The IMA Volumes in Mathematics and Its Applications Volume 8 Series Editors Geroge R. Sell Hans Weinberger Institute for Mathematics and Its Applications IMA The Institute for Mathematics and Its Applications was established by a grant from the National Science Foundation to the University of Minnesota in 1982. The IMA seeks to encourage the development and study of fresh mathematical concepts and questions of concern to the other sciences by bringing together mathematicians and scientists from diverse fields in an atmosphere that will stimulate discussion and col laboration. The IMA Volumes are intended to involve the broader scientific community in this process. Hans Weinberger, Director George R. Sell. Associate Director IMA Programs 1982-1983 Statistical and Continuum Approaches to Phase Transition 1983-1984 Mathematical Models for the Economics of Decentralized Resource Allocation 1984-1985 Continuum Physics and Partial Differential Equations 1985-1986 Stochastic Differential Equations and Their Applications 1986-1987 Scientific Computation 1987-1988 Applied Combinatorics 1988-1989 Nonlinear Waves Springer Lecture Notes from the IMA The Mathematics and Physics of Disordered Media Editors: Barry Hughes and Barry Ninham (Lecture Notes in Mathematics, Volume 1035, 1983) Orienting Polymers Editor: J. L. Ericksen (Lecture Notes in Mathematics, Volume 1063, 1984) New Perspectives in Thermodynamics Editor: James Serrin (Springer-Verlag, 1986) Models of Economic Dynamics Editor: Hugo Sonnenschein (Lecture Notes in Economics, Volume 264, 1986) Harry Kesten Editor Percolation Theory and Ergodic Theory of Infinite Particle Systems With 47 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Harry Kesten Institute for Mathematics and Its Applications Minneapolis, Minnesota 55455, USA AMS Classification: 60K 35 82 A43 Library of Congress Cataloging in Publication Data Percolation theory and ergodic theory of infinite particle systems. (The IMA volumes in mathematics and its applications ; v. 8) Bibliography: p. 1. Percolation (Statistical physics) 2. Ergodic theory. I. Kesten, Harry, 1931- II. Series QCI74.85.P45P48 1987 530.1'595 87-9512 © 1987 by Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1s t edition 1987 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, 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 of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Permission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Springer-Verlag New York Inc. for libraries registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970. U.S.A. Special requests should be addressed directly to Springer Verlag New York, 175 Fifth Avenue, New York, New York 10010, U.S.A. 1987 $0.00 + .2u 987654321 ISBN-13: 978-1-4613-8736-7 e-ISBN-13: 978-1-4613-8734-3 DOl: 10.1007/978-\-4613-8734-3 The IMA Volumes in Mathematics and Its Applications Current Volumes: Volume 1: Homogenization and Effective Moduli of Materials and Media Editors: Jerry Ericksen, David Kinderlehrer, Robert Kohn, and J ,-L. Lions Volume 2: Oscillation Theory, Computation, and Methods of Compensated Compactness Editors: Constantine Dafermos, Jerry Ericksen, David Kinderlehrer, and Marshall Slemrod Volume 3: Metastability and Incompletely Posed Problems Editors: Stuart Antman, Jerry Ericksen, David Kinderlehrer, and Ingo Muller Volume 4: Dynamical Problems in Continuum Physics Editors: Jerry Bona, Constantine Dafermos, Jerry Ericksen, and David Kinderlehrer Volume 5: Theory and Applications of Liquid Crystals Editors: Jerry Ericksen and David Kinderlehrer Volume 6: Amorphous Polymers and Non-Newtonian Fluids Editors: Constantine Dafermos, Jerry Ericksen,and David Kinderlehrer Volume 7: Random Media Editor: George Papanicolaou Volume 8: Percolation Theory and Ergodic Theory of Infinite Particle Systems Editor: Harry Kesten Forthcoming Volumes: 1985-1986: Stochastic Differential Equations and Their Applications Hydrodynamic Behavior and Interacting Particle Systems Stochastic Differential Systems, Stochastic Control Theory and Applications 1986-1987: Scientific Computation Computational Fluid Dynamics and Reacting Gas Flows Numerical Algorithms for Modern Parallel Computer Architectures Numerical Simulation in Oil Recovery Atomic and Molecular Structure and Dynamics CONTENTS Foreword. • • • • • • • . • • • • • . • . • . • • • • • • • • • . • • • • • • • . • • • • . . . • • . • • • • • • . • • • • • • • • . . . • • •• i x Preface ••••.•••••..••.••..•.•.•.•..••...•.•....••...•••.....•.•••.•...••••••• xi Rapid Convergence to Equil ibrium of Stochastic Ising Models in the Dobrush i n Sh I osman Reg ime •••••• , .••••••.•.•.•.•.•• 0 ......................... . M. Aizenman and R. Holley Uniqueness of the Infinite Cluster and Related Results in Percolation ••.•••.. 13 M. Aizenman, H. Kesten and C.M. Newman Survival of Cyclical Particle Systems ........................................ 21 M. Bramson and David Griffeath Expansions in Statistical Mechanics as Part of the Theory of Partial Differential Equations ....................................................... 31 D. C. Brydges The Mean Field Bound for the Order Parameter of Bernoull i Percolation •.•.••.• 49 J.T. Chayes and L. Chayes Recent Results for the ~tepping Stone Model .•.•..••.••.•••••.•..••••••....••. 73 J. Theodore Cox and David Griffeath Stochastic Growth Models ..................................................... 85 R. Durrett and R.H. Schonmann Random Walks and Diffusions on Fractals ...................................... 121 Sheldon Goldstein The Behavior of Processes with Statistical Mechanical Properties .••••••..••.• 131 Lawrence Gray Stiff Chains and Levy FI ight: Two Self Avoiding Walk Models and the Uses of Their Statistical Mechanical Representations ......................... 169 J.W. Halley One Dimensional Stochastic Ising Models ...................................... 137 Richard Holley A Scaling Relation at Criticality for 2D-Percolation ••.••.•••••••••.•.•.••... 203 Ha rry Kes ten Revers i b I e Growth Mode I s on Zd: Some Examp I es .............................. :.213 Thomas M. Liggett Inequal ities for y and Related Critical Exponents in Short and Long Range Perco I a t ion •.•••••••.•.•.•....••..•••..•.••......•.•.•••.•........ 229 C.M. Newman A New Look at Contact Processes in Several Dimensions ....•.•.•....•.•...•.••. 245 Roberto H. Schonmann Fractal and Multifractal Approaches to Percolation: Some Exact and No-So-Exact Results ....••..•..•..•••••...•...•..•.••..••...... 251 H. Eugene Stanley Surface Simulations for Large Eden Clusters .•.•.••..••••.••...••.....•.....•• 301 D. Stauffer Dual ity for k-Degree Percolation on the Square Lattice ..••..•.•...•••...•.... 311 John C. Wierman FOREWORD This IMA Volume in ~athematics and its Applications PERCOLATION THEORY AND ERGODIC THEORY OF INFINITE PARTICLE SYSTEMS represents the proceedings of a workshop which was an integral part of the 19R4-85 IMA program on STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS We are grateful to the Scientific Committee: naniel Stroock (Chairman) Wendell Fleming Theodore Harris Pierre-Louis Lions Steven Orey George Papanicolaoo for planning and implementing an exciting and stimulating year-long program. We especially thank the Workshop Organizing Committee, Harry Kesten (Chairman), Richard Holley, and Thomas Liggett for organizing a workshop which brought together scientists and mathematicians in a variety of areas for a fruitful exchange of ideas. George R. Sell Hans Weinherger PREFACE Percolation theory and interacting particle systems both have seen an explosive growth in the last decade. These suhfields of probability theory are closely related to statistical mechanics and many of the publications on these suhjects (especially on the former) appear in physics journals, wit~ a great variahility in the level of rigour. There is a certain similarity and overlap hetween the methods used in these two areas and, not surprisingly, they tend to attract the same probabilists. It seemed a good idea to organize a workshop on "Percolation Theory and Ergodic Theory of Infinite Particle Systems" in the framework of the special probahility year at the Institute for Mathematics and its Applications in 1985-86. Such a workshop, dealing largely with rigorous results, was indeed held in February 1986. These proceedings contain contributions of almost all the speakers at this meeting. As usual some of the papers are not very close to the talk given by t~e same author. We hope that nevertheless these proceedings give the reader a good impression of the state of the art in 1986. It is an indication of the success of the gathering that several joint papers here report on work which was finished (or even begun) during the workshop. I am grateful to S. Orey and O. Stroock who coordinated the probability year at the IMA for including percolation theory and infinite particle systems in the program. The organizing committee of the workshop consisted of T.E. Harris, R. Holley, T. Liggett and myself. am much obliged to my fellow committee members for their help and support. I also wish to thank the directors of the IMA, H. Weinberger and G. Sell, for their hospitality and help in making the workshop a success. Last but not least thank the staff of the IMA for taking care of practical matters in such a way that things ran smoothly, and Kaye Smith and Patricia V. 8rick for their excellent typing and preparation of these proceedings. Harry Kesten Ithaca, N.Y., Oct 1q86