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

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.

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Frank Spitzer has had a profound influence on probabilIty theory over the past 30 years. On the occasion of his sixty-fifth birthday, as an ex- pression of appreciation and a demonstration of the depth and breadth of his ideas, a group of his friends, colleagues, students, and coauthors agreed to contribute this collection of articles in his honor. By way of an introduction, a selection of Professor Spitzer's seminal papers have been reprinted to show the points of origin for much of the research presented in this book. Basic properties of Brownian motion, fluctuation theory, random walks and interacting particle systems are presented in these papers. The latter topic in particular is presently one of the most exciting and active areas in probability, providing fruitful interaction between statistical physicists and probabilists. Table of Contents About the Cover . . . v Frank Spitzer's Students vi Preface vii Contents . viii Bibliography x REPRINTS OF FRANK SPITZER 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 66 PAPERS DEDICATED TO FRANK SPITZER A Useful Renormalization Argument Maury Bramson and Lawrencc Gray 113 Capture Problems for Coupled Random Walks Maury Bramson and David Griffeath 153 Nonlinear Voter Models J. T. Cox and R. Durrett 189 On the Long Term Behavior of Finite Particle Systems: A Critical Dimensional Example J. T. Cox and Andreas Greven . . . . . . . 203 Large Deviation Lower Bounds for General Sequences of Random Variables A. deAcosta, P. Ney and E. Nummelin 215 Asymptotic Laplace- Transforms H. Dinges . . . . . . 223 Higher Order Hydrodynamic Equations for a System of Independent Random Walks R. Dobrushin and F. Sokolovskii . . . . . . 231 Making Money Prom Fair Games R. Durrett, H. Kesten and G. Lawler 255 Additive Functionals of SuperdijJusion Processes E.B. Dynkin . . . . . . . . . . 269 Interacting Systems, Stirrings, and Flows T .E. Harris .......... 283 The One-Dimensional Stochastic X - Y Model Richard Holley . . . . . . . . . 295 Relations Between Solutions to a Discrete and Continuous Dirichlet Problem Harry Kesten ......... 309 On the Connected Components of the Complement of a Two-Dimensional Brownian Path Jean-Francois Le Gall . . . . . . . . . . 323 The periodic Threshold Contact Process Thornas M. Liggett . . . 339 Bounds on the Critical Exponent of Self-Avoiding Polygons Neal Madras . . . . . . 359 Spitzer's Formula Involving Capacity Sidney C. Port . . . . . . 373 An Integral Test for Subordinators William E. Pruitt .... 389 Microcanonical Distribution, Gibbs States, and the Equivalence of Ensembles Daniel W. Stroock and Ofer Zeitouni 399 Power Counting Theorem in Euclidean Space Norma Terrin and Murad S. Taqqu 425 Etude asymptotique des nombres de tours de plusieurs mouvemr;nts browniens complexes cOTleles Marc Yor . . . . . . . . . . . . . . . . . . . 441