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Random Walk: A Modern Introduction

Gregory F. Lawler and Vlada Limic
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics 123
[Reviewed by
Miklós Bóna
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The title is misleading in two ways. First, there is nothing "introductory" about this book. It is a high-level reference book primarily intended for researchers. Anyone without at least an advanced graduate student's knowledge of the field will find the material difficult to read. Definitions that are below that level are not announced in the book. Second, the subject of the book is a special kind of random walk, namely the random walk corresponding to increment distributions with zero mean and finite variance.

That said, the authors make a serious effort to make the difficult material readable for the expert audience that will use the book. Sometimes they stop the course of formal proofs for a plain English discussion of ideas, or explain why a seemingly simpler approach would not work. This reviewer appreciated these parts, and would have liked to see even more of them.

There are relatively few exercises, and none have their solutions included in the book, which is perhaps not surprising for a book intended as a reference material. See the table of contents for more details about what is in the book.

Miklós Bóna is Professor of Mathematics at the University of Florida.

Preface; 1. Introduction; 2. Local central limit theorem; 3. Approximation by Brownian motion; 4. Green's function; 5. One-dimensional walks; 6. Potential theory; 7. Dyadic coupling; 8. Additional topics on simple random walk; 9. Loop measures; 10. Intersection probabilities for random walks; 11. Loop-erased random walk; Appendix; Bibliography; Index of symbols; Index.