general philosophy that we will encounter throughout the rest of the survey (especially in section 3): if a function f : {O, I} n ----+ R is such that each of its variables has a "very small" influence (i.e. « then f should have a behavior very different from a "Gaussian" one. We will see an illustration

In),

13 299

of this rule in the context of anomalous fl ucLua(,ions {lemma. 3.9). In the Boolean case, these funct ions (such that all t heir variables have very small influence) will be noise sensitive (theorem 3.3) , \vhich is not characteristic of Gaussian nor Wh ite noise behavior.

2.3

Fast review on Percolation

\Ve will only briefly recall what the model is, as well as some of its properties that will be used throughout the text. For a complete account on percolation see [Grigg] and more specifically in our context the lecture notes [Wer07]. \~Ie will be concerned ma inly in two-dimensional percolation and we will focus on two lattices: 'J'} and the triangular lattice 'If. All the results stated for Z2 in this text are also valid for percolations on "reasonable" 2-d translation invariant graphs for which RSVl is known to hold . Let us describe the model on 'Z} . Let lE 2 denote the set of edges of t he graph 'Z}. For any p E [0, I] we define a random subgraph of Z2 as follows: independently for each edge e E IE2 , we keep this edge with probability p and remove it with probability 1 - p. Equivalently, this corresponds to defining a random configuration W E {- I , l} E2 where , independent.!y for each edge e E 1E2, we declare the edge to be open {wee} = 1) with probability p or closed (w{e) = - 1) with probability 1 - p. The law of the so-defined random subgraph (or configurat.ion) is denoted by Pp • In Percolation theory, one is interested in large sca le connectivity properties of t he random configuration w. In particular as one raises the level p, above a certain critical parameter Pc{Z2), an infinite cluster (almost surely) appears. This corresponds to the well-known phase transition of percolation. By a famous theorem of Kesten this transition takes place at Pc(Z2) = ~. Percolation is defined similarly on the triangular grid 'If, except that on this lattice 've will rather consider site-percolation (i .e. here we keep each site with probability pl. Also for this model, the t.ransition happens at the critical point

Po('lI')

~

j.

T he phase transition can be measured with the density function ()Z2 (p) := Pp(O ~ 00) which encodes important properties of the large scale connectivities of the random configurat ion w: it corresponds to the density averaged over the space Z2 of the (almost surely unique) infinite cluster. The shape of the function OZ2 is pictured on the right (notice the infinite derivative at Pc) .

p 1/2

Over the last decade, the understanding of the critical regime has undergone remarkable progress and Oded himself obviously had an enormous impact on these developments. The main ingredients of this productive period were the introduction of the SLE processes by Oded (Sl.-'€ the survey on SLEs by Steffen Rohde in the

14 300

Figure 2.2: P ictures by Oded representing two critical percolation configurations respectively on 1I' and on 7}. The sites of the triangular grid are represented by hexagons. present volume) and the proof of conformal invariance on 1I' by Stanislav Smirnov

([SmiOl]). At this point one cannot. resist to show another famous (and everywhere used) pic~ ture by Oded representing an explorat,ion path on the triangular lattice; this red curve which turns right on black hexagons and left on the white ones, asymptotically converges towards SLE6 (as the mesh size goes to 0). T he proof of conformal invariance combined with the detailed information given by the SLE6 process enabled one to obtain very precise information on the critical and nea7'-critical behavior of '[-percolation. For instance it is known that on the triangular lattice, the density function Oy(P) has the following behavior near Pc = 1/2

O(p) ~ (p - 1/2)""+(", when l'

--+

1/2+ (see [WerD7]).

In the rest of t he text , we will often rely on two types of percolation events: namely the one-arm and jou7'-ann events. T hey are defined as follows: for any radius R. > 1, let A ~ be the event that the site 0 is connected to distance R. by some open pat h. Also, let A11 be the event that there are four "arms" of alternating color from the site 0 (which can be of either color) to distance R (i .e. there are four connected paths, two open, two closed from D to radius R and t he closed paths lie betwecn the open paths). Sl.'€ figure 2.3 for a realization of each event. It was proved in [LSWD2] that t,he probability of the one-arm event decays like IP'[All]

:=

1"2) 0"4(1"2 , 1"3) . A very-useful property known as quasi-mult ipJicat ivity cla ims t hat up to const.ants, these t.wo expressions are the same (this makes the division into several scales practical), This property can be stated as follows . Proposition 2.5 (quasi-multipli cativ ity , [Kes87]) . For any has (both f01' Z2 and 1l' percolations)

where the constant involved in :: such that for any f : Q n ---+ {a, 1} monotone H(J) :::; CA(J)2(1-logA(J)) logn

(the result remains valid if f has values in [0,1] instead). With this result at their disposal, in order to obtain fast convergence of H(Jn) to zero in the context of percolation crossings, the authors of [BKS99] investigated the correlations of percolation crossings fn with Majority on subsets K c [n]. They showed that there exist C, a > universal constants, so that for any subset of the lattice K, IlE[fnMKJ 1 :::; Cn-a. For this purpose, they used a nice appropriate Randomized Algorithm. We will not detail this algorithm used in [BKS99], since it was considerably strengthened in [SSlOb]. We will now describe the approach of [SS10b] and then return to "correlation with Majority" using the stronger algorithm from [SS10b].

°

4.2

The Schramm/Steif approach

The authors in [SS10b] introduced the following beautiful and very general idea: suppose a real-valued function, f : Q n ---+ lR can be exactly computed with a

29 315

randomized algorithm A, so that every fixed variable is used by the algorithm A only with small probability; then this function f has to be of 'High frequency' with quantitative bounds which depend on how unlikely it is for any variable to be used by the algorithm. 4.2.1

Randomized algorithms, revealment and examples

Let us now define more precisely what types of randomized algorithms are allowed here. Take a function f : {-1, l}n ---t JR. We are looking for algorithms which compute the output of f by examining some of the bits (or variables) one by one, where the choice of the next bit may depend on the set of bits discovered so far, plus if needed, some additional randomness. We will call an algorithm satisfying this property a Markovian (randomized) algorithm. Following [SSlOb], if A is a Markovian algorithm computing the function f, we will denote by J c [n] the (random) set of bits examined by the algorithm. In order to quantify the property that variables are unlikely to be used by an algorithm A, we define the revealment ~ = ~ A of the algorithm A to be the supremum over all variables i E [n] of the probability that i is examined by A. In other words, ~ = ~A = supJPl[i E iE[n]

J] .

We can now state one of the main theorems from [SSlOb] (we will sketch its proof in the next subsection) Theorem 4.2. Let f : {-1, l}n ---t JR be a function. Let A be a Markovian randomized algorithm for f having revealment ~ = ~A. Then for every k = 1,2, ... , The "level k"-Fourier coefficients of f satisfy

L

j(8)2 :::; ~A k Ilfll~·

SC[n],lSI=k

Remark 4.3. If one is looking for a Markovian algorithm computing the output of the Majority function on n bits, then it is clear that the only way to proceed is to examine variables one at a time (the choice of the next variable being irrelevant since they all play the same role). The output will not be known until at least half of the bits are examined; hence the revealment for Majority is at least 1/2.

In the case of percolation crossings, as opposed to the above case of Majority, one has to exploit the "richness" of the percolation picture in order to find algorithms which detect crossings while examining very few bits. A natural idea for a left to right crossing event in a large rectangle is to use an exploration path. The idea of an exploration path, which was highly influential in the introduction by Schramm of the SLE processes, was pictured in subsection 2.3 in the case of the triangular lattice. More precisely, for any n :::::: 1, Let Dn be a domain consisting of hexagons of meash lin approximating the square [0,1]2, or more generally any smooth "quad" 30 316

a b

?

c Figure 4.1: T he random interface "tn is discovered onc site at a time (which makes our algorithm !vlarkovian). The exploration stops when "In reaches either (be) (in which case In = 1) or (cd) (j" = 0) .

n

wit h two prescribed arcs OJ, fh c n, see figure 4.1. We are interested in the Left to Right crossing event (in the general setting, we look at the crossing event from OJ to Eh in Dn) · Let in be the corresponding Boolean function and can "t" the "exploration path" as in figure 4.1 (which starts at the upper left corner a). We run t,his exploration path until it reaches either the bottom side (in which case in = 0) or the right side (corresponding to j" = 1). This thus provides us wit h a markovian algorithm to compute j" where the set J = J" of bits examined by the algorithm is the set of ' hexagons' touching "(,, on either sides. T he nice property of both t he exploration path "(,, and its 1/1/.neighborhood J,,, is that they have a scaling limit when the mesh l /n goes to zero, this scaling limit being the well-known SLE6 (this scaling limit of the exploration path was as we mentioned above one of the main motivations of Schramm to introduce these SLE processes) . T his scaling limit is a.s. a random fractal curve in [0, If (or 0) of Hausdorff dimension 7/4 . This means that asymptotically, the exploration path uses a very small amount of all the bits included in this picture. With some more work (see [S \ VOl , \Ver07]), we can see t hat inside the domain (not near the cornel' 01' the sides), the probability for a point to be on "(,, is of order n - 1/ 4+o(l), where 0(1) goes to zero as the mesh goes to zero. One therefore expects the revealment d" of this algorithm to be of order n- 1/ 4+o{l). But the corner/ boundaries have a non-trivial contribution hcrc: for example in this setup, the single hexagon on the upper-left corner of the domain (where the interface starts) will be used systematically by the algorithm making the revealment equal to

31 317

one! There is an easy way to handle this problem: the idea in [881Ob] is to use some additional randomness and to start the exploration path from a random point Xn on the left-side of [0, 1]2. Doing so, this "smoothes" the singularity of the departure along the left boundary. There is a small counterpart to this: with this setup, one interface might not suffice to detect the existence of a left to right crossing and a second interface starting from the same random point Xn might be needed; see [8810b] for more details. Using arms exponents from [8WOl] and "quasimultiplicativity" of arms events ([KesS7, 881Ob]) , it can be checked that indeed the revealment of this modified algorithm is n-1/Ho(1). Theorem 4.4. Let (fn)n?:l be the left to right crossing events in domains (Dn)n?:l approximating the unit square (or more generally a smooth domain n). Then there exists a sequence of Markovian algorithms, whose revealments Dn satisfy that for any E > 0, where C = C(E) depends only on

Eo

Therefore, applying theorem 4.2, one obtains Corollary 4.5. Let (fn)n be the above sequence of crossing events. Let 9 fn denote the Spectral sample of these Boolean functions. Since IIfnl12 ::; 1, we obtain that for any sequence (Ln)n>l' (4.1) lP'[0 < 19fn l < Ln] ::; DnL;'. In particular, this implies that for any

E

> 0, lP'[0 < 19fn l < n 1 / S-

E]

----+

O.

This result gives precise lower bound information about the "spectrum of percolation" (or its "Energy spectrum"). It implies a "polynomial sensitivity" of crossing events in the sense that for any level of noise En » n- 1/ S , we have that lE[fn(w)fn(w En ) ] -lE[fn]2 ----+ O. Remark 4.6.

• Note that equation (4.1) gives a good control on the lower tail of the spectral distribution, and as we will see in the last section, these lower-tail estimates are essential in the study of dynamical percolation . • 8imilar results are obtained by the authors in [8810b] in the case of the Z2-lattice, except that for this lattice, conformal invariance and convergence towards 8LE6 are not known; therefore critical exponents such as 1/4 are not available; still [881Ob] obtains polynomial controls (thus strengthening [BK899]) but with small exponents (their value coming from R8W estimates). 4.2.2

Link with Correlation with Majority

Before proving theorem 4.2, let us briefly return to the original motivation of these types of algorithm. 8uppose we have at our disposal the above Markovian Algorithms for the left to right crossings fn with small revealments Dn = n-1/Ho(1); then

32 318

it follows easily that the events In are very little correlated with Majority functions; indeed let n 2:: 1 and K c Dn ~ [0,1]2 some fixed subset of the bits. By definition of the revealment, we have that IE [IK n Jnl] : : : IKI~n' This means that on average, the algorithm visits very few variables belonging to K. Since

and using the fact that on average, IK n Jml is small compared to deduce that there is some 0: > 0 such that

IKI, it is easy to

This, together with theorem 3.3 and theorem 4.1 implies a logarithmic noise sensitivity for Percolation. In [BKS99], they rely on another algorithm which instead of following interfaces, in some sense "invades" clusters attached to the left hand side. Since clusters are of fractal dimension 91/48, intuitively their algorithm, if boundary issues can be properly taken care of, would give a bigger revealment of order n- 5 / 48+ o (1) (the notion of revealment only appeared in [SSlOb]). So the major breakthrough in [SSlOb] is that they simplified tremendously the role played by the algorithm by introducing the notion of revealment and they noticed a more direct link with the Fourier transform. Using their correspondence between algorithm and Spectrum, they greatly improved the control on the Fourier spectrum (polynomial v.s. logarithmic). Furthermore, they improved the randomized algorithm. 4.2.3

Proof of Theorem 4.2

Let I : {-1, l}n ----7 lR be some real-valued function and consider A, a Markovian algorithm associated to it with revealment ~ = ~ A. Let k E {1, ... , n}; we want to bound from above the size of the level-k Fourier coefficients of I (i.e. 2: S =k }(S)2). j

j

For this purpose, one introduces the function 9 = g(k) = 2: S =k }(S)Xs, which is the projection of I onto the subspace of level-k functions. By definition, one has j

2: S =k j

j

}(S)2

=

j

Ilgll~.

Very roughly speaking, the intuition goes as follows: if the revealment ~ is small, then for low level k: there are few "frequencies" in g(k) which will be "seen" by the algorithm. More precisely, for any fixed "frequency" S, if lSI = k is small, then with high probability none of the bits in S will be visited by the algorithm. This means that IE [g(k) I J] (recall J denotes the set of bits examined by the algorithm) should be of small L2 norm compared to g(k). Now since I = IE [I I J] = 2:k IE [g(k) I J], most of the Fourier transform should be supported on high frequencies. There is some difficulty in implementing this intuition, since the conditional expectations IE [g(k) I J] are not orthogonal. Following [SSlOb] very closely, one way to implement this idea goes as follows:

(4.2)

33 319

As hinted above, the goal is to control the £2 norm of lE [g I JJ. In order to achieve this, it will be helpful to interpret lE [g I JJ as the expectation of a random function gJ whose construction is explained below. Recall that J is the set of bits examined by the randomized algorithm A. Since the randomized algorithm depends on the configuration w E {-I, l}n and possibly some additional randomness: one can view J as a random variable on some whose elements can be represented as w = (w, T) (T extended probability space corresponding here to the additional randomness). For any function h : {-I, l}n ~ JR, one defines the random function hJ which corresponds to the function h where bits in J are fixed to match with what was examined by the algorithm. More exactly, if J(w) = J(w, T) is the random set of bits examined, then the random function hJ = hJ(w) is the function on {-I, l}n defined by hJ(W,T) (Wi) := h(w"), where w" = w on J(w, T) and w" = Wi on {I, ... , n} \ J(w, T). Now, if the algorithm has examined the set of bits J = J(w), then with the above definition it is clear that the conditional expectation lE [h I JJ (which is a measurable function of J = J(w)) corresponds to averaging hJ over configurations Wi (in other words we average on the variables outside of J (w )); this can be written as

n,

I JJ

lE [h

J

hJ := hJ(0),

=

where the integration J is taken with respect to the uniform measure on Wi EOn' In particular lE[hJ = lE[J hJJ = lE[h J(0)J. Since (h 2)J = (h J)2, it follows that

Ilhll~ = lE[h2J = lE[J h}J

=

lE[llhJII~J .

(4.3)

Recall from (4.2) that it only remains to control IllE [g I JJ II§ = lE [gJ(0)2J. For this purpose, we apply Parseval to the (random) function gJ: this gives (for any

wEn),

gJ(0)2 Taking the expectation over

lE[gJ(0)2J

=

IlgJII~

-

L gJ(5)2. ISI>O

w=

lE[llgJII~J

=

(w, T) En, this leads to

-

L

lE[gJ(5)2J

ISI>o

Ilgll~

-

L lE [gJ(5)2J

By (4.3)

ISI>O

"

A(5) 2_ "

~9

~

ISI=k

ISI>O

lE [A (5) 2J { gJ

since 9 is suppor~ed on level-k coeffiCients

_ A(5)2J { by restricting. on < "lE[A(5)2 ~ 9 gJ level-k coefficients ISI=k

Now, since gJ is built randomly from 9 by fixing the variables in J = J(w), and since 9 by definition does not have frequencies larger than k, it is clear that

gJ(5) = { g(5) = /(5), if 5 0,

else.

34 320

n J(w) = 0

Therefore, we obtain

[[IE [g [ JJ [[~

=

IE [gJ(0)2J ~

L g(8)2 JPl[8 n J of- 0J ~ [[g[[~ k d. 181=k

Combining with (4.2), this proves theorem 4.2.

4.3

D

Is there any better algorithm ?

One might wonder whether there exist better algorithms which detect left to right crossings (better in the sense of smaller revealment). The existence of such algorithms would immediately imply sharper concentration results for the "Fourier Spectrum of percolation" than in Corollary 4.5. This question of the "most effective" algorithm is very natural and has already been addressed in another paper of Oded et al. [PSSW07], where they study Random Turn Hex. Roughly speaking the "best" algorithm might be close to the following: assume j bits (forming the set 8 j ) have been explored so far; then choose for the next bit to be examined, the bit having the highest chance to be pivotal for the leftright crossing conditionally on what was previously examined (8j ). This algorithm stated in this way does not require additional randomness. Hence one would need to randomize it in some way in order to have a chance to obtain a small revealment. It is clear that this algorithm (following pivotal locations) in some sense is "smarter" than the one above, but on the other hand its analysis is highly nontrivial. It is not clear to guess what the revealment for that algorithm would be. Before turning to the next section, let us observe that even if we had at our disposal the most effective algorithm (in terms of revealment), it would not yet necessarily imply the expected concentration behavior of the spectral measure of In around its mean (which for an n x n box on 1[' turns out to be of order n 3 / 4 ). Indeed, in an n x n box, the algorithm will stop once it has found a crossing from left to right OR a dual crossing from top to bottom. In either case, the lattice or dual path is at least of length n; therefore we always have [J[ = [In [ ~ n. But if dn is the revealment of any algorithm computing In, then by definition of the revealment, one has IE[[Jn[J ~ O(1)n 2 dn (there are O(1)n2 variables); since [In[ ~ n, this implies that dn is necessarily bigger than O(l)n-l. Now by Corollary 4.5, one has that

L O 0 (since, by definition of X R, lE[XR] = a1(R)). This thus proves the existence of exceptional times. If one had obtained a much weaker control on the correlation than that in (6.4), for example a bound oft- 1Iog(t)-2 a1 (R)2, this would have still implied the existence of exceptional times: one can thus exploit more the good estimate provided by equation (6.4). This estimate in fact easily implies the second part of theorem 6.2, i.e. that almost surely the Hausdorff dimension of the set of exceptional times is greater than 1/6 (the upper bound of 31/36 is rather easy to obtain, it is a first moment analysis and follows from the behavior of the density function () near Pc = 1/2, see [SSlOb] and [Ste09]). The proof of the lower bound on the Hausdorff dimension which is based on the estimate (6.4), is classical and essentially consists into defining a (random) Frostman measure on the set £. of exceptional times; one concludes by bounding its expected a-energies for all a < 1/6. See [SSlOb] for rigorous proofs (taking care of the logarithmic corrections etc .. ). Let us conclude this subsection by briefly explaining why one can achieve a revealment of order a2(r)a1(r, R) for the Boolean function fr,R(w) := l{r~R}" We 55 341

use an algorithm that mimics the one we used in the "chordal" case except the present setup is "radial". As in the chordal case, we randomize the starting point of our exploration process: let's start from a site taken uniformly on the 'circle' of radius R. Then, let's explore the picture with an exploration path I directed towards the origin; this means that as in the chordal case, when the interface encounters an open (resp closed) site, it turns say on the right (resp left), the only difference being that when the exploration path closes a loop around the origin, it continues its exploration inside the connected component of the origin (see [Wer07] for more details on the radial exploration path). It is known that this discrete curve converges towards radial SLE6 on 11', when the mesh goes to zero. It turns out that the so-defined exploration path gives all the information we need. Indeed, if the exploration path closes a clockwise loop around the origin, this means that there is a closed circuit around the origin making fr,R equal to zero. On the other hand, if the exploration path does not close any clockwise loop until it reaches radius r, it means that fr,R = 1. Hence, we run the exploration path until either it closes a clockwise loop or it reaches radius r. Now, what is the revealment for this algorithm? Neglecting boundary issues (points near radius r or R), if x is a point at distance u from 0, with 2r < u < R/2, in order for x to be examined by the algorithm, it is needed that the exploration path did not close any clockwise loop before radius u; hence there is an open path from u to R, this already costs a factor ctl (u, R). Now at the level of the scale u, what is the probability that x lies on the interface? The interface is asymptotically of dimension 7/4, so in the annulus A( u/2, 2u), about U 7/4+o(1) points lie on I and the probability that the point x is in I is of order U- 1/ 4 . In fact it is exactly up to constant ct2(U). Hence, the probability IP[x E J], for Ixl = u is of order ct2(U)ctl(U, R). Now recall that the revealment is the supremum over all sites of this probability; it is easy to see that the smaller the scale (r), the higher this probability is; therefore neglecting boundary issues, one obtains that 8 ~ ct2(r)ctl(r, R). 6.3.2

Consequences of the Geometric approach ([GPSIO])

We now present the applications to dynamical percolation of the Sharp estimates on the Fourier spectrum obtained in [GPS10]. Since the results in [GPSlO] are sharp on the triangular lattice 11' as well as on 7i}, the first notable consequence is the following result Theorem 6.3 ([GPSlO]). If (Wt)t>o denotes some trajectory of dynamical percolation on the square grid Z2 at Pc ~ 1/2, then almost surely, there exist exceptional times t E [0, 00), such that Wt has an infinite cluster. Furthermore there is some ct > 0, such that if E C [0,00) denotes the random set of these exceptional times, then almost surely, the Hausdorff dimension of E is greater than ct. Remark 6.4. The paper [SSlOb] came quite close to proving that exceptional times exist for dynamical critical bond percolation on Z2, but there is still a missing gap if one wants to use randomized algorithm techniques in this case. On the triangular lattice 11', since the critical exponents are known, the sharp control on in [GPSlO] and especially on its Lower-tail enables one to obtain

iR

56 342

detailed information on the structure of the (random) set of exceptional times £. One can prove for instance Theorem 6.5 ([GP81O]). If £ C [0, (0) denotes the (random) set of exceptional times of dynamical percolation on 'lI', then almost surely the Hausdorff dimension of £ equals

*.

This theorem strengthens theorem 6.2 from [881Ob]. Finally, one obtains the following interesting phenomenon Theorem 6.6 ([GP81O]). Almost surely on 'lI', there exist exceptional times "of the second kind" t E [0,(0), for which an infinite (open) cluster coexists in Wt with an infinite dual (closed) cluster. Furthermore, if £(2) denotes the set of these exceptional times; then almost surely, the Hausdorff dimension of £(2) is greater than 1/9.

We conjectured in [GP81O] that £(2) should be almost surely of dimension 2/3; but unfortunately, the methods in [GP81O] do not apply well to non-monotone functions, in particular in this case, to the indicator function of a two-arm event. Let us now focus on theorem 6.5. As in in the previous subsection, in order to have detailed information on the dimension of the exceptional set £, one needs to obtain a sharp control on the correlations JPl[O ~ R, 0 ~ R]. More exactly, the almost sure dimension 31/36 would follow from the following estimate (6.5) which gives a more precise control on the correlations than the one given by (6.4) obtained in [8810b] (which implied a lower bound on the dimension of £ equal to 1/6). As previously, proving such an estimate will be achieved through a sharp understanding of the Fourier 8pectrum of fR(w) := 10~R (but this time, in order to get a sharp control, we will rely on the "geometric approach" explained in section 5). Indeed, recall from (6.2) that (6.6) Before considering the precise behavior of the Lower-tail of QfR' let us detail a heuristic argument which explains why one should expect the above control on the correlations (estimate 6.5). Recall Wt can be seen as a noised configuration of Woo What we learned from section 5 is that in the Left-Right case, once the noise E is high enough so that "many pivotal points" are touched, the system starts being noise sensitive. This means that the Left to Right crossing event under consideration for WE starts being independent of the whole configuration w. On the other hand (and this side is much easier), if the noise is such that pivotal points are (with high probability) not flipped, then the system is stable. Applied to our radial setting, if R is some large radius and r some intermediate scale then, conditioned on the event {O ~ R}, it is easy to check that on average, there are about r 3 / 4 pivotal points at scale r (say, in the annulus A(r, 2r)). Hence, if the level of noise t is such 57 343

that tr 3 / 4 « 1, then the radial event is "stable" below scale r. On the other hand if tr 3 / 4 » 1, then many pivotal points will be touched and the radial event should be sensitive above scale r . As such, there is some limiting scale L(t) separating a frozen phase (where radial crossings are stable) from a sensitive phase. The above argument gives L(t) :::::: r 4 / 3 ; this limiting scale is often called the characteristic length: below that scale, one does not feel the effects of the dynamic from Wo to Wt, while above that scale, connectivity properties start being uncorrelated between Wo and Wt·

Remark 6.7. Note that in our context of radial events, one has to be careful with the notion of being noise sensitive here. Indeed our sequence of Boolean functions (fR)R>1 satisfy Var(fR) ---+ 0; therefore, the sequence trivially satisfies our initial definition of being noise sensitive (definition 1.2). For such a sequence of Boolean functions corresponding to events of probability going to zero, what we mean by being noise sensitive here is rather that the correlations satisfy lE[f(w)f(w E )] ~ O(1)lE[f]2. This changes the intuition by quite a lot: for example, previously, for a sequence of functions with L 2 -norm one, analyzing the sensitivity was equivalent to localizing where most of the Fourier mass was; while if IIfRI12 goes to zero, and if one wishes to understand for which levels of noise tR > 0, one has lE[fR(W)fR(w ER )] ~ O(1)lE[fR]2, then it is the size of the Lower tail of iR which is now relevant. To summarize and conclude our heuristical argument, as in the above subsection, one can bound our correlations as follows

p[o ~ R, 0 ~ R]

~

p[o ~ L(t)] P[L(t)

~

R, L(t)

~

R] .

Not much is lost in this bound since connections are stable below the correlation length L(t). Now, above the correlation length, the system starts being noise sensitive, hence one expects P[L(t) ~ R, L(t) ~ R] ~ O(l)P[L(t) ~ R]2 = O(l)al(L(t), R)2. Using quasi-multiplicativity and the values of the critical exponents, one ends up with

p[o ~ R,

0 ~ R]

< O(1)al(L(t))-lal(R)2 < r 5 / 36 al(R)2,

as desired. From now on, we wish to briefly explain how to adapt the geometric approach of section 5 to our present radial setting. Recall that the goal is to prove an analog of theorem 5.12 for the radial crossing event fR(W) = 10~R" Before stating such an analogous statement (which would seem at first artificial),we need to understand some properties of Y fw First of all, similarly as in section 5, it is easy to check that W[IYfRl] ::::: R2a4(R) 1 1 (recall that PfR IS the renormahzed Spectral measure IlfRl12QfR = Ql(R)QfR)' The second moment is also easy to compute. One can conclude from these estimates that with positive probability (under W fR ), IYfRI is of order R2a4(R) (:::::: R 3 / 4 on 1I'). A

•

•

A

58 344

A

Now, following the strategy explained in section 5, we divide the ball of radius R into a grid of mesoscopic squares of radii r (with 1 < r < R). It is easy to check that for any such r-square Q, one has

Furthermore, and this part is very similar as in the chordal case, one can obtain a "weak" control on the dependencies within Y fR , namely that for any r-square Q and any subset W such that Q n W = 0, one has

where Q is the concentric square in Q of side-length r /2 and c > constant.

°

is some absolute

As in section 5, if S c [- R, Rj2, we denote by S(r) the set of those r x r squares which intersect S. Thanks to the above weak independence statement, it remains (see section 5) to study the lower-tail of the number of mesoscopic squares touched by the Spectral sample, i.e. the lower-tail of Iy(r) I (only the very bottom-part of this distribution needs to be understood in a sharp way). Hence, one has to understand how Y(r) typically looks when it is conditioned to be of very small size (say of size less than log R/r, i.e. much smaller than the average size of Y(r) which is of order (R/r)3/4). The intuition differs here from the chordal case (analyzed in section 5); indeed recall that in the chordal case, if Iy(r) I was conditioned to be of small size, then it was typically "concentrated in space" somewhere "randomly" in the square. In our radial setting, if Iy(r) I is conditioned to be of small size, it also tends to be localized, but instead of being localized somewhere randomly through the disk of radius R, it is localized around the origin. The reason of this localization around the origin comes from the following estimate: if Qo denotes the r-square surrounding the origin r

< JPl[YfR c B(O, 2)C] A

1

r

A

O:l(R) Q[YfR c B(O, 2Y]

O:l~R)lE[lE[fR I B(O,R) \ B(O,r)]2] 0(1)

By lemma 5.3

2

< O:l(R) O:l(r, R)O:l(r) ::; 0(1) O:l(r)

°

(In the last line, we used the fact that lE[fR I B(O, R) \ B(O, r)] ::; O:l(r) lr+-+R). Since 0:1 (r) goes to as r - t 00, this means that for a meso scopic scale r such that 1 « r « R, Y(r) "does not like" to avoid the origin. In some sense, this shows that the origin is an attractive point for Y fw Still, this does not yet imply that the origin will be attractive also for y(r) conditioned to be of small size. If one assumes that, as in the chordal case, Y(r) tends to be localized when it is conditioned to be of small size (i.e. that for k small, W[IY(r)1 = k] ~ W[Iy(r)1 = 1]), then it not hard to see that it has to be 59 345

localized around the origin: indeed, one can estimate by hand that once conditioned on IY(r) I = 1, y(r) will be (with high conditional probability) close to the origin. To summarize, assuming that y(r) tends to "clusterize" when it is conditioned to be of small size, one has for k "small"

Q[IY(r)1

=

kJ ~ Q[Iy(r)1

1J ;:::: Q[Y(r) ;:::: Q[0y!oYfR cB(0,r)J =

;:::: lE[lE[fR [ B(0,r)J 2 J -

=

QoJ

Q[YfR

=

0J

;:::: 0(1)0:1(1')0:1(1', R)2 - 0:1(R)2 1

;:::: -(-) 0:1 (R)2 using quasi-multiplicativity. 0:1 l' Using an inductive proof (similar as the one used in the chordal case, but where the origin plays a special role), one can prove that it is indeed the case that y(r) tends to be localized when it is conditioned to be of small size. More quantitatively, the inductive proof leads to the following estimate on the lower-tail of Iy(r) I Proposition 6.8 ([GPSlO], section 4). There is a sub-exponentially fast growing function g(k), k ~ 1, such that for any 1 :::; l' :::; n

(6.7) Using the same technology as in section 5 (i.e. the weak independence control, the above proposition on the lower tail of the meso scopic sizes and a large deviation lemma which helps implementing the "scanning procedure"), one obtains that Q[O < IYfRl < r 2 0:4(r)J ;:::: QfR [Iy(r) I = 1J ;:::: al~r)O:l(R)2, which is exactly a sharp

JR'

lower-tail estimate on More exactly one has the following theorem (analog of theorem 5.12 in the chordal case) Theorem 6.9 ([GPSlO]). Both on the triangular lattice 'f and on 7!}, if fR denotes the radial event up to radius R, one has

where the constants involved in ;:::: are absolute constants.

On the triangular lattice 'f, using the knowledge of the critical exponents, this can be read as follows

or equivalently for any u ;S R 3 / 4 ,

60 346

Recall our goal was to obtain a sharp bound on the correlation between fR(wo) and fR(wt). From our control on the lower-tail of iR and equation (6.6) one obtains

p[O ~

R, 0 ~ R] < 6[0 < l.9'fRI < 1ft] < riBCY1(R)2,

which, as desired, implies that the exceptional set E is indeed a.s. of dimension ~ (as claimed above, the upper bound is much easier using the knowledge on the

density function 811').

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Lucio Russo. A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 43(1):39-48, 1978.

[Rus82]

Lucio Russo. An approximate zero-one law. Z. Wahrsch. Verw. Gebiete, 61(1):129-139, 1982.

[SchOO]

Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221-288, 2000.

[Sch07]

Oded Schramm. Conformally invariant scaling limits: an overview and a collection of problems. In International Congress of Mathematicians. Vol. I, pages 513-543. Eur. Math. Soc., Zurich, 2007.

[SS10a]

Oded Schramm and Stanislav Smirnov. On the scaling limits of planar percolation. Ann. Probab., to appear, 2010.

[SS10b]

Oded Schramm and Jeffrey Steif. Quantitative noise sensitivity and exceptional times for percolation. Ann. Math., 171(2):619-672,2010.

[Smi01]

Stanislav Smirnov. Critical percolation in the plane: conformal invarianee, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sir. I Math., 333(3) :239-244, 2001. 63 349

[SW01]

Stanislav Smirnov and Wendelin Werner. Critical exponents for twodimensional percolation. Math. Res. Lett., 8(5-6):729-744, 2001.

[Ste09]

Jeffrey Steif. A survey of dynamical percolation. Fractal geometry and stochastics, IV, Birkhauser, pages 145-174, 2009.

[Ta194]

Michel Talagrand. On Russo's approximate zero-one law. Ann. Probab., 22(3): 1576-1587, 1994.

[Ta195]

Michel Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Etudes Sci. Publ. Math., (81):73-205, 1995.

[Ta196]

Michel Talagrand. How much are increasing sets positively correlated? Combinatorica, 16(2):243-258, 1996.

[Tsi99]

Boris Tsirelson. Fourier-Walsh coefficients for a coalescing flow (discrete time). 1999.

[Tsi04]

Boris Tsirelson. Scaling limit, noise, stability. In Lectures on probability theory and statistics, volume 1840 of Lecture Notes in Math., pages 1106. Springer, Berlin, 2004.

[TV98]

B. S. Tsirelson and A. M. Vershik. Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations. Rev. Math. Phys., 10(1):81-145, 1998.

[Wer07]

Wendelin Werner. Lectures on two-dimensional critical percolation. lAS Park City Graduate Summer School, 2007.

64 350

KOISE SENSITIVITY OF BOOLEAN FUNCTIONS AND APPLICATIONS TO PERCOLATION by h ;\1 BENJAMINI, Gil, KALAl, DllW SCHRAMM

AJ:l'iTR At:T It is shown that a 1afl,'C dass of e\'cllts in a product probahility !opal''' all'" highl y scn~it;w 10 nuis the uniform ",,,a.'oll". A uuss;IIg i. a p'.l h Ihal juin' lh," kfl aurl light sidr:~ of Ih,- .-c.ioi5e 5t"" nsil il"ity "" Ilm:e example'! 1."1.. r" flucnt"""s of ,·".riabICll I.:t WdghU'd majority 1.4. Stability.

6

1

"

1tJ

10

I.;). ~ou ri(:r·W~bh ': "p~lI:;iou 1.1;. St,m" rd"t,·,] and fUIUIT; work 1.7. Thl' strUCtuTt' of this paprr.

"""

:l. St:lnitivilY to nuis.:

3. Com::lation with m suppose thaI Y (YI, .··,Yn) is a random perturbation of X; that is, for every j E {I, ... , 1l}, Yj Xi with probability I-e, independently for distinctj's. Here e E (0, I) is some small fixed constant. This random perturbation of x will be denoted N£(x). We may think of N£(x) as x with some noise. Based on the knowledge of Nr(x), we would like to predict the event x E •._,g. Since the joint distribution (x, N£{x)) is the same as that of (N£(x), x), an equivalent problem is to predict Ne(x) E ,4; knowing x. The event . , ,~ is noise sensitive if for all but a small set of x, knowing x does not significantly help in predicting the evelll Ne(x) E . .~~. More formally, ~/~ is noise sensitive, if for some small 0> 0,

=

(1.1)

=

,)(.'6, e, O).= p{x: Ip (N.(x) E. J~

I x) -

P(. '6)1

>o} 0 such that (l.l) holds. This will be called the sensitivity gauge of ... ~. A sequence of evcnlS •. ~l", C On.. will be called asymptotically noise sensitive if lim c>(.A"" E) =O ,

Remark 1.1. and only if

(1.2)

As shown

In

'Ie E (0, 1/2). Section 2, ,/t are asymptotically noise sensitive if R,

lim var[p(N,(x) E Am lx)) = O.

A simple example of a sequence of events which are not noise sensitive is To dictatorship. The ftrs( bit dictator is the event 1?~ = { (Xl> ... , xn) E n~ : Xl = verify that {EW"n} is not asymptotically noise sensitive, consider some evr.nt ,~ C On. The:n for k > n we may obviously consider .4, as a subset of n.b by ignoring the t"xtra variables. Note (hal this does nol change the value of 4>(•. ~,E). Consequently, $(!¥.. £) =~(!¥, , £) of 0 fo' all n > I.

1}.

352

NOISE SEr.;SITI VID' OF BOULEA.:" FL'r.;CTIOr.;S

A;'\~D

API'L1CATIO:\"S TO I'ER.COLATIOr.;

7

Let us examine now the example of majority. Pick some e E (0, 1/ 2). Let

At ne On denote the majority event, that is,

The probability that LjXj - 11/ 2 > ..fii is bounded from below as 11 - 00. Given such an x, the probability that N£(x) E viI.," is greater than P[.A't .J + SI for some constant 01 > 0, depending on e. We conclude that majority is not asymptotically noise sensitive as n _ 00. Majority and dictatorship are not only noise insensitive, they are actuaHy "noise stable", in a sense defined in Subsection 1.4 below. It turns out that the noise insensitivity of majority and dictatorship is atypical, and many natural and interesting events are asymptotically noise sensitivf'. Our third example is bond percolation on an m+ 1 by m reclangle in the ordinary square grid Z2. A configuration is an element in n = {O , I}£, whcn,: E is thc scI of edges in this rectangle. Let ro E n be a random configuration, selected according lO the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges e with w(e) = 1. Let 'ffm be the event that there is some no~sing of this rectangle. By duality, it is not hard to see that P[Wm l = 1/ 2.

¢I(jof~,

Theorem 1.2. e) - 0 as m -+

The crossing events if", are asymptotical!J noise sensitive; that

i~,

00.

This theorem will appear as a corollary of a general result. To introduce the morc genaal ~tatemellt, we need the notion of influence. 1. 2 . Influences

of vanablu

Sct [n] = {l , ... , n} . Given x E n andj E [nl, let crjx=(~, ... ,i,.), wh~re ~=Xk when k t j and .~ = I - Xj. The influence of the k-th variable on a fun ct ion f: n - R is defined by (1.3)

I,(f) ;

II f(a,.,) -

f(x)l l, .

In other words, IJ..( f) is the expected absolute value of the change in f when the x'th bit Xl is nipped. We shall often not distinguish between an event ...,15 and its indicator function X. t. In particular, for events . /~, IA(A) = lk(x. -1,). Note (hat lAt. /1 ) is the probability (hat precisely one of the two clements x, O".tX is in ..4~. This notion of influt':nct': was introduced by Ben-Or and Linial [41. Ka hn , Kalai and Linial [23} (see also [1 0, :H]) showed that for every, 4, C n" with P (....~] = 1/ 2 th ere is aj E fnl with I;{.~);>. c1ogn/ ll, for some constant (> 0, a nd that there always

353

,

n :'\i

BE..\JA~-IJ:\I ,

(;11. KALAl, ODED SC HRAM:'.I

exists a set S C [n] with lSI ~ c(E)n/ log n whose cumulative infiuencf: is > 1 - E; that is, the measure of the set of inputs for variables in [n] - S which determine the value off is less than E. Put

lU )

H U)

= L, 1,(/),

= L, l,ul'

Theorem 1.3. --_. ut . ' ~ m C ON," be a sequence of events and suppose that H (..·.g m) - 0 as m _ 00. Then {. ,6 .. } is a.ryT/lprotically noise sensitive. Equivalenlty, there is J'ome amtirJUous JUlIction ¢l salisfling ~O J E) = 0 such thai 41(. /l, E) ~ ~(.A~ ), e) for every event .A!, in some On. On nn, we use the usual lattice order: (XI. ,." x~) ~ 0'1> "',Yn) iff Xj ,;;; Yj for all j E InJ. A function f: On - R is monotone if f (x) ~ f (y) whenever x ~ y. An event .~ -t C On is monotone if its indicator fun ction X. -I- is monotone. For monotone events, Theorem 1.3 has a converse :

Theorem 1.4. Let ,.4..", C On.. be a sequence infH(. A~ m)

if monotolll

events with

> O.

m

Then

{~~"' }

is not asymptntically noise sensitive.

The assumption that the events ~/6", are monotone is necessary here. (For example, take ,A~m to be a uniform random subset of Om, o r parity: '~ m := {x E n m, Ilxll, is odd}.) Suppose that , .~ is a monotone event where the influences of all the variables arc the same. The influence 11(, J~) then measures the sensitivity of, 4.- to flips of a single. variable. NOlf' Ihat, qu ite paradoxically, ,4:, is ka~t sensitive to noise wh (~n l l ('~) is largest. We now give a q uantitative version of Theorem 1.3 under the assumption that H (. -(1., ,,,) goes to zero fast enough.

Theorem 1.5. - I.£t ,/~ C On, and .~uppose that Then there exist CI , C2 > 0, depending only on a so that

H(,A~) ~ n -~,

where a E (0 ,

1/21.

'ieE(O,I/4).

Consequent!y, if . .-'l m C On.. is a sequence if evenLr satisfying H (... .t m) sequence in (0, 1/ 4) such that Emlog n m - 00, then $(•.'6 m , em) _ O.

354

:$;

(n",) - Q and Em is a

NOISE SE"'SlTIVITY OF BOOL.f'.A."J F1JNCnONS A,l) APPIJCAT IO:"S 10 PERCOL'\TIO:"

9

It turns out that for monotone events noise insensitivity is also closely related to correlation with majority functions. ut K C [n] and define the tnajority function on K by

MK(X)= sign ~)2') - I); jEK

that is,

L,EKxi < IKI / 2 ; if L ""xi= IKI/2 ; if LicKxi > IKI/2 . if

(1.4)

For f:

O:~ -

R

!>Ct

AU) = max{IEUMKII' K C [nJ} . Theorem 1.6. -

Let/: On - [0,

lJ

be monotone. Then

H (f) < CA(f)2 ( I - logAUI) logn, where C ir some universaL constant. Consequent!.v, if .A3 .. C O~," ir a sequence (1.5)

of tnJ)rJotollL events with

lim A(Aml' (I - logA(Am») lognm= O.

m_=

Then {...,..g",} is asympwtUalb' noise sensitive. One cannot get rid of the logn", factor (see Remark 3.1 0), except by using weighted majority functions. For positive weights w = (w\, lffi.!, ••. , w~) consider a weighted majority function , which is defined by

Mw(x!,

X2, .. .•

x~) = sign (~)2.tj. - I) u:;).

FinaUy write

7hLorem 1.7. - Let Am C A" be a sequence of tnJ)notollL events. Then {A .., } is • arym/JltJtically no~, sensitive if and only if lim._= AlA.) = o. For a monotone event Jt, C Uh , which is symmetric in the n variables, its correlation with unweighted majority is enough to determine if it is noise sensitive.

355

to

ITA! BEJ'.!JAMTN I, G IL KAlAl, OIlElJ SCHRAMM

1.4. Stability

We now define the notion of stability, which is the opposite of noise sensitivity. Suppose .~ CO", and let x E (1" be random-uniform. For t: > 0, let N(...,..g denote the event Nc{x) E .A. It is then clear that P[.~.0.NE....,.gJ -+ 0 as £ -+ O. (.!J96.~ denotes the symmetric difference, (.99 - ..",g) U (..."g - .99).) The faste r P[~6 Nt'-'.g] tends to zero, the morc noise-stable A is. More precisely, let {.~} be a collection of events, where v& en".. We say thac {v-&} are uniformly stable if the limit lim t _ O P[x E ~ l:::. N~] ;;: 0 is uniform in i. For w E R" and $ E R , let ,/lllw. , be the (generalized) weighted majority event

Let rot denote the coUection of such events: 9Jl:={At w " :n=I,2,,,., w E R", SE R }. In Section 3 we show that Theorem 1.B.

- 9Jl

is uniformly stable. Moreol.'n; Jar every..$&

E 9Jl

where C is a univmal consto.nt inrkpendent qf ..de, .

Note that an infinite sequence {.Ah} with P [..-"~] bounded away from 0 and 1 cannot be asymptotically noise sensitive and uniformly stable. We also observe (Lemma 3.8) that when {.~}, (~ c n~J , is asymptotically noise sensitive and {~}, (.93; C .QR). is uniformly stable, then OJ-&- and ~ are asymplotically uncorrelated. One can say, somewhat imprecisely, that the noise sensitive events are asymptotically in the orthocomplement of the uniformly stable events. Stability and sensitivity are two extremes. However, there are events that are neither sensitive nor stable. For example, if W is the event of a percolation crossing, as described above, and J6 is the majority event, then WnJ~ is neither asymptotically noise sensitive, nor uniformly stable. 1.5. Fourier-Walsh expanswn

For a boolean function f on {O, l}n, consider the Fourier-Walsh expansion j= L:s cl~17 (S)us, where, us(T)= (_I)lsnTI. Here and in the following, we identitY any vector x E Q~ with the subset {j E [n] : X;.= I}, of [n] = {I, 2, ''', n}. Consequently, Ixl denotes the cardinality of that set; that is, Ixl = \lXlll for x E .Q~.

356

NOISE SENSI1MTY O F BOOLEAN FUNCTIONS AND APPLICATIONS "ID PERCOLATION

II

n..,

be a sequence ojevents, and set ~ = X.-t . Then {.Am} m if asympwtical(y notse sensitive ifffar etlery finite k Theorem 1.9. -- Let ' ~m C

(1.6)

(1.7)

limL::{i.(S)' , S C in), I ~ lSI d •

lim sup L{i

4 _0 I , a (..,g) = log I(..,g)1 log n, ~(..,g ) =

- log J(...1!)f1ogn.

For events A we clearly have 0 ~ ~(~ ) , and I3{.A) ~ a(. ~ ), provided that P [.A ];;: 1/ 2. When .A is monotone a(.A»,,;: 1/ 2. Perhaps some words of explanation are needed. I(.AS) measures the Slim of the influences of the variables. For monotone events it is maximal for majority, where 1(./6) ~ .fii and thus a (A) --+ 1/ 2. In the terminology used in pt': rcolation theory, 1(....1$) is ilie expected number of pivotal edges. For the crossing events W of percolation (in arbitrary dimensions) it is conjectured that 1(W) behaves like a certain fractional power (a critical rxponen~ of n. It is conjectured that in dimension 2, as n tends to infinity, a (W) tends to 3/8. Thus, this critical exponent generalizes and has a Fourier-analysis interpretation fo r arbitrary Boolean functions. a(~) is larg(: if there are substantial Fourier coefficients ] (S) for larg(: lSI. In contrast, P(...".g) is large if there are no substantial Fourier coefficients J (S) for S of small positive sizc. We conjecture iliat for the crossing events for percolation, as n tends to infinity P(W) tcnds to a positive limit. We are curious to know whether this limit is strictly smaller than the limit for a (W).

1.6. Some relalid and future work There are interesting connections bet\veen noise sensitiVIty and isopcrimetric inequalities of thc form described by Talagrand in [32]. These connections and

357

ITA! HENJAMINI, GIL KALAl, ODED SCHRAMM

1'./

applications for first passage percolation problems will be discussed m a subsequent pap" [6]. Our notion of noise senSlUvHy is related to the study of noises by 'Isirelson [34, 35]. "Noise", in Tsirelson's sense, is a type of a-field filtration. Uniform stability seems to correspond, in the limit, to the noise being white, while asymptotic sensitivity seems to correspond to the noise being black. 1.7.

The structure

rif this paper

Theorems 1,:1 and 1.4 are proved in the next section. Our proofs combines combinatorial reasonings with applying certain inequalities for the Fourier codflCients of Bonami and Beckner which were used already in [23]. However, to get the resulrs in the sharpest forms we have to rely on a sophisticated "bootstrap" method of 1.33] and on the main results of that paper which rely on this method. TaJagrand's remarkable paper [33] has thus much influence on the present work. Weighted and unweighted majority functions are considered in Section 3. An applications to percolation is described in Section 4 followed by some related open problems in Section 5. In Section 6, we will work out two examples (due to Ben-Or and Linial). In one of these a(.~) - I - Jog:! 3 and il(.~) - 1 - log2 3. In Section 7 wt: consider relations with complexity theory. A simple description of noise-s!:nsitivity in terms of random walks is given in Section 8, In Section 9 we consider perturbations with a different SOrl of noise, where the number of bil" that are changed is fixed. The: conclusions arc similar to those above, but there is an amusing and slightly unexpected twist.

For simplicity we consider here the uniform measure on On. More generally, one may consider the product measure PI>' where Pp{x : Xi = I } = p. Our results alld proof apply in this setting. (All that is needed is to replace the Fourier-Walsh transform by its analog as given in Talagrand's paper [311 and the proofs go through without change.) However, the case when p itself depends on n is interesting, but will not be considered here. Sin!.:t: the flfst version of this paper was distributed, a few of the problems we posed were settled by several people, not always in the direction anticipated by us. These developmenl~ are mentioned briefly in a few "late remarks" throughout the paper.

AcknowWgmmts. It is a pleasure to thank Xoga Alon , Ehud Friedgut, Ravi Kannan, Harry Kesten, Yuval Peres, Michel Talagrand and Avi Wigderson for helpful discussions.

358

"'OI~E

St:',,\SITIVTIY OF BOOI.F-Ai\" mNCnONS AXD

APPUCXrlOf\~

'ID Pt:RGOLATIOX

13

2. Sensitivity to noise

We now put the noise operator Nt defined in the introduction into a somewhat more general framework. That will allow us to deal, for example, with the situation where the I bits are immune to noise but the 0 bits are noise prone. Consider the following method for selecting a random point x E O n. Let ql, ... , 9" be independent random variables in [0,1], with Eq,- = 1/2, for j = I, ... , n, and let (0 E [0, l] n be random uniform. Set

if I - roj < f{j, otherwise. Then x is distributed according to the uniform measure of O n; it will he denoted by N (w, q).

Let v he the measure on [0, I] " such that v(X) =P[(ql> ... , qn) E Xl- We think of q = (ql, ... , q.) is selected according to v. -Chis q gives a product measure Pq on {O, I} " that satisfies Pq{'t E O n : 't(j ) I} fjj. Then x is chosen according to the measure P q. For example, suppose z E On. Define q = q(z) E [0, W by q,- = I - E if Zj = I and fjj = £ if Zj = O. Then for every Z E an, the p o} .

For y E V' , the triviaJ estimate l C4x~.g - p[~4!] I:5;; I holds. Therefore,

va" .A!, £) ~ P[Y'J + 0' ~ 2 < II!II,.".

lLmma 2.4 (Bonam;, Beckne D we have var(g, E) ~ var(A, E). Moreover, g takes only the values a and I. By applying Theorem 2.6 for g, and using Theorem 2.2, Theorem 1.3 immediately follows. 0

Proolof 1.4, (2.11)

ObseIVe that for a monotone f: On -

I,(f)=2IJ(Uj)l.

365

R

20

!TAI Bf...'\IAMJJ'\J, GIL KALAl, ODED SCI IRAMM

and therefore H (f) = 4

(2.12)

LJ\{j })'. j

Hence 1.4 follows from Theorem 1.9. 0 Note that (2. 12) implies the well-known inequality

(2.13)

H (.A\)

;

Sjx:;;

(~) (I (k) -J(, - k))

(Yl , ... ,y~) where Jj == I -

I(f) = 2- '

LL •

.

and 'y;:;; X; for

Xj

i:f j.

Then

If (x) - f ('jX)l ·

j

Sincefis monotone,f(x)-J (Sjx) ~ 0 when Xj== I andf(x)-J(sjX) ~ 0 when ~' ::;O. Hence the expression for I(f ) simplifies, I(!) = 2- '

= 2-'

(3.2)

L f(x)( 21xl •

,)

~ (:Y(k)(2k -

= z-' L

,)

(:) \I(k) - J(, - k)) (2k -

, ).

*> ~

Fo, any I. ~ 0 write l(1.) = (, + I.Vn)/2. Since 0 ~J(k) ~ I, by compa,ing (3.2) and (3.1 ), we obtain the following estimate. l(f)

~ (2l(1.) -

n) E(fM) + T'

(3.3)

< I.fn E(fM) + 2 -,

L (') (J'(k) - J(, - k)) (2k -

k>(fi..)

L k>Ji}.)

(n)k (2k - ,).

Because there are constants C h C2 > 0 such that

367

k

n)

ITAI RENJAMINI, GIL KAlAl, OIlEI) SCHRAMM

22

holds for every nand k, by choosing A. = C s '; large constant, we get

T'

log E(] M), where C 3 is a sufficiently

L (:) (2k - n) ~ C, In E(fM),

k>4().)

and the theorem follows from (3.3). 0 Given a set K C [nl , let MK denote the majority function on the set K; that is,

A1so set,

IK(f)=

LIM). 'EK

CQrollo.ry 3.2. -

Let K C [11] and suppose thatf: On -

[0, I] is monokmt. Then

where C is some universal constant.

Proof -

Set m = IK I, and assume, that K= {1, .. " m}. Given;: E O"" set

1«,) = 2m - '

L

.,En._..

f(z,y).

Then JK is monotone and I(/"K) = IKlf). Consequently, the corollary follows from Theorem 3. 1. 0

Proof W1.6. -

Assume, with no loss of generality, that

for allj E {l •... ,n- I}. Cor. 3.2 implies that

(3.6)

• L [,{f) ~ C, A(f) (I + J -IOg A(f)) Ii ) =

1

368

XO ISE SE:-.ISITIVIlY OF BOOI.EA-"\'

~UNCll0NS

AXD AP rUCATIOI"S TO PERCOIAT10r-;

23

for some constant C 1 and every k E (n]. Subject to these constraints and (3.5), H (J) is maximized if equality occurs in (3.6) for every Ie Therefore,

, =O(I)A(f)' (I - logA(f))I>- ' A; ", I

= O( I)A(f)' (I - log A(f)) log n. This proves the first part of Theorem 1.6. The second part now follows from Theorem 1.3. 0 Theorem 1.6 tells us that if A(A",) -- 0 fast enough for monotone events Am, then they are asymptotically noise sensitive. Conversely, if a sequence of (not necessarily monotone) events satisfies inf", A(A",) > 0, then it is not asymptotically noise sensitive. This can be proven directly, and also follows from Lemma 3.8 below. It is interesting to note that

Theortm 3.3. -

Maion!;, maximiqs I among monoume events ... ,g C O.,

This follows from [15] , although the explicit statement does not appear there. It also follows from the classical Kruskal-Katona theorem. Sec also (l8, Lern. 6. 1]. 3.2, Genera! weights

We will investigate now some relations between noise-sensitivity and weighted majority fun ctions. Several of the propenies we need for weighted majority functions are easy to establish if the distribution of weights allows us to use a normal approximation for J (x) = L j w But, as it turns out, working with arbitrary weights is harder. Our ftrst goal is to show that weighted majority functions are uniformly noise stable. This will imply the "only if" part of Theorem 1. 7. For this, the following easy (and quite standard) lemma will be needed.

n.

lLmma 3.4 . . - Let w = (w" ..., w,) (3.7)

*

0 and J (x) =

Ej U;{2xj -

P[IJI ~ '1l wll,l ~ 3,',

and

(3.8)

P[IJI ~ 0.311wll,l ~ 0.92. A much stronger estimate than (3. 7) is known (see [28]).

369

I). Thm

ITA] BENjAMINI, Gil . KAlAl, ODED SCHRAMM

Proof -

311 wll: - 211wll:

Without loss of generality, we assume that II wll2 = I. Then E[ Pl = ~ 3. H ence (3.7) foUow",

P(lf l ~ I] = P[f' ~ t] ~ I-'E[f'] = 31- ·'. This implies

Hence

We choose

t= 10,

and obtaln

lD - 9.9PU'

~ I / lD] =

10PU' > I / lD] + P[J'

~ I / lD] / IO

, E[l U'''o, J'] ~ 9/ lD, which gives (3.8).

[1

Lemma 3.5 . .- ' Let h > 0, kt li t, lid ~ h, and kt g = Lj= I (mB/ 4)']

< 4{mW'E [(K - EK)'] =4(mW' ( E [K' J - (EK)')

(3.13)

< 4{mW' E[K] < 4m Let L be the ,et of

k E [m] m eh that IY(.J,)I

>

' B '.

aVS/ IO. By (3.8), applied to Y(j,) in

place of j, P[k E L

I.-"\,J

~

8/100.

Moreover, conditioned on all the]l. the events a calculation similar to (3 .13) gives

P[iLI < mB/ 100 I K ~ mB/4]

{k E L} are independent Consequently,

< O(I)m- 'O-'.

When we use this and (3. 13) together, we get

(3.14)

P[ILI < mB/ 100]

< O(l )m- ' B- ' .

If we condition on L, on all Y(J.) for Ie f/- L and on all [YUi)1 for k E L, then what remains to determine f are only the signs of Y(Jk) with k E L. Mo reover, these signs arc independent, and are + or - with probability 1/ 2. H ence we may apply Lemma 3.5 with

b,= aVS/ IO, d'=IL I, "='0 -

E", Y(j,), g= E", Y(j,),

and take 0= (0,) to be

the sequence (IYUi)1 : k E L). The conclusion is that for t ~ 1

p [lf- ' ol 0 such that (w, IW) n = I, 2, ... and every wE R n with nonnegative coordinates and IIwll2= 1.

~

c for every

*

Proof - Set fix) = 2:;. ,12,) - I)wj fo' x E n •. Then 11{ j})=Wj fo, j E [n] and liS) = 0 foe 5 C En] , 15 1 J. On the othe' hand, I,{-A'6.) = M•.({ j} ), whee< M I

29

Let W:::; ~ c n be the event that there is a left-right crossing of R; that is, W is thc sct of all configurations that contain a path joining the left and right boundaries of R. An easy and well known application of duality shows that P[g"]:::; 1/ 2. Kesten (24J gives an estimate from above for the probability that an edge near the middle of R is pivotal for W. Similar estimates for edges near the boundary can probably be extracted from Kesten's paper. These give an inequality of the form lAW..) ~ m - I -~, c > 0, for eachj. Then Theorem 2.5 implies 1.2. However, we prefer to present another proof, based on Theorem 1.6. The only percolation background needed to understand the proof is that in our situation the probability that a vertex in R is connected in the configuralion to some vertex at Euclidean distance r is at most Cr- I/ P , for some constants C, p > O. This follows from the celebrated Russo-Seymour-Welsh Theorem [29, 30] (see also [19]).

Proof of 1.2. - Let E, be the sel of edges in the right half of R, with edges exactly centered included. Let K C E,. We now estimate E(x/'; MK )' Consider the following algorithmic method of randomly selecting a configuration. K Let ro and (j}K be two independent elements of i1 IK1 and n "_ IKI' respectively. Let VI be the set of vertices on the left boundary of R , and set VISITED:::; 0. As long as there is some edge [v, u] f/ VIS[TED joining a vertex v E VI to a vertex u f/ VI, choose some such edge e:::; [v, u], and do the following. Append e to VISITED. If e E K, let y{e) be the first bit in the sequence roK that has not been previously used by the algorithm, while if e f! K let y (e) be the first bit in the st~quence @K that has not been previously used by the algorithm. If y(e) =" I, then adjoin to VI the vertex u. This procedure defmes y for all e E VISITED. Let ~ E n be random, uniform, and independent of y, and let x = y on VISITED while x:::; z on E - VISITED. This defmes a configuration x E Q . The following is obvious: Lemma 4.1. - The cotifiguration x given by the abol)( algon·thm is uniform{y distribull:d in n. The event x E W is equal to the event that at the end qf the algorithm V I intersects the right boundary and is independent from ~ (can be defn'mined by y). 0

Let us estimate the probability that K n VJSITED is large. An edge e E K is in VISITED iff there is in x a path joining a vertex of e to the left houndary of R. Since K C E" it follows from the above stated consequence of the Russo-Seymour-Welsh Theorem that the probability for the latter event is bounded by em- I / p, for some constants C, p > O. Consequently,

E lK n VISITED j ~

q Klm- I / "

which implies

375

30

ITi\l Bti'UAMIr-.I, Gil. KALAl , ODED St:HItAMM

where

Let

.~ is

,..,.g 2

the event

be the event that there is an integer j in the range [ ~j ( IK lm- 'l/(3 P) such

that

It is easy to see that the

P (~ ]

dC':cays super.polynomially in m; in pa rticular,

P['/6,] ~ O(m- '/ p). As P~ d, U..d,j ~ O(l)m- '/'3', we have (4.1)

E(x..", U.", X", M K ) ~ O(I ) m- '/'3' .

Now suppo~ mat the algorithm produced a y such that Al U ' ~~ 2 does not hold . Then it follows that

IVISITED n K I

2

L

re v[srn:u n K

y (,)

~

O( I)VI K lm- ' /(3P: logm.

This implies that E[MK(X) Iyj ~ O (I) m- ,/,s" logm,

Since x E W can be determined from y , we get

E((I -

X..• , U •..,lx." M K ) ~ 0 (1) m- '!O" log m.

In view of (4.1 ) this implies E(x-c MK) '" O (l )m ' 1/(3p) logm,

and Cor. 3.2 gives

(4.2)

IK(W) ~ 0( I ) JiKjm- ' /(3'~ogm)3/'

for every K c E" since W is monotone. 8y symmetry, this would also hold for K C E - E" and therefore for every K C E. Consequently, by the proof of Theorcm 1.6

An appeal to Theorem 2.5 completes the proo(

376

0

:'\fOISt: SF..'\'SITIVI'll' OF BOOLEA."\; H":NCTIONS

'\'"\;J)

APPUCATIO:'\fS TO PERCOtATIOJ\"

3t

Remark 4.2. -- Since I(W)= 2::. 1.(3") is also the expected number of pivotal edges for 'tf, (4.2) sho\·...s that the expected number of pivotal edges is bounded by 0(1 ) ml - I/(3P)Qog m)l/'l.. Although this is better than the general bound ofO(I)m that follows from Theorem 3.3, a somewhat better bound can be extracted from Kesten's [24].

Caro/Iilry 4.3. - - There is a constant c > 0 with the following proper!y. If E = cl log m, then for large m, with probabilirJ at least 1/4, {x, Ndx) } n = I. That is, if each edge is .fwitchttf with probabiliry cl log m, independentlY, then the crossing is /wry to be created ar destroyed.

I

'lfl

The corollary follows from (4. 3) and Theorem 1.5. The details are left to the reader. 5. Some conjectures and problems concerning percolation

5.1. Other sensitiuiry co,yeclum

Consider the crossing event W", for an (m + I) x m rectangle in the square grid Zd. By Theorem 1.2 and Section 2, from knowing which edges are open for all but a small random set of edges, we have almost no information whether crossing occurs. This suggests that for some deterministic subsets of the rectangle R = R"" knowing the configuration restricted to that configuration typically gives almost no information whether crossing occurs. It follows from the Russo~Seymour-We1sh Theorem [29, 30J that E" the set of edges in the right half of the rectangle, is not such a subset. Yet we believe that all the horizontal edges (or all the vertical edges) is such a subset. That is, let x,y E n be two independent uniform~random con figurations. Let d,e) = x(e) for horizontal edges e, and d,e)=y(e) for vertical edgcs. Let P(ffi)= P[Z E WIX=ffiJ. Co,yecture 5.1.

- For any

£

> 0, for all sufficiently large m,

p{ro E 110 iP(rol - 1/21

> 0

so that lirn",_oc 41( 8"'"" m- Il ) = 0.

It is known [25, 241, respectively, that for some reals 0 < hi < b2 < I,

mbl ~ I(W",) ~ m~, and it is conjectured (see, e.g., [14] , p. 91 ) that 1(3";") behaves like m3/~ .

Problem 5.3. - - Is it true that iirn",_oo 4>(W""

Em) =

0 when Em = o(m- 3/ 4 )?

Recall that OUf proof of noise sensitivity for ~ us D set xl{e):=xo{e) if the number of points in (D, I] n X, is even, and xl{e):= I - :cote) if the number is odd. This gives a continuous time stationary Markov chain XI in n ={O, I} ER. Write ji for the probability measure governing this process. For each ftxed t, the random variable XI can b(~ thought of as ordinary {Bernoulli (1/2») percolation in Z'l . An interesting problem raised by [20] is weather there are (exceptional, random) times t in which there is an inftnite percolation duster in XI. The result described below might be relevant. As before, leI Wm denote the set of conftgurations in n that have an open left· right crossing of R.". For all t, P[x/ E Wm] = 1/ 2. Let Sm be the set of switching times; that is, S", is the boundary of {t ~ 0 : XI E 3""m }. As a corollary of Theorem 1.2, we have,

CQroilary 5.1.

- IS.n [0, III ~

00

in probability.

378

r-:QISE SENS1T1Vny OF B001.FA">: FI;r-:CT10NS i\..... O APPl.l(:AT10:-;'S TO PERC01AT10:-;'

:n

Prorf. - Suppose $ > t ~ O. Obsenre that the distribution of the pair (x" x,) is the same as the distribution of the pair (XcI, Ndxo)), where ( is a function of $ - t and ( > 0 when $ > t. (Actually, ( /($ - ~ _ I as $ - t -~ 0.) Let k be some positive integer, and set (;;;: ({Il k}. Let ~: ~jl k. Let all/ be the set or ro E n such that IP tNr(ro) E W] - 1/ 21 > 1/ 4. Then P[o///] - -lo 0 as m _ 00, by Theorem 1.2. Let Z (a, b) be the event that S n [a, b] ;;;: 0. Observe that for ro f/. 0//1'", we have

because £(~, ~I) is disjoint from the event I {x~, X~' f } the following estimate,

n

Wi;;;: 1.

lienee we can make

PI£(O, ~.oI] = P[£(O, ;) n £(;, ;.,)] = LP [£(O,;)n£(;,;.,)I x, =wl p{w} 00,0

x,

= L p [£(O,;) I = wl p [£(;, ;.,) 00, 0

Ix., = wlp{ w}

(by the Markov property for XI )

< P(o/1'']+

p [£(O,;)lx,= w] p [£(~,;,, )l x,= wj P{w}

L WEO - '7//

p[£(O,~) I x, = wlp(w}

< P(O/F) + (3/ 4) L we U- 7J//

x,

< P[O/1''] + (3/ 4) L p [£(O, I) I = wl p(w} ",, 0

= P[o///j + (3/ 4)P [£(O, ;)l .

9]

Using tllis inequality and induction gives ii [z (o, ~ 4P[O/#] + C~/4Y. By stationarity, for every t ~ 0, the same estimate for the probability of Z (t, t + j l k) holds. Since k may be chosen arbitrarily large, and P[o//I] - 0 as m - t 00, the corollary easily follows. 0 5.4. limits and coriformal

mvanance

The motivating questions behind this work were the conjecture regarding the existence of the limit and the conformal invariancc conjecture for two-dimensiona1 percolation. These conjectures say, roughly, that the crossing probabilili(:s inside a domain between two boundary arcs have a limit as the mesh of the grid goes to zero, and the

379

rrAI BENjMII:":J. (;11. KAlAl, ODED SCHRAM:>.I

limit is invariant under conrormal transformations of the domain and the boundary arcs. tor more netails, see Langlands, Pouliot and Saint-Aubin [26J. Consider a triple ;9'" ;:: (G, A, B), where G;:: (V, E) is a finite planar graph with m edges, and A, B c V. Let p:.;" be tht: probability that there is an open crossing from A to B in a uniform-random configuration x E .Q ;:: {O, I y. u :t .9(1 ;:: (H , A', B') be a triple obtained from G by the following operation: for every edge e of G delete e with probability (I - ~ / 2 contract e with probability (I - ~/ 2 and leave e unchanged with probabil ity t, independently of the olher edges . .7tS is a random variable which takes values in planar graphs with two distinguished vertex sets. Of course, F-{P.M ) ;:: p.~., and noise sensitivity, when it applies, asserlS that the value of p.~ is concentrated around the mean. Noise sensitivity enables one to relate the crossing probabilities of percolation on different graphs and we had hoped that it will be relev 0 som(: fixed constant) and let A and B be its left and right boundaries. It follows from Theorem 1.5 and (4.3) that PJlJ - Pffl --+ 0 in probability, provided that t· log In - 4 00. (Conjecture 5.2 would give it even when t· m~ --+ 00 for some p > 0.) It is conjectured thai the (,;fOssing probability tends to a limit as m tends to infmity and an approach to this conjecture would be to relate the distribution of such random planar graphs starting from similar rectangles of different sizes. ( fh(: values of t should depend on the size but be large enough that noise sensitiviry applies). In a different direction, the random planar graphs .~ obtained when you start with the (m+ 1) x m grid and apply certain random contractions and deletions may be related to models of random planar graphs in mathematical physics [1]. 5.5. Fourier-Walsh codJicimts qf percolo.tion It is a natural question to try to understand the Fourier-Walsh coefficients of boolean functions given by percolation problems. Consider (again) the event go';:: W'" of a left-right crossing of an {m + I} x m rectangle R = R", of the square grid, Z2. Let 1m :;:: Xy,:,,· The Fourier coefficients of1m are indexed by subsets of E R , the edges in R",.

-,

The values J can be regarded as a measure on the space of subgraphs of R",.

Problem 5.4. -

Describe this measure!

It follows from Theorem 1.5 and our estimates for H(if~), that all but a negligible part of the 0 weight of the Fourier coefficients (S), where S is non-empty, is for

J

380

l\OISF. SE:-'::SITIVITY OF BOOU::A.'\;

ft:NCTI O.~S

A:XD APPL!CATIO:'\S TO I'J:;RCOIAno:x

3.5

lSI> c1ogm. Conjecture 5.2 is equivalent to the ass(:rtion ch at, in fact, this lSI > mil for some p > O. Conjecture 5. 1 is equivalent to the statement that

is true for lor all but a negligible pan of these Fourier coefficient s, the number of vertical edges in S tends to infmity with m. 5.6. Other modelt of statistic{/l mechanics

It would be of interest to extend the results of this paper as well as earlier results on influence (123, 18]) to other models of statistical mechanics, such as the Ising and Potts models. Many of the results on influence and on noise sensitivity should be extendible to measures on .on for which th e coordinate variables are positively associated, namely, measures for which every two monotone real fun ctions are positively correlated. 6. SoDl.e further eXaDlples

We will discuss now four examples, the first two were considered by Ben-Or and Linial (4]. 6.1. Tribes

Consider n boolean variables divided into t tribes T j, T 1 ... , '1'/ of size s each, and let J be the boolean fun ction which take the value I if for some j, I ~ j ~ I , aU

prJ= I] ~ ~. Also note that f will b~ immune to

variables of T j equal I . If s = log n - log log n + log log 2, then

that I ~(J) ,..., log n/ n for ('very k. It is easy to show directly (;.noise when e = o( J/ log n) and will be devastated bye-noise if £ log 11

_

00.

Thus, J(f ) - log nl n,

6.2. &cursivt rIUIjon"ry on the ternary tr« Consider n = 31 boolean variables which form the leaves of a rooted ternary tree of height t. A boolean function f is defined as follows: Giw:n values for the variable on the leaves compute for each other vertex its value as the majority of the values of its sons and set the value off (0 be the value of the root. Ben-Or and Linial showed that Ik(f) '" n- ~'l./ 1C'F(3 for every k and lhus P(f) ~ I - log 2/ log 3, This follows at once from the iollov.'ing observation: for l = I , if we switeh the value of each leaf with probability p independently, then for small p the probability that the outcome will be switched is (3/2)p + o{p).

a(f)

~ I - log 2/1og 3 as I ~ 00, It is easy to sec that also

381

36

ITAI

IlE~JAMI;\"I ,

GIl KALAl , onEI) SCHRAMM

There is an absolute constant 130 < 1/ 2 (fLIld it!) such that for every monotone Boolean function J, 13(1) " flo.

Conjecture 6.1. -

Late Remark 6.1. - It was pointed out by Massel and Peres by considering certain recursive majorities on larger trees that this conjecture is false. 6.3. Number qf runs We considered mainly monotone events. Here is an interesting noi.se stable nonmonotone event. Given a string of n bits XI, X2, ••• , Xn, let R(x], X2, .. •xn ) be the number of runs. Thus, R is one plus the number of pairs of consecutive variables with different values.

Thr event that R (xl, X2, ... X~) is larger than its median is noise stable. Indeed, writc)'j ::;Xj(£l xi+l , i= l , ... , n-l . and note that they;'s are independent, and R isjuSl the majority event on the is. (Here ED is addition mod 2; that is, xor.)

6.4. Mqjoriry

0/ tritmgles

We considered only the case where

p is

a constant. VVhcn

p tends to

zero with 71 , vertices with edge probability p= 71- a, a> 0 and the event that the number of triangles in the graph is larger than its median. This is a noise stable event but its correlation with majority (or any weiglued majority) tends to 0 as 71 tends to infinity. new phenomena occur. Consider, for example, random graphs on

71

7. Relations with cODlplexity theory There is an interesting connection between the complexity of boolean fun ctions and the notions studied in this paper.

7.1. AGO and iriflutnCes An important complexity cla..~s ACO of Boolean fun ctions arc those which can be expressed by Boolean circuits of polynomial size (in the numbcr of variables) and bounded depth. Boppana [9] proved that iff is expressed by a depth-c circuit of size ~ then

(7.1)

I(J) < C, log'- ' N.

Earlier, Linial, Mansour and Nisan [27] proved that the Fouricr coefficients of functions which can be expressed by Boolean circuits of polynomial (or quasipolynomial) size and bounded depth in ACO decays exponentially above polylogarithmic "frequ encies". Both these results rely on the funda mental Hastad Switching Lemma, see [21 , 2].

382

~OISt: St;~SrnV ITY

OF BOOLB,..J\I F1j;":CTIO:-':S

A.~D

:\Pl'I.IC:\TIOr-.:S TO PERCOLAT IU:\'

37

Recall (hat a monotone circuit is one where all the gates are monotone increasing in the inputs; i.e., there are no "not" gates. The Hastad lemma for monotone boolean circuits is easier and was proved already by Boppana f8]. Wf' conjecture that a reverse rdation to 7.1 also holds.

Conjf'Cture 7.1 (Reverse Hastad). - For ('very £: > 0 there is a K =K(£:) > 0 satisfying the following. l'or every monotone ,..m c nil, there is a ,ji!J C nil such that P [,,1S6.J9] < £: and .JfJ can be expressed as a Boolean circuit such that Qog NY- ' < KI(A ), where c and :\I arc Ih e depth and si7.e of the circuit, respectively. MonOlOne Boolean functions with bounded influence were characterized by Friedgut ['16, 17]. The results of (I Il arc also relevant to this conjecture. Ha Van Vu raised the question if there is a spectral way to distinguish between hounded depth circuits of polynomial size and bounded depth circuits of quasi .. polynomial size, In particular, he was looking for a way to show that the graph property "having a clique of size log n" for graphs with n vertices, cannot be expressed by a bounded depth circuit of polynomial size. (Here the sel of variables correspond to the G) possible edges.)

Conjecture 7.2. - Let £: > 0 be a fixed n:al number. Let ,/1 be a monotone property expressed by a depth-2 circuit of size M and Jet f ~ X, i ' Then there is a set ..7' of polynomial size in M (where (he polynomial depends on , and £:) so that

L{]'(S) : S ~Y} ~

£.

This conjecture may also apply to TCO, see br.low. It would be of great interest to characteri,.e Boolean runctions ror which most of the weight of the fourier coefficients is concentraled on a set of polynomial size in n. 7.2. TeO and noise sensitiviry Noise sensitivity seems related to another class of boolean functions - threshold circuits of bounded depths see [36, 22). In a threshold circuit each gate is a weighted majority runction. For the study of spectral properties of signs of low degree polynomials sec Bruck fl21 and Bruck and Smolensky [13].

Conjecture 7.3 . .- Let f be a boolean function gIVen by a monotone threshold circuit of depth c and size M. Then

(7.2)

383

:m

ITAI BEI'{,AMtNI, (aL KALAl, ODED SCHRAMM

1'hus, for 1/ £ :::;; 0 (1) (log M y- I we: expect that var(); £) is bounded away from zero. Also here it is a tempting conjecture that a reverse relation holds. We conjecture further that all functions f that can be expressed by a depth6( monotone threshold circuit where all the threshold gates are balanced are uniformly stable. (And in particular, J(f ) = 0 (1).) Possibly, functions in this class of {unctions approximate arbitrary well arbitrary uniform stable monotone Boolean functions. Conjecture 7.3 implies theorems of Yau [36] and Hastad and Goldmann [221They proved that the and/or tree (or equivalently lhe example of ternary tree of Section 6) does not belong to monotone TeO; i.e., it cannot be expressed as a monotone bounded depth circuit of polynomial size. The results of Yau and H astad arc still open for the non·monotone case. This would follow if relation 7.2 holds even for every monotone boolean fun ction f given by a (general) threshold circuit of depth c and size M. 8. Random walks

For nonempty A C On, consider a random walk defined as follows: start with a point chosen at random uniformly from .~, and at each step, stay where you arc with probability 1/2, and with probability 1/(2 11) move to anyone of the neighboring vertices. Let P~--t he the measure on .o.~ given by the location of the walk after t steps, and sct

Here IIP:.-t - PII is the measure ([.1) norm of the difference between uniform measure.

Theorem 8.1. - Sup/Xlse Iiu.ll inf ~ Pl./ l m] > O.

. ,.~6~1

C Qn,. is a seqU£T/ct

1. {"~m} are asymptoticallY noise sensitive

lfW.

Proof -

SCl.f,(x),= 2" P: ""[{x}l . Note ",at 'm

f~, = (1/2)f, + (2 n.r'

and the

satisfYing

iff lim", W(.,4;, £)/ nm=Ojor everyfued £

~.) . ~, then W(A"" E) ~ n H -Gur alld C. KAlAl, Evt:ry mu nu!une graph p"'perty has a sharp threshold, PlI)(. Am"'. Malll. Sue. 124 (1996), 2993-3002.

[191

G. G\UMMETT, Pn, l'IIbl. 1.11.£,5., 81 ( J995), 73-205.

[33J

M.

K !;ST['.'~,

&a1ing

T...u.GAANll,

rdation~

for 2f)·pt"rcolation, Cornm. Malll. ""YJ. 109

~H-:

( J~7 ),

109-156.

J.

Sl4tUl.

J'/oiJ.

43 (1978), 39-48.

How murh arr. increasing

set~ ~iti\ldy

388

if !'lob.

mrreialnl?

22 (1994), 1:'76-15/37.

Q,mhi~alOluo

16 (1'J96), 243-258.

NOISE Sf.NSITMLl' OF BOOLEAN

~1JNCTIONS

ANn APPIJCATIONS 'ID PERCOlATION

(H}

B. T~JKi:J.sm•. Fourier-Walsh cor.fficien!' for a coalescing

P5J

B.

TSIIlY.I.\.O~,

The F'wtc

nois.-~,

(d;5Cretc lime), prcprim, ma{h.PR /9903068.

prepr;m.

l36J A. Y"o, Cin:u;u and local computation, (1989), 11:16-196.

now

4:\

PnxuJi"l!~

'!!

2/~1

Annual ACM o/mposium on Thto,),

'!!

Computing,

LB. The Weizmann Institute of Science, Rehovot 76100, Israel [email protected] http://'MYW.wisdom.weizmann.ac.il/ ",itai/ G. K. The Hebrew University, Givat Ram, Jerusalem 91904, Israel [email protected] http://www;ma.huji.ac.il/rvkalail

o. s. The Wiezmann Institute of Science, Rehovot 76100, Israel [email protected] http://'MYW.wisdom.weizmann.ac.il/ ",schramm/

Manuscn't refU IL 4 janvier 1999.

389

Annals of Malhematic,. 171 (2010).619--672

Quantitative noise sensitivity and exceptional times for percolation By ODED SCHRAMM and JEFFREY E . STElF

Abstract One goal of this paper is to prove that dy namical critical site percolation on the planar triangular lau ice has exceptional times at which percolation occurs. In doing so, new qllQmirarive noise sensitivity results for percolation are obtai ned. The latter is based on a novel method for controlling the "level k" Fouri er coeffici ents via the construction of a randomized algori thm which looks at random bits, outputs the value of a particular function but looks at any fi xed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff di mension of the set of percolating times. We then study the problem of except ional times for certain "karm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no ti mes at which there are two in fin ite "white" clusters, obtain an upper bound on the Hausdorff d imension of the set of times at which there are both an infini te wh ite cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infi nite clusters are present. I. 2.

3. 4.

5.

Introduction Noise sensitivity of algori thmically dilutc funct ions 2.1. Noi se sensitivity background 2.2. Proof of Theorcm 1.8 Percolation background and notation Noise sensitivity for percolation 4.1. The simply connected case 4.2. Annulus case Exceptional timcs

Research supponed by the Swedish Research Council and the Gomn Guslafsson Foundation (KYA). 619 I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probability C Springer Science+Business Media, LLC 2011 and Statistics, DOI 10.1007/978-1-4419-9675-6_13, 391

620

ODED SC HR AMM and JEFFREY E. STEI F

6.

Hausdorff dimension of exceptional times

7.

Exceptional times for k-arm events

8.

Upper bounds fo r k-arm times

9.

The sq uare lattice

10.

Some open questions

Appendix A.

Quasi-multiplicat ivity

References

I. Introduction Consider bond percolation on an infinite conn ected locally finite graph G , where for some P E [0, '], each edge (bond) of G is, independently of all others, open with probability P and closed with probability 1- p. Write Jrp for this product measure. The main questions in percolation theory (see r12]) deal with the possible ex istence of infinite connected components (clusters) in the random subgraph of G consisting of all sites and all open edges. Write ~ for the event that there ex ists such an infinite cluster. By Kolmogorov's 0-\ law, the probability of '6 is, for fi xed G and p, either 0 or I . Since Jrp (C(5) is nondecreasing in p, there ex ists a c ritical probability Pc = Pc(G) E [0, I] such that

Jrp(~) =

f0 lI

for p < Pc for p > Pc·

At p = Pc we can have e ither Jr p ('€ ) = Oar Jr p ('€) = I, depending on G. Haggstrom, Peres and Steif [131 initiated the study of dynamical percolation. (The notion of dynamical percolation was invented independentl y by I. Benjamini. While the present paper was moti vated by l I3], the question studied here had previously been asked by Be nj amin i, as we recently became aware.) In thi s mode l, with p fixed , the edges of G sw itch back and forth according to independent 2 state continuous time Markov chains whe re closed switches to open at rate p and open switches to closed at rate 1- p. Clearl y Jrp is a stationary distribution for this Markov process. The general question studied in r131 was whether, when we start with di stribution Jrp , there cou ld exist atypical times at which the percolation structure looks markedly different from that at a fixed time. Write 'll p for the underlyin g probability measure of this Markov process, and write ((5 / for the event that there is an infinite cluster of open edges at time t . Two results in l13 J which are relevant to us are PROPOS ITIO N 1. 1. For allY graph G ,

I

'II p( C£ / occursfor every t) = 1

\II p ( -. ((5 / ) occurs for every

t)

392

= 1

if p > p,(G) , if p < p,(G) ,

QUANTITATIVE NOISE SENSI TI VITY AND EXCEPT IONAL TIM ES FO R PERCOLATION

62 1

THEOREM 1.2. For d ::: 19, the integer lattice 7L d satisfies

'II Pc ( -,«6z) occurs for every t ) = 1. One important aspect of the proof of the latter result is that it uses the fact, proved in lI4] , that for d ::: 19,

( I.I )

JTp(O is in an infinite open cluster) = O(lp - Pel) .

It is proved in [211 that (J. I) does not ho ld for d = 2. Therefore, the question of whether Theorem 1.2 is true for d = 2 becomes interesti ng. At thi s point, we menti on that site percolation is the analogous model where the vertices (rather than the edges) are open o r closed independe ntly, each w ith probability p, and dy namical percolation is defin ed in a completely analogous manner. Our main result says that Theorem J.2 does not ho ld for site percolation o n the planar tri angular grid . The triangular gri d is the graph whose vertex set is the subset of C = 1R2 consisting of the po ints

il+ exp(2)ri /3)il ~ {(k +i/2 , Jii/2) : k ,i E

il}

and two suc h po ints have an edge between the m if and o nly if the ir distance is l. Explicit ly stated. our main result is TH EO REM 1.3. Almost surely. the set of times t E [0, Ij .5lIch that dynamical critical site perco/atioll on the triangular lattice has all infinite open eluster is nonempty. There are no other transitive g raphs for which it is known that d yna mical critical perco lation has such exceptional times. (In 131. it was argued that the event discussed in Theorem 1.3 is measurable. A similar comment appli es to our other results below. Thus. measurability issues will not concern us here.) We are convinced that Theorem 1.3 is true for bond percolation on the square lattice. However, our proof uses the existence and exact values of certain so-called critical exponents, which are o nl y known to ho ld fo r site perco lation on the tri angular lanice. These are believed to be the same for bond percolati on on the square lattice. but even the ir existence has not yet been established in that case. However. the method s of thi s paper seem to come quite close to a proof for the square grid as well: it seems that there are several ways in which th is can perhaps be achieved without detennining these critical exponents. These issues will be furth er discussed in Section 9. It is interesting to note that by l 13. Cor. 4 .2], as. at every lime t the set of vertices that are contai ned in some infinite cluster has zero density. On a heuri stic level, for Theorem 1.3 to hold, it is necessary that the configuration "changes fast" in o rder to have "many chances" to percolate so that we will in fact have a percolaring time . Mathematically, "changing fast" can be interpreted as

r

393

622

OD ED SC HR AMM and JEFFREY E. STEIF

hav ing small correlati ons over short time intervals. whi ch then suggests the use of the second moment method which we indeed will use. In othe r words, one needs to know that the confi gurati on at a g iven time te ll s us almost nothing about how it w ill look a short time later. The notion of noise sensitivity introduced in [2J is the relevant tool which descri bes this pheno menon. We now bri efl y explain this. Gi ven an integer nI, a subset A of {O, l}m and an f; > 0, define

I

N(A , e):= var[p[( YI , . .. , Ym) E A Xl , ... , XmJ] where {Xi} I ~ i !O m are i.i.d. with p[ X j = 1] = 1/2 = p[ Xi = 0] a nd condi tional on the {Xi }'s, {Yj }1 I be an increas ing sequence in N going to 00 and let Am be a subset of {O, I}n", for each m . We say that the sequence {Am}m >, is noise sell.5itive if for every r:; > 0 , ( 1.2) This says that for large III knowi ng the values of X" ... , X R ", g ives us almost no information concern ing whether (Y" . .. , YRIII ) E Am . Thi s is not the exact definiti on of noise sensitivity given in 121 but is easily shown to be equi valent ; see page 14 in that paper. It is also shown in 12] that if ( 1.2) holds for some e E (0, 1/ 2), then it holds for all such 8 and in additio n that N(A, 8) is decreasing in 8 on [0, 1/ 2]. Let 11m be the number of edges in an (m + I) x III box in 1'..2 and let Am be the event of a left to right crossing in such a box. By duality, p[ Am] = 1/2 for every m (see 112 ]). In 12}, the following result is proved. TH EO REM 1.5.

Th e sequence {Am} m> I is liaise sensitive.

A by-product of the tools needed to prove Theorem 1.3 wi ll imp ly the following more quantitative version of Theorem 1.5 , which was conjectured in 12]. T HEO REM 1. 6.

Th ere exists y > 0 so that lim N(Am , ,,,- l' ) = O.

m~oo

We have the same resu lt for the triangular lattice but wi th a better y, since cri tical exponents are known in th is case. TH EOREM 1.7. For critical site percolatioll on the trianglliar lattice, let A~, be the event oj the existence oj a left-right crossing in a domain D approximating a square oj sidelength m. Th enjor all y < 1/8,

lim N (A:n , m-l') = O.

m~oo

394

QUANTITATIV E NOISE SENSIT IVITY AN D EXCEPT IONA l. T IM ES FOR PE RCOl.AT ION

623

In proving our quantitative noise sensiti vity results (Theorems 1.6 and 1.7 as well as those later on necessary for obtaining Theorem 1.3), one of two key steps will be Theore m 1.8, which gives estimates of certain quantities involving Fourier coeffici ents of a fun ction based o n the properties of an algorithm calcul ating the fun ctio n; the other key step wi ll be the constructio n of an appropriate al gorithm. Precise definition s of undefined terms will be given in Section 2. where the connection with noi se sensitivity will also be recalled. TH EOREM 1.8. Let" EN and set Q = 0 11 := {O, l}/I. LeI f : Q -+ II\\: be a junction. Suppose that there is a randomized algorithm A jar determining the value oj I which examines some of the inplll bits of I olle by one. where the choice of the next bit examined may depend on the bits examined so lar. Let J S; [II] := {I , 2, ... , II} be the (random) set of bits examined by the algorithm. Set 0 = OA := sup{P[i E J] : i E [/I J}. Th en, for every k = 1. 2, ... , the Fourier coefficie11fS oj I satisfy (1.3) S~[/ll ,

ISI=k

where III II denotes the L 2 norm sllre on Q.

01 I

with respect 10 the IIl1iform probability mea-

This result mi ght have some applicati ons to theoretical computer science. We will cali oA the revealment of the algorithm A. The restriction of x to J (the set of bits examined by the algorithm) is a witness for the fu nction I, in the sense that it determines I(x). As ex plained in Section 2.2, Theorem 1.8 extends to some other types of witnesses. In the case k = I, the inequality ( 1.3) cannot be improved by more than a facto r of O( I / Iogl/): there is an example showing this with 0::: 1/ - 1/3 10g(II) , which appears in [4, §4]. The paper [4] investigates how small the revealment can be fo r a balanced boolean funct ion on {O, I }". When the fun ction is monotone, it is shown that the revealment cannot be much smaller than 11 - 1/ 3 and in general it cannot be much smaller than 1/ - 1/2. Examples are given there which come within logarithmic factors of meeting these bounds. We do not know if ( 1.3) is cl ose to being optimal for k » I. One is tempted to specul ate that the inequality can be improved to L ISI:5k j(S)2::: 0(1) k 0 11/112. We do not know any counterexample to this inequality. However, the AND function I(x) = x) gives an example where

n;=1

0(1 )

L

j(S)'" Jk811/11'

ISI~k

for k satisfyi ng Ik - / I / 21= 0(1/ 1/ 2). (It is easy to check that the best revealment possible for thi s I is exactly (2 - 2 1- /1) / 1/.) 395

624

ODED SCHRAMM and JEFFREY E. STEIF

Once Theore m 1.3 is established, it is natural to ask: how large is the set of "exceptional" times at whi ch percolation occurs? In thi s direction , we have the foll ow ing result. THEOREM 1.9. The Hausdorff dimension of the set of times at which dynamical critical site percolation 011 the trianglilar lartice has an infinite eluster is all almost sure cO l/stanl which lies ill [~ , ~!l.

We conjecture that ~! is the correct answer. In a different di rection, once we know that there are exceptional times at which percolation occurs, it is natural to ask how many clusters can exist at these exceptional times. The followin g provides the answer. THEOREM 1. 10. On rhe trianglila r laTtice, a.s. ,here are I/O limes at which dynamical critical site percolation has two or more infinite open clusters.

For the square grid, we can onl y prove

On 7L. 2 , a.s. there are no times at which dynamical critical bond percolarion has three or more infinite open clusters. THEOREM 1.11 .

In some of the fi gures, we will represent open sites by white hexagons on the dual g rid, and cl osed sites by black hexagons. Thus, percolation clusters correspond to connected components of the union of the white hexagons. These wi ll also be called white clusters. Likewise, we may also consider black clusters, which are connected components of black hexagons . Aski ng whether two infinite w hite clusters can coexist at some time is very different from asking whether two infinite cluste rs of differellt colors can coexist at some time. We conjecture that there are in fact exceptional times at which there are both a wh ite and a black infinite cluste r and that the Hausdorff dimension of such times is 2/3. We can however prove the followin g. TH EOREM 1.1 2. 011 the triangular lattice, a.s. the Hausdorff dimension of the set of times ar which there are both an infinite white cluster and an infinite black cluster is at most 2/3.

We also have the foll owing two results concerning the upper half-plane. THEOR EM 1. 13. On the triangular lattice intersected with the IIpper halfplane, a.s. the Hausdorff dim ension of the set of times at which there is an infinite cluster is at most 5/9.

1.14. On the triangular lattice intersected with the lIpper halfplane, a.s. the set of times at which there are both an infinite white cluster and an infinite black cluster is empty. THEOREM

396

QUANTITATI VE NOISE SENSITI V ITY AND EXCEPTIONAL TI MES FOR PERCOLATI ON

625

Theorems 1.1 0, 1.12, 1.1 3 and 1.1 4 will fo llow immediately fro m generalizati ons presented in the last part of the paper, which are concerned with studying dynamical percolati on on two other two-dimensional objects, namely wedges and cones . For every 8 E (0, (0) , we let We denote the wedge of angle 8 and Ce denote the cone of angle 8. For Ce, we wi ll require that 8 is a multiple of 1r /3. The prec ise defi nitions of these will be given in Section 3. First, we menti on that for all 8, the critical value for s ite percolati on on We and on Ce is 1/2, as for site percolation on the tri angular grid and bond percolation on 71.. 2 . The following results prov ide upper and lower bounds on the critical angle for which there are exceptional times for certain k-arm type events as well as provide estimates fo r the Hausdorff dimen sion of the set of exception times for a given angle. In these resu lts, if an upper bound on the Hausdorff dimension is negative, this means that the set in questi on is empty. We will on ly do the case where the arms are alternat ing in color (and hence for the case of cones, there will be one or an even number of arms). We do this partially because it is easier than the general case and because it is all that is needed in order to make statements concerning the number of infinite clusters. By a k -arm event, we mean an event of the form "there are k disjoint infi nite paths having a spec ified color sequence"; for a wedge, the color sequence is well defined wh ile for a cone, it is well-defined up to cycl ic permutations. TH EOREM 1. 15. Fix the wedge We andfor illleger k ~ I , let Atve be the event that there are k infinite disjoint paths in We whose colors alternate. Th en a.s. the Hau sdorff dim ension, H~e ' of the set of exceptional times at which Atve occurs satisfies

1-

4k(k + I)rr < H", k _2k-'('ck",+;-Ic)-_rr < 138 " 0 98

In particular, for any k ~ 1, rhere are exceptional times for rhe event Atve for () >

4k(kil)Tr

alld there are

I/O

exceptiollaltimesfor ()
1 even, A~e be the event that there are k infinite disjoim paths in Ce whose THEOREM

colors alternate (ifk > I ). Then a.s. the Hausdorff dimension. H~e ' of the set of exceptional times at which A~e occurs satisfies 51r 38 -

I 51r < 1- , 188

I --< He

alldfor k

~

2

\-

4(k' - I)rr k :: 2", (k,,',,-;-I!C )rr c. 38 -< He, < - l98 397

626

ODED

SC ~IRAMM

and J EFFREY E. STE IF

In par/iell/a rJar k = I , there are exceptional times for the event Ab, fo r (j > 5:

and there are /10 exceptionalrimesjor 8 < ~~. \ll1I i/efor k ~ 2, there are exceptional rimes for the event A ~8 f or () > 4(k\~- I)1r and there are 110 exceptiollal rimes fo r

8
fr(z, e) = z. On Ihe surface X we defin e the metric dx as the pullback of the euclidean metric of 1R2 under 1jJ, namely, dx(x , y) is the infimum of the length of IjJ 0 y for any continuous path y C X connecting x and y. Let Coo denote the completion of (X. elx). Since 1R2 is complete, it is easy to see that Coo \ X consists of a single poin t, which we denote by O. We extend the map IjJ by setting 1jJ(0) = O. Let V be the set of points in Coo that are mapped to vertices of the triangular grid under 1jJ. The triangu lar grid on Coo has vertices V and an edge between any two vertices at distance I apart. Now Ihe wedge We C Coo ;, defined by We := {OJ U {(z. e') E X: e' E 10. e) }. The triangu lar grid on We is just the intersection of the triangul ar grid on Coo with We. On Coo we may define the rotation R e by Re(O) = 0 and

Re(z , e') = (e ie z, e + e'). Thi s is cl early an isometry of Coo. The cone Ce is defined as the quotient Cool Re; that is, the set of equi valence classes of points in Coo. where two points are considered equi valent if one is mapped to the other by a power of R e. Now suppose that = II If 13 where II E N+ . Then Re restricts to an isomorphism of the triangul ar grid on Coo. In thi s case we defin e the tri angul ar grid on Ce as the quotient of the grid on Coo under Re. In other words, the vertices are equivalence classes of verti ces in Coo and an edge appears between two equi valence classes if there is an edge connecting re presentatives of these classes. Note that C21t is just the euclidean plane with the triangular lauice.

e

406

QUANTITATIVE NO ISE SENS ITI VI T Y AN D EXCEPTIONAL TI MES FOR PER COLAT ION

635

We end th is section by describi ng the so-called full and half-plane exponents for k-ann events derived in l3 1]. For integer k ~ I, let Ak (r , R) be the event that there are k disjoint crossin gs of the annulus {z E R2 : r :::: Izl :::: R} with a speci fied color seque nce (up to rotat ions), where we require that both colors appear in the color sequence. For k:: 2, and 10k, it was proved in l31J that

r ::

(3 .7)

as R ~ 00 wh ile r is fi xed. (The result for a2(R) in (3.5) above is a special case of thi s.) Next, whe n A~ (r, R) is the event that there are k di sjoi nt paths in the upper half-plane from Izl:::: I' to Iz i :: R with any speci fied color seq uence, then for k :: I, and I' :: 10k, it was proved in [31 J that (3 .8)

k a:(r, R):~ P[A+(r, R)l ~( R / r)

-k(k ±l)

6

()

+0 I.

as R ~ 00 while r is fixed. (The result for O'±(R) in (3.5) is a spec ial case of thi s.) Just as we said that the proofs of (3.5) actually yield (3.6), it is also the case that the proofs of (3.7) and (3.8) also yield versions when R l r ~ 00 while 1':: 10 k is not necessari ly fixed.

4. Noise sensitivity for percolation 4. 1. The simply connected case. To apply Theorem 1.8 to percolation, we will need to describe algorithms achiev ing small revealment. One result of that nature is Let Q = QR be the indicator fill/ctionfor the eve1lt that critical site percolarion on the standard triangular grid collfains a left to right crossing in some grid-approximating domain D to a large square of side length R. (For example, we could take D to be the union of the hexagons in the dual grid that are contained ill the square.) Then there is a randomized algorithm A determining Q stich that OA :::: R- 1/ 4 ±O( I) as R ~ 00. For critical bond percolation on the squa re grid, there is sitch all algorithm satisfying 0 :::: C R- a for some COll.5tants a, C > o. THEOREM 4. 1.

Remark. Theorem 4. 1 says that there is an algorithm fo r the relevant event whic h exposes on average at most R 7 j 4 ±O(J) bits. Since the probabili ty of points not too close to the boundary be ing pivotal is about R - s/ 4± o( l ) (this is the 4-ann event) and for a monotone fun ct ion f, j ({i}) is the probabi lity that Xi is pivotal, the case k = I in Theorem 1.8 implies that the revealment is at least R- 1/ 2±o(l). As po inted out in Peres, Schramm, Sheffield and Wilson [251 , this can also be obtained usi ng an inequali ty of O' Donnell and Servedio. 407

636

ODED SOI RAMM and JEFFREY E. STEI F

Figure 4.1. Following the interface from the come r. Theorems 1.8 and 4.1 immediately give COROL L ARY 4.2.

For every e > 0 there is a

L

CO l/stallt C

10 suffices. If a is smaller, then the esti mate we are now striving fo r is trivial.) Since the re is no hexagon intersecting both 5\ and -5\, it fo llows that the conditional di stribution of the colors of the cell s meet ing S1 is uniform i.i.d. Consequentl y, the condi tional probability that fit hits H is bounded by a2((a I\b)/2). Thus,

p[ H In the case b

p[ H

~

a

~

visited] :5 O( I) a(2 a, R) a,«a "b )/2) .

2 r, we may use independence on disjoint sets to conclude that

visited]

:5 0( I)a,(r)a,(2r,a/2)a(2a , R) :5 0( I)a,(r)a(2r,a/2)a(2a , R) (3.2 )

:5 0(1) a,(r) a(r, 2 r) a(2 r, a 12) ala 12, 2 a) a(2 a , R)

(3.1 )

:5 0(1) a,(r) a(r. R).

On the other hand, if b ~ a and a :s: 2 r, the n we use our assumption a ~ r - 0(1) and (3 .5) to get a2(a /2) :s: ,. - 1/4+0(1) :s: a2(r) r°(1), which is also sufficient. In the case b < a, a similar argument (and simil ar to the proof of Lemma 4.3) shows that

p[ H

visited IC] :5 0(1) a,(bI2)a+ (2b

+ 0( 1) , c -

b - 0(1)).

Next, picki ng a constant q E ( 1/ 4 , 1/3), we the n have by the above and (3 .6)

P[ H visited

Ic]:5 0( I)a,(bI 2)(clb) - q

As in the proof of Theorem 4. 1, we have P[2 j:::: c < 2 j + I ]:::: 0(1)2j / R. Jt easi ly fo llows that

p[H

visited] :5 0(1) a,(bI2) ( R lb) - q

If b /2 > r, then we may estimate

a,(b 12) :5 a,(r) a,(2 r, b 12) :5 a,(r) a(2 r, b 12) and (3.6)

( Rlb)- q :5 0( I)a(bI2. R) and we get from (3. 1) and the above

p[H

visited] :5 0(1) a,(r) a(r, R).

If b/2:s: r, we use instead (3.6)

a,(bI2)(Rlb) - q:5 0(I)a,(bI2)a,(bI 2 , r)a(r, R) :5 r°(t)a,(r)a(r, R) .

o

Thi s completes the proof. 417

646

ODED SOI RAMM and JEFFRE Y E. STEI F

Remark 4.9. It is easy to see that if w e assume the analogue o f (3 . I ) for (12 proved in the append ix, then the r°(l) te nn in (4. 1) can be replaced by O( I).

5. Exceptional times in this sect ion we prove Theorem 1.3. We point out that the absolute key necessary step is to get a good bound on the correlation for an event occurring at two differe nt but close-by times. Once this is done, the rest is fairly standard . Proposition A 16 in Lawler l22J indicates this general type of argument. Proof a/ Theorem 1.3. By Kolmogorov's 0- 1 law, it suffi ces to prove that with posit ive probability there are times in [0, I ] w hen the origin is in an infinite cluster. Fix R > 2 large and let Vt ,R be the event that at time I there is an open path from

the ori gin to d istance R away. We then let

X = XR

:=

fo

I V, .R dt

I

be the Lebesgue measure of the set of times in [0 , I] at which V"R occurs. The firs t moment of X is given by

E[X] =

10' P[V"R ]ill = P[VO,R] = a(R) .

The second moment is

(5 .1) E[ X']

=

E[f 10'

I V,. R I V". R d s ds']

= 10' ].' P[V" Rn V" ,R] ds ils' .

For each site v we let

X~:=

- 1 I

1

v is open at time s otherwi se,

and for a fi nite set of sites S set s .= Xs·

n

X'v·

"eS

Fix s, s' E [0. I], and set t := Is - s'l. Recall that the state of a site v fl ips between closed and open w ith rate 1/ 2. Equival entl y, we may think of the state as being re-random ized with rate I. Consequentl y, p[ X~' = X~ ws ] = e- t + ( I - e- ' )/2 = (I + e- t )/2, and hence.

I

E[X;

Xn = exp(-I),

E[X~

Xn =

n

E[X;

X:' ] = exp(-IISI) .

"eS

Moreover, if S ::j:. S', then E[Xs X~, ] = O. Consequentl y, if f is a fun ction depending on the states of finitel y many lattice poi nts and has the expansion f(w) = 418

QUANT ITAT IVE NOISE SE NSITIVIT Y AN D EXCEPTIO NAL TIMES FOR PERCOLATIO N

647

LS ](S) Xs(w), then

1l[J(w,) I(w,,) ] =

(5,2)

L ](S)' exp(-r lSI) . s

Let 1/ (w ) be as in Theorem 4.4 . Fix some 1 E (0.1] and let r E [2 , R). Clearly, 0 ::: (w) ::: 10 (w) 12~ (w) for every w. Consequentl y,

1l

Il[JOR (w,) lOR (w,,)] " 1l[J0' (w,) I,~ (w, ) fo' (w" )f,~(w,,)] 1l[J0' (w,) 10' (w,,)]1l[J,~ (w,) I,~(w,,)] " 1l[J0' (w,) ]1l[J,~ (w,) I,~ (w,,)].

P[V" R n V", R] = =

(To obtain the second equality, we have used the independence on disjoint sets of sites.) Thus,

P[V" R n V" ,R] " a (r) 1l[J,~ (w,) I,~(w,, ) ] = a(r)

L e-' I S I ],~ (S)' S

°

The latter sum restricted to 5 with 151 = k for fixed k =j:. is estimated using Corollary 4.5 , while for k = 0, we use j2~(0) = a(2/", R). Thi s yields

P[V" R n V" ,R] "a(r) (a(2 r, R)' + r,( I )

L e- k , k a(2 r , R )' a , (r)). 00

k= !

Lk=l

It is easy to check that k e- k l :s: 0(t - 2). Thi s and the inequaliti es (3.1) and (3.2) allow us 10 wri le thi s estimate as

(5.3 )

P[ V"R

n V",R] "

' + r ,(I ) r-' a,(r)) .

O( I ) a(R)' a(,r (I

We proved the above claim for all r E (0, 1] and r E [0 . R) but now we observe Ihal (5.3) is also Iri vially true when r ~ R as well. We now choose,. = 2,-8 = 2 1s - s'I- 8. Applyi ng Ihis in (5.3) with (3.3) and (3.5) gives

P[V" R n V" ,R] " 0(1) a( R)'ls _ s'I - 5 / 6 +,( 1)

(5.4 ) Hence, (5 .5)

J.' J.'

P[ V"R n V" ,R] ds ds' " 0(1) a(R)'-

The Cauchy-Schwarz inequality lells us that

Il[ X]' p[x > 0] ::: Il[X' ] ' Consequentl y, the above inequality, the fact that E[ X] = a(R), the expression (5.1) for Il[ X'] and (5.5) show that infR >Op[ X R > 0] > O. Let TR := {r E 10, 11 : V" R holds}. Fatou's lemma tell s us that with positive probability TR =j:. 0 for 419

648


infinitely many R E N. Since T R :> T R' when R' > R , this impli es that

p[n R>O {TR

¥ 0 }] > O.

Our goal is to show that p[n RT R ::j:. 0] > O. Since the TR'S are not closed sets, n R>O {TR '# 0 } does not immedi ately imply TR '# 0 . (The reason that TR is not necessarily closed is that the set of times at which an edge is open is nol a closed set since we have a right continuous process.) Thi s technicality is taken

nR

care of by the fo llowing lemma from f 131 . L EMMA 5.1 (f 131). Let 0

< p < I and let G be any graph where

1tp

«(5) = O.

Let {WI} rep resent ollr dynamical percolation process in that Wr (v) is the state of vertex v at time 1. Consider the process {wr} obtained from {WI} by seltillg,jor every vertex v. the set {t : W, (u) = I } to be the closure of the set {t : WI (v) = I}. Theil \1# p-a.s.,for every vertex V {t E [0, 00) : v percolates in WI} = {t E [0 , 00) : v percolates in WI}' III particular, 0 .5 . this set of times is closed.

Returning to our proof, let TR be the closure of T R. It is easil y checked that

n

TR ={tE[O , I] :

°

percolates in wl},

R>O where {WI} is defin ed as in Lemma 5. 1. By compactness, if the TR'S are all nonempty, it fo llows that T R is nonempty. This implies that there is some time at which WI percolates and hence by Lemma 5. 1, some time at whi ch the original process WI percolates. 0

nR

For future reference. we note that Lemma 5. 1 implies that a.s.

n

(5.6)

TR =

R >O

n

TR.

R>O

6. Hausdorff dimension of exceptional times In thi s section, we prove Theorem 1.9. Thi s is separated into two theorems, Theorem 6. 1 and T heorem 6.3, where lower and upper bounds are given. We point out however that the lower bound is simply a refin ement of the argument for proving that there ex ist exceptional times . First note the fact that the Hausdorff dimension is an almost sure constant foll ows immedi ately from ergodici ty. TH EO REM 6 .1. As stich, the Hau.5dorjJ dimension of the set of exceptional

times is at least 1/ 6. Proof. Fi x y < 1/6. It suffi ces by ergodicity and countable additi vity to show that with pos iti ve probability, the set of exceptional times in 10. I] at which the 420

QUANTITATIVE NO ISE SENS ITIVITY AND EXCEPTIONAL T IMES FOR PERCOLATION

649

origin percolates has Hausdorff dimension at least y. For each integer R , let, as before, V( ,R be the event that at time l there is a path from the origin to distance R away and define a random measure OR on [0 , 1] by aR(S) = a/R)

is

I VLRdt

for each Borel set 5 C [0, I]. The results in the previous section immedi ately give that E[lloRII] = I and E[lIoRII 2] ::: 0(1) where IIORII denotes the total variation of the measure oR. Cauchy·Schwarz gives

E[lIaRII,] ' / 2 P[lIaRIl > I /Z]"'::: E[ lIaRIIl ]]GR ]]> ' /'] ::: E[lIaRIlI - I /Z = I /Z. Consequently, P[IIORII > m on [0, I] and y >0, let

1/2] ~ C 1 for some constant C 1 > 0.

Given a measure

'liy (m) = f f it -sl-Y dm(t) dm(s) . Note that

E['Ii

(aR)]

= E[ [' ['

Y

1010

daR(t)daR(s)] It -slY

r'

= ['

P[V"R n V"R] dt ds. 10 10 a(R)'lt -slY

Therefore, by (5.4) and y < 1/6,

C, := sup E['li y (a R)]
I /Z} n {'liy (aR) ~ C,T} ,

by the choice of T, we have that

By Fatou's lemma,

P[I;m sup UR] ::: C,/Z . R ~oo

We now show that on the event lim sup R U R , the Hausdorff di mension of the set of percolating times in [0. I] is al least y . Let T R again be the closure of the set of times in [0, 1] at which there is a path from the ori gin to di stance R away. Clearl y OR is supported on TR. By (5.6), it suffices to prove that nR > O TR has Hausdorff dimension alleast y on the event lim suPR UR. Th is is achieved in the fo ll ow ing 0 (deterministic) lemma, which completes the proof. 421

650

ODED SC IIR AMM and JE FFREY E. STE Il'

L EMMA 6.2 . Let DI :2 D2 ;2 D 3 .. . be a decreasing sequence of compaCI

subsets of [0, 1J, alld let j.q , fJ.2, ... be a sequence of positive measures lVith J-Ln supported 011 Dn . SI/ppose that there is a constallt C sitch that fo r infin itely man)' vailies of fl, (6. 1) Then/he Ha usdorff dimension

ainn

Dn is at feas t y.

Proof C hoose a sequence of integers {Ilk} for w hich (6.1) holds. Note that IItJnk f ::: < 8y (J1.nk)!: C. By compactness. choose a further subsequence {II~} of {Ilk} so that tLnk converges weakl y 10 some positive measure JLo.::i' C learly J.Loo is supported on Dn and III-Looll ~ 1/ C. For all M , we have that

II Ix -

nn

yl -Y

A

M dJ.l.oo(x) dJ.l.oo(Y) = k_oo lim

II Ix -

yl - Y A M dJ.l.,'k (x) dJ.l.,'k (y)"

c.

Now let M -» 00 and apply the monotone convergence theorem to conclude that

II Ix -

yl -Y dJ.l.oo(X) dJ.l.oo(Y ) "c.

S ince IIttoo II > 0, it now follows from Frostman's theorem (see fo r example. ll7 J) that the Hausdorff dimension of Dn is at least y. 0

nn

T HEOREM 6.3 . As sllch, the Hausdorff dimension of the set of exceptional times is at most 3 1/36.

Proof Let Vn be the event that there is a time in [0. 1/ 11] fo r which the o rigin percolates. Since the set of vertices which are open for some t E [0, 1/11] is an LLd. process with density 1/ 2 + (1 _e- I / (2n ») / 2 :S: 1/ 2 + 1/ 11 . it is immediate that

p[U,l " "l+;\('I6o). where (60 is the event that the origi n percolates. By page 3 of [311 , for every there is a C so that

£;

> 0,

(6.2)

Now let

, Nn = L I Uj.,,' j= 1

where

Uj,11

is the event that there is a time in [(j - 1) / 11. j/II] for which the origin

percolates (so that VI,n = Vn above). By the above, we have that E[ N n ] :s: ell * +e. 422

QUANTITATIVE NO ISE SENSI TI VITY AND EXCEPTIONAL TIMES FOR PERCOLAT ION

It fo llows that

lim n

E[N,] 1/ *+ 2£

and so from Fatou's lemma, we gel E [lim in f n

Therefore

= 0

;n1

1/ .

+ 2£

N lim inf 11 n n

65 1

11 36+2e

= O.

= 0

a.s. Thi s says that a.s. for infinitely many 11, the set of exceptional times in [0, I] at which the orig in percolates can be covered by 1/ 1i+ 2e intervals of length 1/1/. Hence, the Hausdorff dimension of the set of these exceptional times is at most ~! + 28 a.s. By countable additi vity, we are done. 0 Remark 6.4. The upper bound wi ll be proved again by a different argumen t when we prove Theorem 1.16. The above proof is included here, because it is shoner. One should nonetheless point out that the above argument uses (6. 2), while the argument below is more self-contained.

7. Exceptional times for k-arm events In thi s section, we give the proofs of the lower bounds in Theorems 1. 15 and 1. 16. but generall y omit those detail s which are the same as in the corresponding proofs of Theorems 1.3 and 6. 1. For > 0 and integer k ~ I, let A~e (r, R) be the event that we have k disjoint crossings of alternating colors (with black most clockwise) between distances rand R of the ori gin in We and let afvtO (r, R) = p[ AfvtO (r , R)]. if ,. is suppressed, then it is taken to be 10k. We will , of course, need the asymptotics of a~e (r. R). For this purpose, conforma l invariance will be used. Although when > 2rr the surface We is not planar and conformal invariance is usually stated for planar domain s. the proof of conformal invari ance cenainly holds in this setting. The asy mptotic decay as R /r -;. 00 of the probability of k di sjoint crossings between di stances rand R in We from the ori gin in the percolation sca ling limit is detennined by confonnal invariance. SpeCifically, the map Z 1---+ z7r /e maps We to the upper half-pl ane, and we may conclude from (3.8) that the decay (for the percolation scaling limit) is of 1fk~k ± 1) ( ) the form (R / r ) & +0 1 • as R / r -;. 00 while k stays fi xed . Then, one can conclude, as for the other exponents we have di scussed, that for R ~ r ~ 10k,

e

e

(7. 1) 423

652

ODED SC I'IRAMM and JEFFREY E. STEf F

as RI r --+ 00 w hile k is fi xed . by the argument in [3 1J. We will also use the fact that the quasi-multiplicativity relari ons (3. 1) and (3.2) hold fo r a~o and fo r U2. This is proved in the appendix; see Remark A.6. Proof of the lower bOl/lld ill Th eorem 1. 15. We first handle the case k = I and the refo re abbreviate te mporarily u :V/r , R ) by a Wa(r, R). (A different approac h will be needed for k ~ 2.) Let D be the unio n of the hexagons in We that contain points whose distance from the orig in is in [r, R]. Let R D and r D denote the set of po ints in aD that are at d istance::: R (respectively. ::: r) from the o ri gin. Also, we denote by aO D and BD, the components of aD n awo that are at angle about o(respecti vely, about 8) in radi al coordinates on We. The algorithm we use to determine if there ex ists a crossin g of D is essenti ally the same as the algorithm determining the ex istence of a left to ri ght crossing of a D square, where URD plays the ro le o f the rig ht side of the square and where pl ays the role of the left side of the square. (Thi s is of course crucial; if we reversed things, then the hexagons near the inner c ircle would be revealed with too hig h a probability.) It is clear that this algorithm works and so we now need to compute its revealme nt. We will show that the revealment is

a

a

a

ur

(7.2)

0 (1) a,(r) aw, (r, R).

Using (7. 1) and (3.6), one can show that thi s is essentiall y (i .e., up to some 0 ( 1) factor) monotone decreasing in r in the relevant range () > 8 Jr f3. We just look at the fi rst interface ari sing in the a lgorithm, the one which terminates whe n it hits r D u e D, since the estimates fo r the second interface will be essentially the same. Fix some hexagon H C D . Let 5 = dist(H , uRD uao D U{O}) with odenoting the orig in . We also use Ipi to de note distance from 0 in We. We di stin guish several di fferent cases.

a

a

Case I : dist(H , {O}) = s. For H to be visited , we need o ur 2-arms event holding withi n d istance 5 f2 of H and a c rossing of the desired color between d istance 25 and di stance R from the origin . T hese are independent and we get that H is visited with probability at most a2(sf 2)awo(2s , R). By the analogues of (3. 1) and (3.2) for a2 and awo' this is compatible with our claimed revealment (7.2). Ca.5e 2: dist(H , UR D)=s. As in the proof of T heorem 4.4 with c:=dist(po , H ) 1\ ( R f2), we obtain

p[H

visited

I pol" 0 ( 1) a + (2s + 0 ( 1) , c - s -

0 ( 1)) a,(s/ 2).

Proceeding as in that proof, we see that thi s is also compatible w ith o ur claimed revealment (7.2) . 424

QUAN TITATIVE NO ISE SENSI TI VITY AND EXCEPTIONAL TIME S FOR PERCOLAT ION

653

Case 3: dist(H ,aOD) = s. Let w eao D so th at dist(H , w) =S. We separate Case 3 into three subcases. Case 3(a): s ::: Iw1/2. Then the triangle inequality gives di st(H , O) ~ 3s. For H to be visited, we need our 2-arms event holding within di stance sl 2 of H and a path of the desired type between distance 4s and distance R fro m the origin. These are independent and we get that H is visited with probability at most a2(sI2) awo (3s , R) , which is compatible with our claimed revealment (7.2). Case 3(b): s ~ Iwl/2 ~ R 14. For H to be visited, we need our 2-arms event holdi ng within di stance s 12 of H , a white crossing in the half-annulus centered at w with outer radius Iwl and inner radius 2s (which is identical to a half-annulus in a half-pl ane; Iwl ~ R I 2 guarantees that the above half-annulus does not intersect JRD) and a white crossing between di stance 21wl and di stance R fro m the origin. These are independent and we get that H is visited with probability at most

a,(s/2) a +(2s.lwl) aw, (2Iwl . R). Since up to an 0(1) factor, a2(s) a+ (s , Iwl) is increasi ng in s, the product of the first two terms is at most 0 ( I)a2(lwl) and since Iwl::: 2s::: r, the whole product is at most O(I)a,(r)aw, (r. R). Case 3(c): Iwl ::: R 12; s ~ Iw1/2. For H to be visited, we need our 2-arms event holding within distance s/2 of H and if 2s < d(po , w) it is also necessary that a white cross in g occurs between distances 205 and d(po , w) /\ Iwl from w. (Note that the latter event takes pl ace in the upper half-pl ane.) These are independen t and since Iwl::: R I2 we get

P[H visited

Ipol :o O( I)a+(2s. d(po .w)/\(R/2))a,(s/2).

As in Case 2, thi s is compatible with (7.2). Thi s covers all possible cases, and hence establishes that the revealment is as claimed. We now proceed to di scuss the algorithm and the revealment when k > I. It turns out si mplest in fact to modify the event A1v (r, R) as foll ows. Partition the o outer boundary JRD into k arcs of roughly equal di ameter YI , Y2... .. Yk (ordered counterclockwise) and let o (r, R) be the event that for every odd (respecti vely, even) i E {I , 2, . .. , k} there is a black (respectively, white) crossing in D from J r D to Yi . Thus. instead of look ing at the set of times for which At,/:I (ro , R) occurs

A1v

(where ro = 10 k. say), we will look at the set of times at which Clearly,

A1vo (ro , R) occurs.

A1vo (r, R) C At,o(r, R), and therefore this is justified. We wi ll also use 425

654

ODED SOI RAMM and JEFFREY E. STEIF

the relation (7.3)

fo r some constant Cf, depending only on k and 8, which ho lds by Remark A.7. If Y ea R D is an arc, let A Hr, R) (respectively, Ayl (r , R) be (he event that there is a white (respectively, black) crossi ng from Y to arD in D . Suppose that for each i = 1, 2, ... ,k, we have a partiti on Yi = Yj,+ U Yi,- of Yi into two arcs Y; ,+ and Y;,_ , Since A ~. I(r, R) = A~/+ (r, R ) U At/- (r , R) , we have

A w, (" R) =

(7.4)

n

U

-k

k

. ( 1)'

Ay~.,; (', R) .

ye{ _,+ }k i = !

The algorithm starts out by picking points Xi E Vi. randomly, un iformly and independently. Then Y;,+ and Yi ,_ are chosen as the two components of Yi \ {Xi }. For each of the 2k possible Y E {- , + }k , the algorithm then proceeds to detennine if the corresponding component

n k

A(y):=

.

AH )' (" R)

i= ]

Y")'i

of (7.4) has occurred. For that purpose, interfaces are started at each of the points Xi, and are followed until the event has been determined one way or the other. (Of course, the in terface wi ll have e ither white on the left and black o n the right or vice versa, depending on the color of crossing it is meant to detect and whether the correspond ing arc Yi ,± is to the left or right of Xi.) However, the order in which the inte rfaces are extended is somewhat important. A simple rule that works is that among the hexagons necessary to extend the k interfaces o ne more step, the algori thm chooses the one that is farth est away from O. The event A(y) is decided positively o nly if all k interfaces reach arD. The revealment of this a lgorithm is at most k 2k times the maximum probabi lity that the inte rface, started at Xi, visits a hexagon H before the determi natio n of the correspond ing A(y) is tenn inated. Here, the max imum is over all hexago ns H C D and all i E {I , 2, ... ,k}. The correspondi ng bound is attained as in the case k = 1. but now a tvo is replaced bya fvo' Our rule of thumb for selecting w hic h interface to extend guarantees that we never exam ine a hexagon H un less (di st(O, H ) + 0(1) , R) occurred. As in the case k = I, when es-

A1vo

timat ing the revealment it is important that a2(r, R) .::: O(l)afvo(r, R). In the range () > 4Jr k (k + 1) /3, w hic h is the relevant range for the lower bound in Theorem 1. 15. this fo llows frolll (7.1) . The remainder of the proof goes through as before. D 426

QUANTITAT IVE NOISE SENS ITIV ITY AN D EXCEPTIO NAL T IMES FOR PER COLATION

655

Proof of 10IVer bOllnd in Theorem 1.16. Here we simpl y say that the proof for the lower bounds in Theorem 1.15 can be carried out in a similar way. In fact, for k ::: 2, the proof is simpler topologicall y than the k = 1 case for the plane, since we do not need to worry about interfaces making complete circuits around the origin (if thi s ever happens, the event in question cannot occur and we stop the algorithm). 0

8. Upper bounds for k-arm times The following result, which will be useful for the proofs of the upper bounds in Theorems 1.15 and 1. 16, is abstract: the graph stmcture does not pl ay any role. Let A be an event involvin g inde pendent Bernoulli ( 1/ 2, 1/ 2) random variables Xl , X2, . .. , X m . Recall that the influe nce of the index i on A , denoted l i(A), is the probability that Xi is pi votal; namely, that changing the value of Xi changes whether A occurs or not. The sum of the influences is denoted by I ( A) = Li Ii (A).

TH EO REM 8.1. Let {Adn :! 1 be some .~equellce of events in {O.l}v , each depending on olily fin itely mally coordinates. Assume that lim n --+ oo P[AII ] = O. Let {wr } be the Markov process 0 11 {O, I} v where indepelldently O's go to I at rate 1/2, I 's go to at rate 1/ 2 and started according to its stationary distribution Jr , . Let T be the set of exceptiollal times 1 at which Wr E nll~ 1 An. If lim infll -40oo I(AII) < 00, then T = 0 a.s. Othen vise, the Hausdorff dimellsioll of T is a.s. at most

°

(8. 1)

. . (

11m mf 111-4000

IOg P[Ad) - '

() log I All

.

Proof Let Til := {t E [0, I]: WI E Ad, let aTII be the boundary points of Tn in (0, I) and set Nn:= la Tn l. We cl aim that (8.2)

E[ Nn ] = I (A n) / 2.

To see thi s, write Nil = L vN;( where N;( counts the number of eleme nt s in aTII at which time the vertex v flipped . We now need to show that, for each vertex v, E[N;( 1 is I v( An)/ 2. Gi ven a time interval [t, r + d t ], the probability that there is a is equal to I v( An) dt / 2+0(dt) time point in the interval which contributes to and the probability of k ::: 2 such time points is clearly O (dtk). From thi s, (8.2) easily follows. For any e > 0, let T,f be the e-neighborhood of Tn intersected with [0, 1]. Since T: ~ Til U Ux ear" [x - e. x + ej,

N::

(8.3)

where f.L de notes Lebesgue measure. For any set U and e > 0, let X (U, e) de note the number of e intervals needed to cover U. From the above, using the fact that 427

656


r;

the intervals comprisin g all have length at least E, it follow s that .N(T; , e) :::: 2 M(T;) £- 1, and so, llsin g (8 .3), .K(Tn . e) :::: .K(T: , e) :::: 2J-L(T,,} £ -

1

+ 4 Nfl .

Therefore, by Fubini 's theorem and (8.2), E[X (T" £)] ,, 2 P[A,] £- 1 + 2/(A,).

(8.4)

We temporaril y assume that lim infn _ oo leAn} = which goes to 0 as 1/ -+ 00. By (8.4). we have

(8.5)

00.

Let an =

p[ An] / I(An) ,

E[X (T, G,)] " E[X (T"G ,)] " 4/(A, ).

By pass ing to a subsequence if necessary. we assume with no loss of generality that the lim io f in (8. 1) is a limit. Let L denote the value of that limit. It is ele mentary to check that for every e > 0, for all sufficiently large",

I(A »)L H I(A ) < ( - ' , - P[A,] This together with (8.5) implies that the Hausdorff dimension of T is at most L + E a.s. As e is arbitrary, this completes the proof in the case I (An) -+ 00. Since TIJ f 0 impli es that NIJ ~ I or TIJ :2 (O.I), it follow s by (8.2) and

Markov's inequality that

Thus, T = 0 a.s. when lim inflJ I (AIJ) = O. The case liminflJ I{AIJ) E (O,oo) requires a differe nt argumenl. Let elJ = / P[AIJ1 . By (8.4) , we have lim inflJ-+oo E(N{TIJ.elJ)] < 00. Since EIJ ~ 0, the cardinality ITI of T is bounded from above by lim inflJ -+oo .N"{TIJ ,EIJ) ' Fatou 's le mma yields E[ IT I] < 00 and hence p[ IT I < 00] = I. We fin all y conclude that P IT ,. 01 = by comb;n;ng [ II , Th. 6.7J and [ 10, (2.9)J. 0

°

Proof of Theorem 1.16. Since the lower bounds have been establi shed in Sect ion 7, it remains to prove the upper bounds. Fix k = I or k > I even. Let AR be the event th at there are k disjoint crossings of the annulus DR := {z E C() : 10k ::: Izl ::: R} , where we require that the colors be alternating if k f I. Here, Izl denotes the di stance to 0, wh ich is the apex of the cone C(). One can prove that k = I,

(8.6)

k> I ,

in the very same way that we have justifi ed (7. 1). By Theorem 8.1 (and easy algebraic manipulation), it therefore suffices to prove that (8.7) 428

QUANTITATIVE NO IS E SENS ITI VITY AND EX CE PTI ONAL TIMES FO R PERCOLATION

657

Let H be a hexagon in Ce, and let 5 = s(H) be the distance from H to O. For H to be pivotal it is necessary that there be k di sjoint (alternating, if k > I) crossin gs from di stance 10 k to 5/2 from the origin (unless 5/ 2 ~ 10k) and between di stances 2s and R (unless 2s ~ R). Likewi se, there should be four alternating cross ings between H and di stance (5/ 2) /\ dist(H , aD R) fro m H. These events are independen t Using the quasi-multiplicati ve property of the k-arm crossing events (Remark A.6) and (3.7) with k = 4, thi s gives (when 5 < 8 R / 9), (8.8)

IH(AR) " O(I ) P(AR] S-SI4+o (l )

where this 0(1) factor (as well as those appearing below) may depend on k and 8 . Since the number of hexagon s in Ce satisfying 5 = 5( H) < p is 0(p2) , an easy calcul ation yields

I:

IH(AR ) " O ( I)P(AR] R '/Ho( I).

H :s( H) < 8 R / 9

Now suppose that H is a hexagon satisfying s(H) ~ 8 R /9. For H to be pivotal , it is necessary that there be k (alternating, if k > I) crossi ngs in Ce between {Izl = 10k} and {Izl = R / 2} , there should be four alternating crossings between H and di stance di st(H , iJDR) / 2 from H , and there should be three alternating crossings between di stance 2 dist(H, aD R) and di stance R / 2 from a point on iJD R closest to H. The latter event is governed by the 3-ann half-plane exponent, whose asymptotic behaviour is described by (3 .8). Since s + dist(H , aD R) = R + 0(1), we get

Si nce for b ~ I there are O(b R) hexagons at di stance easy calculation gives

I:

~

b from

{Izl =

R}, another

IH(AR) " O(I)P(AR]R 3/ 4+o(l).

H :s( H) ;::8R /9

Together, thi s yields (8.7) and the proof is complete.

o

Proof of Theorem 1.15. The lower bound was proved in Sect ion 7, and so onl y the upper bound needs to be justifi ed. The proof proceeds like the proof of the upper bound in Theorem 1.16. except that the influence estimates are slightl y different. Let DR = {z E we: 10k ~ Izl ~ R} , AR be the k-ann event in We between {z : Izl = 10k} and {z : Izl = R}, and H CDR be a hexagon. lei s=s(H) = dist(O, H) , and let b = b(H) = di st(H , iJDR), where we write aDR for the boundary of DR in Coo ; i.e. , the points on aWe are included. For H to be pivotal for AR , it is necessary that the k-arm event hold s between distance 10k and s/2 from 0 429

658

ODED

SC ~ IRAMM

and J EFFREY E. STEfF

(unless 5/2::::: 10 k) as well as between distances 2s and R (unl ess 2s::: R), that the altern ating 4-ann event holds between H and di stance bl2 away from H , and that the a ltern ating 3-arm event must hold between d istances 2b and s / 4 away from a po int in aD R closest to H (unless 2b:::: s/4). There are O(b's') hexagons H sati sfying b (H ) ::::: b' and s(H ) ::: s'. The rest of the proof proceeds like that of

Theorem 1.16, and is left to the reader.

0

Proofof Theorems 1. 10, 1. 12, 1.1 3 and 1. 14. At any time at which there are two infinite white cl usters in the plane, we must also have the 4-ann event occurring (w ith altern ating co lors) but by T heorem 1.16 (with k = 4 and = 2rr). there are

e

no such times. Thi s proves Theorem 1. 10.

AI any t'i me at which there are two in fi nite d ifferent colored clusters, we must also have the 2-arm event occurring (with d iffe rent col ors) but by Theorem 1.16 (with k = 2 and () = 2Jr), the set of such times has Hausd orff dimensio n at most 2/3. This proves Theorem I 12. The other two theore ms are similarly proved. 0

9. The square lattice We start thi s secti on by proving Theorem 1. 11. Afterwards, possible ways in whi ch our arguments for T heore m 1.3 may be improved to appl y to 71.. 2 as well , will be di scussed. In the proof of Theorem 1. 11 we wil1use the fact that the six alternating arms exponent is larger than 2, or, more preci sely, that the probability fo r six alternating arms between radii I' and R is bounded above by 0( \ ) (rl R)2+e fo r some E > O. Thi s is essenti all y d ue to [20. Lemm a 5 1, but a proof is also given in the append ix (Corollary A. 8).

Proof of Th eorem I. I I. For 0 < I' < R , let S(I', R ) be the event that there are three differe nt c luste rs that connect the circles of radi i I' and R about O. By the above mentioned bo und on the altern ating 6-ann probabilities, We may choose some fixed e > 0 and some functi on p = per) > r such that fo r static critical bond percolation on 71.. 2 , for all 1' ,

p[ 5(1', p) ] "p->-< .

(9. 1)

Consider some bo nd e, and let F(e) be the event that e is pivotal for S(r, p). Then P[ F(e)] is j ust the influence 1,(5(1', pl). Assume that P[ F(e) ] '" O. Note that the events F(e) and {e is open} are independent events. Thi s implies that P[ S(r.p) F (e) ] = 1(2, which one may wri te

I

p[ F(e) n S(r, p) ] = p[F (e) n ~S (r. p)]. S ince thi s applies to every bond e, we conclude that the expected number of pivotals on the event S(I', p) is half of the total influence I (S(I', p». However the number 430

QUANTITAT I VE NO ISE SENSI TIV I TY AND EXC EPTIO NA L TIMES FOR PERCOLATION

659

of pivotals for S(r, p) is bounded by the total number of edges intersecting the disk of radius p about the origin , which is certai nly 0(p2). Thus,

1(5( r, p)) '" 2 p[ 5(r, p) ] O(p' ) = O(p-' ), Con sequentl y, by Theorem 8.1 , fo r every ro > 0 a.s. there are no exceptional times in which nr>ro S(r. p(r» holds. This proves ou r theorem. 0 Remark 9. 1. An alternative way to prove the above result is based on using the fa ct that the 6-arm exponen t is strictl y larger than 2 together with the fact that the number of different configurations (counti ng repetitions) that appear in a ball of rad ius II d uring the time interval [0, I] has a Poisson distribution with a parameter which is at most 0 ( 1)1/ 2 . As we wi ll briefl y ex plai n below, the proof of Theorem 1.3 al most works fo r bond perco lation on the square grid . In fact , there are several alternative routes by which the result might perhaps be extended to 71. 2 ; ( I) Establi shing (9,2)

for 71. 2 for some fi xed

E

> 0,

(2) improvi ng the estimate (1.3), (3) proving the ex istence of an algorithm (or a witness whic h would still permit the use of Theorem 1.8) with smaller revealment. (4) extending Smirnov 's theorem to 71. 2 . Note that the weake r vers io n of (9 .2) (12(r) ::: (1(r)2 fo ll ows from e ithe r the Harri s-FKG inequality or Reimer's inequality [27J. Kesten and Zhang have proved some related strict inequaliti es between exponents [211 , but it seems that their methods are not sufficient to prove (9.2). We now explain why (9.2) in the 7L 2 setting implies exceptio nal ti mes fo r 7L 2 . First we want to have the revealment for the algorithm determi ning bounded by 0(1) (12(r) ex(I", R). One problem seems to be that the bound on the revealment for the tri ang ul ar grid involves the summand featurin g (1 +, wh ic h is relat ively negligible, while on 7L 2 , we do not know how to prove that the other summand dominates. The fix is to replace the determ in istic R by a random R ~ E [R , 2R] . The random variable R ~ will depend on some ex tra random bits, that we add, and these random bits also evolve in ti me. We construct the depende nce of R' on these bits so th at R' can be calcul ated by an a lgorithm with very small revealment. Thi s is rather easy to arrange, because we are not limited in the number of bits that we may take. If we consider an edge whose di stance from the

1/

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ODED SC I'IRAMM and J EFFRE Y E. STEf F

origi n a is in the range [RI2,2Rj, then the probability that the edge is examined given R' is at most O(I)az(R'-a) IW >a - ]. By (A. I), thi s is at most O(I)a2(R)a2(R' -a , R)- l lR'>a- l. The pr;bability that IR' -a l ::: 2 j is at most 0(1) 2j I R. We al so knowt hat 0'2(r\ , r2) - 1 :::: O( 1) (rz/ r1) 1- e' for some £' > 0, by Reimer's inequality [271 and (A.S) . It fo llows that the probability that such an edge is examined is 0(1) C12(R). The r Q(1) fac tor in Theorem 4.4 is easi ly

replaced by an 0( 1) factor, if we use Proposit ion A. I in the course of the proof. Then we get (5 .3) for the square grid , but without the 1'0(1) facto r. We may then choose the dependence betwee n rand t such that a(r) ~ t ,.e/ 2, where £; is the constant in (9.2). The rest is immediate from (5.3), since clearly r - E/ 2 ::: 0(1) (E' fo r some 8' > O. A consequence of thi s argument, which appli es without assuming (9 .2), is th at for bond percolation on 7L 2 we have

P[ V"R n VO ,R] :': O(t - I ) P[V"RJ'. This gives yet another illustration as to how close the result for 71. 2 seems to be - if the ( - I tenn was improved to ( - I +t , that would have been enough. Consequently, significant improvements in the algorithm or in (1.3) wou ld also be suffic ient. 10. Some open questions Foll owing are a few questions and open problem s suggested by the present paper. ( I) For the results in Theorems 1. 15 and 1.1 6, what is the Hausdorff dimension of the set of exceptional times in question? We tend to believe that the answer is the upper bound. In particular, is the upper bound of 31/36 in Theorem [.9 the correct answer? (2) Prove that there ex ist exceptional times for percolat ion on the sq uare lattice (see Section 9 for a di scussio n). (3) What is the best y for which Theorems 1.6 and 1.7 ho ld? (4) What is the best revealment of an algorithm determining the even t QR in Theorem 4.1 ? (5) What is the sharp form of Theorem 1.8? (6) What are the properties of the infinite cl uster at an exceptional time at which it ex ists? For example, what is the g rowth rate of the number of vertices in the Euclidean di sk of radius r around the orig in which belong to the cluster of the o rig in at the first time ( :: 0 in which the c luster is infinite? Is the growth rate the same at all exceptional times? 432

Q UAN TITAT IVE NOISE SENS ITI VI T Y AND EXCEPTIONAL TIMES FOR PERCOLATION

661

(7) What is the relati onshi p between the exceptional infi nite cluster and the incipient infi nite cluster? Note added in proofs. Questi ons ( I), (2), and pan of (3) have been answered by C. Garban, G. Pete, and O . Schramm . Question (7) has been answered by A. Hammond, G. Pete. and O. Schramm .

Appendix A. Quasi-multiplicativity In this appendix, we discuss the k-arm probabi lities and prove that they satisfy the corresponding analogue of the relation (3. 1). For R > 0, let H R be the union of the hexagons intersecting 8(0. R). For R > r > 0 let Aj (r, R) denote the event that there are at least j cross ings from oH r to aHR , of alternating colors. The fo llowing result refers to critical site percolation on the triangul ar grid and critical bond percolation on the square grid . PROPOSIT tON A. I. Let j > 0 be even. There i !i a cOIISta1lt on j, stich that for all r < r' < rl/

e, depending only

p[ Aj(r, r") ] :s p[ Aj(r, r') ] p[ Aj(r' , r") ] :s C p[Aj (r, r") ], and p[Aj (r, 2 r )] > lie if P[ Aj(r, 2 r)] > 0 (i.e., if r is large enough to allow j

(A 1)

C- '

crossings). Moreover, a corresponding statement I/Olds for critical bond percolation 011 the squa re grid which alternate between primal and dual crossings.

T his theorem would have been a useful tool in [3 11, had it been avai lable. In stead, the authors of that paper proved a weaker form of th is which was good enough for their purposes. Our proof below uses techn iques from l 19J, l24J and l3 1J. Indeed, the entire resul ts of the appendix do fo llow from the ideas of l 19J. We include them here fo r compl eteness, and for ease of reference. Additionall y. though the bas ic ideas are the same, in several respects our treat ment is a bit different

from [19] . Below, we wi ll work in the setting of the triangular grid . The proof for the square grid is essentiall y the same. In the setting of the triangular grid , there is the color exchange trick [201 . [II . which shows that the probability for havi ng altern at ing cross ings is comparable to the probab il ity of any color sequence as long as both colors are present. In the setting of the square grid, as far as we know, such a trick does not ex ist. At the end of the appendi x we wi ll explain how the proof of Proposition A.I can be generali zed to any color sequence. In the foll owing, an interface from aH r to oH R is an ori ented sim ple path in the hexagonal grid that has one color of hexagons adj acent to it on the right, and the opposite color adjacent to it on the left. Thus, it is the common bou ndary of a black crossing and a white crossing. 433

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OD ED SCHRAMM and JEFFREY E. STEIF

au

For R > ,. > I. consider all (he interfaces cross ing from r to aH R. and define s(r, R) to be the least di stance between any pair of e ndpo ints of two interfaces on aH R. If there are no interfaces, we take s(r, R) = 00. Note that s(r, R) is monotone nonincreasing in r. This quantity will roughl y measure the "quality" of the interfaces; when s(r, R) is comparable to R , the interfaces are well separated , and, as we wi ll see, easier to eX lend. LEM MA A.2. For all a E (0, 1), R > 0, 8 > 0,

P[s(a R, R)

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