Progress in Probability Volume 28

Series Editors Thomas Liggett Charles Newman Loren Pitt

Random Walks, Brownian Motion, and Interacting Particle Systems A Festschrift in Honor of Frank Spitzer

Rick Durrett Harry Kesten Editors

Springer Science+Business Media, LLC

Rick Durrett Department of Mathematics Cornell University White Hau Ithaca, NY 14853

Harry Kesten Department of Mathematics Cornell University White Hall Ithaca, NY 14853

Library of Congress cataloging-in-publication data

Random walks, Brownian motion, and interacting particle systems / Rick Durrett, Harry Kesten, editors.

p. cm. — (Progress in probability : v. 28) Festschrift in honor of Frank Spitzer. Includes bibliographical references and index. ISBN 978-1-4612-6770-6 ISBN 978-1-4612-0459-6 (eBook) DOI 10.1007/978-1-4612-0459-6 1. Random walks (Mathematics) 2. Brownian motion processes.

3. Probabilities. 4. Statistical physics. 5. Spitzer, Frank, 1926-I. Durrett, Richard, 1951- II. Kesten, Harry, 1931-

in. Spitzer, Frank, 1926- . IV. Series: Progress in probability

Printed on acid-free paper. © Springer Science+Business Media New York 1991 Originally published by Birkhäuser Boston in 1991 Softcover reprint of the hardcover 1st edition 1991 Al l rights reserved. No part of this publication may be reproduced, stored in a retrieval

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: 28. QC174.85.R37R37 1991 519.2-dc20

91-31173 CIP

ISBN 978-1-4612-6770-6

Typset by authors in TeX.

9 8 7 6 5 4 3 2 1

ABOUT THE COVER

The editors wish to thank Robert Fisch and David Griffeath for the graphic on the cover. The graphic depicts convergence to equilibrium in a system of particles which perform random walks under the "exclusion" restriction that there may be at most one particle per lattice site. Initially the particles are packed in a tight ball and successive "still frames" show the state of the system at times 10, 100, 1000 and 10000 (on a 256 x 256 lattice with wrap-around) .

The picture was produced using the hardware and software of T. Toffoli and N. Margolus. A Matrix film recorder, funded by the National Science Foundation, made the transparency used in the printing process.

STUDENTS OF FRANK SPITZER

1957 J. W. Lamperti, On the asymptotic behavior of recurrent and almost­recurrent events.

1964 W. W. Whitman, Some strong laws for random walks and Brownian motion.

1965 J. C. Mineka, The existence and uniqueness of positive solutions to the Wiener-Hopf equation with positive kernel.

1969 R. A. Holley, The motion of a heavy particle in an infinite one-dimen­sional gas of hard spheres

1972 F. C. Solomon, Random walks in a random environment.

1974 K. G. Logan, Time reversible solutions in statistical mechanics. R. L. Thompson, Equilibrium states on thin energy shells.

1976 J. T. Cox, Entrance laws for Markov chains and one-dimensional diffusions. D. S. Griffeath, Coupling methods for Markov processes.

1977 L. F. Gray, Controlled spin-flip systems. A. Kleinerman, Limit theorems for infinitely divisible random fields.

1979 C.-T. Hsiao, The stochastic time evolution of Gaussian interacting systems.

1984 N. N. Madras, A process in a randomly fluctuating environment.

PREFACE

This collection of articles is dedicated to Frank Spitzer on the occasion of his 65th birthday. The articles, written by a group of his friends, colleagues, former students and coauthors, are intended to demonstrate the major influence Frank has had on probability theory for the last 30 years and most likely will have for many years to come. Frank has always liked new phenomena, clean formulations and elegant proofs. He has created or opened up several research areas and it is not surprising that many people are still working out the consequences of his inventions.

By way of introduction we have reprinted some of Frank's seminal articles so that the reader can easily see for himself the point of origin for much of the research presented here. These articles of Frank's deal with properties of Brownian motion, fluctuation theory and potential theory for random walks, and, of course, interacting particle systems.

The last area was started by Frank as part of the general resurgence of treating problems of statistical mechanics with rigorous probabilistic tools. It is presently one of the most exciting and active areas in probability, and is one of the reasons for the fruitful interaction between statistical physicists and probabilists nowadays. About one-third of the papers in this collection are in this area. The variety of topics of these papers illustrates how far the subject has developed over the last fifteen years. Among the subjects of the other articles directly related to Frank's work are potential theory of Brownian motion, winding of one or more two-dimensional Brownian motions and aspects of random walk. Despite the age of these topics, they keep generating new questions.

We regret that limitations of space prevented us from inviting more of Frank's friends to contribute to this Festschrift. Frank always enjoyed explaining his ideas to others and has been generous in sharing them and attracting coauthors. We are sure we speak for all mathematicians who have worked with Frank or who have been influenced by his work, when we thank him for introducing us to beautiful mathematics.

Rick Durrett Harry Kesten

Table of Contents

About the Cover ...

Frank Spitzer's Students

Preface .

Contents

Bibliography

REPRINTS OF FRANK SPITZER

v

vi

vii

. viii

x

A combinatorial lemma and its application to probability theory 3

Some theorems concerning 2-dimensional Brownian motion 21

Recurrent random walk and logarithmic potential . . 33

Electrostatic capacity, heat flow and Brownian motion 53

Interaction of Markov processes

PAPERS DEDICATED TO FRANK SPITZER

A Useful Renormalization Argument Maury Bramson and Lawrence Gray

Capture Problems for Coupled Random Walks Maury Bramson and David Griffeath

Nonlinear Voter Models J. T. Cox and R. Durrett

On the Long Term Behavior of Finite Particle Systems: A Critical Dimensional Example

J. T. Cox and Andreas Greven . . . . . . .

Large Deviation Lower Bounds for General Sequences of Random Variables

A. deAoosta, P. Ney and E. Nummelin

Asymptotic Laplace-Transforms H. Dinges ...... .

viii

66

113

153

189

203

215

223

Higher Order Hydrodynamic Equations for a System of Independent Random Walks

R. Dobrushin and F. Sokolovskii . . . . . .

Making Money from Fair Games R. Durrett, H. Kesten and G. Lawler

Additive FUnctionals of SuperdiJJusion Processes E.B. Dynkin . . . . . . . . . .

Interacting Systems, Stirrings, and Flows T .E. Harris ......... .

The One-Dimensional Stochastic X- Y Model Richard Holley . . . . . . . . .

Relations Between Solutions to a Discrete and Continuous Dirichlet Problem

Harry Kesten ........ .

On the Connected Components of the Complement of a Two-Dimensional Brownian Path

Jean-Fran~is Le Gall . . . . . . . . . .

The periodic Threshold Contact Process Thomas M. Liggett . . .

Bounds on the Critical Exponent of Self-Avoiding Polygons

Neal Madras . . . . . .

Spitzer's Formula Involving Capacity Sidney C. Port ..... .

An Integral Test for Subordinators William E. Pruitt ....

Microcanonical Distribution, Gibbs States, and the Equivalence of Ensembles

Daniel W. Stroock and Ofer Zeitouni

Power Counting Theorem in Euclidean Space

231

255

269

283

295

309

336

339

359

373

389

399

Norma Terrin and Murad S. Taqqu . . . . . . . . 425

Etude asymptotique des nombres de tours de plusieurs mouvements browniens complexes correles

Marc Yor . . . . . . . . . . . . . . . . . . . 441

ix

PUBLICATIONS OF FRANK SPITZER

(THE ARTICLES MARKED WITH AN ASTERISK

ARE REPRINTED IN THIS VOLUME.)

1. On a clIIU of random variables, Proc. Amer. Math. Soc. 8 (1955),494-505.

2. On interval m:vm:nt BUmI of independent random vCll'iobles, Proc. Amer. Math. Soc. 7 (1956), 164-171.

*3. A combinotoriollemmo ond its opplictJtion to probobility thllOfll, TraDII. Amer. Math. Soc. 82 (1956), 323-339.

4. The Wiener-Hopf equotion whose kernel is 0 probobility density, Duke Math. J. 24 (1957), 327-343.

*5. Some theorems concerning 2-dimensionol Brownion motion, TraDII. Amer. Math. Soc. 87 (1958), 187-197.

6. (with A. Calderon and H. Widom), Inversion of Toeplitz mAtrices, Dlinois J. Math. 3 (1959), 490-498.

7. Some probobilitylimit ~mI, Bull. Amer. Math. Soc. 85 (1959), 117-119.

8. A Thuberion theorem ond its probability interpretotion, TraDII. Amer. Math. Soc. 94 (1960), 150-169.

9. The Wiener-Hopf equAtion whose kernel is a probobility density, II, Duke Math. J. 27 (1960), 363-372.

10. (with C. Stone), A cllJ88 of Toeplitz fOfTfl8 ond their opplicotion to probability, D1inois J. Math. 4 (1960), 253-277.

11. (with P. Schmidt), The Toeplitz motrices of on orbitrary LAurent pol!lnomiol, Math. Scand. 8 (1960), 15-38.

12. Some properties of recum:nt random wol1c, Dlinois J. Math. 5 (1961), 234-245.

13. (with H. Widom), The circumference of 0 conllt% polygon, Proc. Amer. Math. Soc. 12 (1961), SOIh'i09.

*14. Recum:nt random wolle ond /ogtlrithmic potentiol, Proc. 4th Berkeley Symposium on Probability and Statistics II (1961), 515-534.

15. (with H. Kesten and D. Ornstein), A general property of random wolle, Bull. Amer. Math. Soc. 88 (1962), 526-528.

16. Hitting probobilities, J. Math. Mach. 11 (1962), 59:Hi14.

PUBUCATIONS OF FRANK SPITZER

17. (with H. Kesten), Ratio theorems for random walk8, I, J. Analyse Math. XI (1963), 285-322.

18. Principles of random walk, D. Van Nostrand Co., 1964; 2nd edition, Springer Verlag, 1976; Russian translation Mir, Moscow 1969; French translation Dunod, Paris,1970.

*19. Electrostatic capacity, heatjlow, and Brownian motion, Z. Wahrsch. verw. Gebiete 3 (1964),110-121.

20. (with H. Kesten), Random walk on countably infinite abelian groups, Acta Math. 114 (1965), 237-265.

21. (with P. Ney), The Martin boundary for random walk, Trans. Amer. Math. Soc. 121 (1966), 116-132.

22. (with H. Kesten and P. Ney), The Galton Watson process with mean one and finite variance, Teor. Veroyatnost i Primenen 11 (1966), 579-611.

23. Promenades aleatoires recurrentes sur les groupes discrets, Universite de Strasbourg, 1966.

24. Two explicit Martin boundary constructions, Symposium on Probability Methods in Analysis (Loutraki, 1966), Springer Lecture Notes in Math. 31 (1967), 296-298.

25. Renewal theorems for Markov chains, Proc. 5th Berkeley Symposium on Probability and Statistics II, part 2 (1967),311-320.

26. (with A. Joffe), On multitype branching processes with p ~ I, J. Math. Anal. and AppJ. 19 (1967), 409-430.

27. Uniform motion with elastic collision of an infinite particle system, J. Math. Mech. 18 (1969), 973-989.

28. Random processes defined through the interaction of an infinite particle system, Probability and Information Theory (Proc. International Symposium, McMaster University, Hamilton, Ontario, 1968), Springer Lecture Notes in Math. 89 (1969), 201-223.

*29. Interaction of Markov processes, Adv. Math. 5 (1970), 246-290.

30. Markov random fields and Gibbs ensembles, Amer. Math. Monthly 78 (1971), 142-154.

31. Random fields and interacting particle systems, Notes on lectures given at the 1971 MAA Summer Session, Williams College, Williamstown, Mass., Mathematical Association of America, Washington, D.C ..

32. A variational characterization of finite Markov chains, Ann. Math. Statist. 43 (1972), 303-307.

33. Contribution to discussion of ·Subadditive ergodic theory- by J. F. C. Kingman, Ann. Probab. 1 (1973), 883-909.

34. Recurrent random walk of an infinite particle system, Trans. Amer. Math. Soc. 198 (1974), 191-199.

35. Introduction aw: processus de Markov a parametre dans ZII' Lectures given at Saint Flour Summer School 1973, Springer Lecture Notes in Math 390 (1974), 114-189.

36. Random time evolution of infinite particle systems, Proc. Intern. Congress of Math. (Vancouver 1974) 2, 169-171.

37. Markov random fields on an infinite tree, Ann. Probab. 3 (1975), 387-398.

38. (with H. Kesten), Controlled Markov chains, Ann. Probab. 3 (1975),32-40.

PUBLICATIONS OF FRANK SPITZER

39. RAndom time evolution 01 infinite particle 1JI8tem.!, Adv. Math. 18 (1975), 139-143 ; Also appeared in Surveys in Applied Mathematics, (N. Metropolis, S. Orazag and G.-C. Rota eels.), Academic Press, 1976.

40. Phase transition in one-dimensionAl nearut-neighbor .tenu, J. Thnct. Anal. 20 (1975), 240-255.

41. (with H. Kesten and M. V. Kozlov), A limit low lor "'Mom tDGlk in A "'Mom enllironment, Com­positio Math. 30 (1975), 145--168.

42. (with H. Wan, Jr.), The chAructerizAtion 01 optimlllsAlIing ProgNms in A quAdrAtic model, J. Math. Econom. 3 (1976), 43--79. '

43. StochAstic time evolution of one dimensionAl infinite pArticle 'l/8tenu, Bull. Amer. Math. Soc. 83 (1977),~.

44. Contribution to ducussion of -AbhAngige Venchiebungen, 111-, Serdica" (1978), 135-163.

45. (with H. Kesten), A limit theorem relAted to A new clo88 of ,eq-similor proce88e8, Z. Wahrsch. YelW.

Gebiete 50 (1979), 5--25.

46. Infinite IJIICems with locAlly inteNcting components, Ann. Probab. 9 (1981), 349-364.

47. (with T. M. Liggett), Ergodic theorems for coupled ",ndom tDIJlks And other 8ystems with locAll" interActing components, Z. Wahrsch. YelW. Gebiete 58 (1981),443--468.

48. (with H. Kesten), Convergence in dUtribution of product. 01 random mAtricu, Z. Wahrsch. YelW.

Gebiete 87 (1984), 363-386.

49. A multidimensionAl renewAl theorem, Probability, statistical mechanics, and number theory (G.-C. Rota, ed.), Academic Press, 1986, pp. 147-155.