random walk - a modern introduction - g. lawler, v. limic (cambridge, 2010) ww

This page intentionally left blank CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 123 Editorial Board B. BOLLOBÁS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO Random Walk: A Modern Introduction Random walks are stochastic processes formed by successive summation of indepen- dent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling. Gregory F. Lawler is Professor of Mathematics and Statistics at the University of Chicago. He received the George Pólya Prize in 2006 for his work with Oded Schramm and Wendelin Werner. Vlada Limic works as a researcher for Centre National de la Recherche Scientifique (CNRS) at Université de Provence, Marseilles. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board: B. Bollobás, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: http://www.cambridge.org/series/sSeries.asp?code=CSAM Already published 73 B. Bollobás Random graphs (2nd Edition) 74 R. M. Dudley Real analysis and probability (2nd Edition) 75 T. Sheil-Small Complex polynomials 76 C. Voisin Hodge theory and complex algebraic geometry, I 77 C. Voisin Hodge theory and complex algebraic geometry, II 78 V. Paulsen Completely bounded maps and operator algebras 79 F. Gesztesy & H. Holden Soliton equations and their algebro-geometric solutions, I 81 S. 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Soliton equations and their algebro-geometric solutions, II 115 E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics 116 D. Applebaum Lévy processes and stochastic calculus (2nd Edition) 117 T. Szamuely Galois groups and fundamental groups 118 G. W. Anderson, A. Guionnet & O. Zeitouni An introduction to random matrices 119 C. Perez-Garcia & W. H. Schikhof Locally convex spaces over non-Archimedean valued fields 120 P. K. Friz & N. B. Victoir Multidimensional stochastic processes as rough paths 121 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Representation theory of the symmetric groups 122 S. Kalikow & R. McCutcheon An outline of ergodic theory Random Walk: A Modern Introduction GREGORY F. LAWLER University of Chicago VLADA LIMIC Université de Provence CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-51918-2 ISBN-13 978-0-511-74465-5 © G. F. Lawler and V. Limic 2010 2010 Information on this title: www.cambrid g e.or g /9780521519182 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org eBook ( EBL ) Hardback Contents Preface page ix 1 Introduction 1 1.1 Basic definitions 1 1.2 Continuous-time random walk 6 1.3 Other lattices 7 1.4 Other walks 11 1.5 Generator 11 1.6 Filtrations and strong Markov property 14 1.7 A word about constants 17 2 Local central limit theorem 21 2.1 Introduction 21 2.2 Characteristic functions and LCLT 25 2.2.1 Characteristic functions of random variables in R d 25 2.2.2 Characteristic functions of random variables in Z d 27 2.3 LCLT – characteristic function approach 28 2.3.1 Exponential moments 46 2.4 Some corollaries of the LCLT 51 2.5 LCLT – combinatorial approach 58 2.5.1 Stirling’s formula and one-dimensional walks 58 2.5.2 LCLT for Poisson and continuous-time walks 64 3 Approximation by Brownian motion 72 3.1 Introduction 72 3.2 Construction of Brownian motion 74 3.3 Skorokhod embedding 79 3.4 Higher dimensions 82 3.5 An alternative formulation 84 v vi Contents 4 The Green’s function 87 4.1 Recurrence and transience 87 4.2 The Green’s generating function 88 4.3 The Green’s function, transient case 95 4.3.1 Asymptotics under weaker assumptions 99 4.4 Potential kernel 101 4.4.1 Two dimensions 101 4.4.2 Asymptotics under weaker assumptions 107 4.4.3 One dimension 109 4.5 Fundamental solutions 113 4.6 The Green’s function for a set 114 5 One-dimensional walks 123 5.1 Gambler’s ruin estimate 123 5.1.1 General case 127 5.2 One-dimensional killed walks 135 5.3 Hitting a half-line 138 6 Potential theory 144 6.1 Introduction 144 6.2 Dirichlet problem 146 6.3 Difference estimates and Harnack inequality 152 6.4 Further estimates 160 6.5 Capacity, transient case 166 6.6 Capacity in two dimensions 176 6.7 Neumann problem 186 6.8 Beurling estimate 189 6.9 Eigenvalue of a set 194 7 Dyadic coupling 205 7.1 Introduction 205 7.2 Some estimates 207 7.3 Quantile coupling 210 7.4 The dyadic coupling 213 7.5 Proof of Theorem 7.1.1 216 7.6 Higher dimensions 218 7.7 Coupling the exit distributions 219 8 Additional topics on simple random walk 225 8.1 Poisson kernel 225 8.1.1 Half space 226 Contents vii 8.1.2 Cube 229 8.1.3 Strips and quadrants in Z 2 235 8.2 Eigenvalues for rectangles 238 8.3 Approximating continuous harmonic functions 239 8.4 Estimates for the ball 241 9 Loop measures 247 9.1 Introduction 247 9.2 Definitions and notations 247 9.2.1 Simple random walk on a graph 251 9.3 Generating functions and loop measures 252 9.4 Loop soup 257 9.5 Loop erasure 259 9.6 Boundary excursions 261 9.7 Wilson’s algorithm and spanning trees 268 9.8 Examples 271 9.8.1 Complete graph 271 9.8.2 Hypercube 272 9.8.3 Sierpinski graphs 275 9.9 Spanning trees of subsets of Z 2 277 9.10 Gaussian free field 289 10 Intersection probabilities for random walks 297 10.1 Long-range estimate 297 10.2 Short-range estimate 302 10.3 One-sided exponent 305 11 Loop-erased random walk 307 11.1 h-processes 307 11.2 Loop-erased random walk 311 11.3 LERW in Z d 313 11.3.1 d ≥ 3 314 11.3.2 d = 2 315 11.4 Rate of growth 319 11.5 Short-range intersections 323 Appendix 326 A.1 Some expansions 326 A.1.1 Riemann sums 326 A.1.2 Logarithm 327 A.2 Martingales 331 A.2.1 Optional sampling theorem 332 viii Contents A.2.2 Maximal inequality 334 A.2.3 Continuous martingales 336 A.3 Joint normal distributions 337 A.4 Markov chains 339 A.4.1 Chains restricted to subsets 342 A.4.2 Maximal coupling of Markov chains 346 A.5 Some Tauberian theory 351 A.6 Second moment method 353 A.7 Subadditivity 354 Bibliography 360 Index of Symbols 361 Index 363 [...]... perturbations of random walks, or with particle systems and other models that use random walks as a basic ingredient, often need more precise information on random walk behavior than that provided by the central limit theorems In particular, it is important to understand the size of the error resulting from the approximation of random walk by Brownian motion For this reason, there is need for more detailed... coupling a random walk and a Brownian motion on the same probability space, and give error estimates The dyadic construction of Brownian motion is also important for the dyadic coupling algorithm of Chapter 7 Green’s function and its analog in the recurrent setting, the potential kernel, are studied in Chapter 4 One of the main tools in the potential theory of random walk is the analysis of martingales... that T has an exponential distribution with parameter λ, i.e P{T > λ} = e−λ Let Fn denote the σ -algebra generated by Fn and the events {T ≤ t} for t ≤ n Then {Fn } is a filtration, and Sn is a random walk with respect to Fn Also, given that Fn , then on the event that {T > n}, the random variable T − n has an exponential distribution with parameter λ We can do similarly ˜ for the continuous-time walk. .. ♣ In many ways the main focus of this book is simple random walk, and a first-time reader might find it useful to consider this example throughout.We have chosen to generalize this slightly, because it does not complicate the arguments much and allows the results to be extended to other examples One particular example is simple random walk on other regular lattices such as the planar triangular lattice... denote the σ -algebra generated by the union of the Ft for t > 0 If Sn is a random walk with respect to Fn , and T is a random variable independent of F∞ , then we can add information about T to the filtration and still 1.6 Filtrations and strong Markov property 15 retain the properties of the random walk We will describe one example of this in detail here; later on, we will similarly add information without... Preface xi argument introduces the reader to a fairly general technique for obtaining the overshoot estimates The final two sections of this chapter concern variations of one-dimensional walk that arise naturally in the arguments for estimating probabilities of hitting (or avoiding) some special sets, for example, the half-line In Chapter 6, the classical potential theory of the random walk is covered... the random walk This will not affect the distribution of the random walk, provided that this extra information is independent of the future increments of the walk A (discrete-time) filtration F0 ⊂ F1 ⊂ · · · is an increasing sequence of σ -algebras If p ∈ Pd , then we say that Sn is a random walk with increment distribution p with respect to {Fn } if: • for each n, Sn is Fn -measurable; • for each n... research As has been gradually discovered by a number of researchers in various disciplines (combinatorics, probability, statistical physics) several objects inherent to a graph or network are closely related: the number of spanning trees, the determinant of the Laplacian, various measures on loops on the trees, Gaussian free field, and loop-erased walks We give an introduction to this theory, using an approach... that if k ≤ d and L is a k-dimensional lattice in Rd , then we can find a linear transformation A : Rd → Rk that is an isomorphism of L onto Zk Indeed, we define A by A( xj ) = ej where x1 , , xk is a basis for L as in the proposition If Sn is a bounded, symmetric, ∗ irreducible random walk taking values in L, then Sn := ASn is a random walk with increment distribution p ∈ Pk Hence, results about walks... walks on Zk immediately translate to results about walks on L If L is a k-dimensional lattice in Rd and A is the corresponding transformation, we will call | det A| the density of the lattice The term comes from the fact that as r → ∞, the cardinality of the intersection of the lattice and ball of radius r in Rd is asymptotically equal to | det A| r k times the volume of the unit ball in Rk In particular, . page intentionally left blank CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 123 Editorial Board B. BOLLOBÁS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO Random Walk: A Modern Introduction Random. Szamuely Galois groups and fundamental groups 118 G. W. Anderson, A. Guionnet & O. Zeitouni An introduction to random matrices 119 C. Perez-Garcia & W. H. Schikhof Locally convex spaces. symmetric groups 122 S. Kalikow & R. McCutcheon An outline of ergodic theory Random Walk: A Modern Introduction GREGORY F. LAWLER University of Chicago VLADA LIMIC Université de Provence CAMBRIDGE