Название: Random Walks on Infinite Groups
Автор: Steven P. Lalley
Издательство: Springer
Серия: Graduate Texts in Mathematics
Год: 2023
Страниц: 373
Язык: английский
Формат: pdf (true), epub
Размер: 20.1 MB

This text presents the basic theory of random walks on infinite, finitely generated groups, along with certain background material in measure-theoretic probability. The main objective is to show how structural features of a group, such as amenability/nonamenability, affect qualitative aspects of symmetric random walks on the group, such as transience/recurrence, speed, entropy, and existence or nonexistence of nonconstant, bounded harmonic functions. The book will be suitable as a textbook for beginning graduate-level courses or independent study by graduate students and advanced undergraduate students in mathematics with a solid grounding in measure theory and a basic familiarity with the elements of group theory. The first seven chapters could also be used as the basis for a short course covering the main results regarding transience/recurrence, decay of return probabilities, and speed. The book has been organized and written so as to be accessible not only to students in probability theory, but also to students whose primary interests are in geometry, ergodic theory, or geometric group theory.

The first seven chapters form the core of the book; they could be used as the basis of a short course in the subject. These chapters deal with the most basic properties of random walks: transience/recurrence, speed, entropy, decay of return probabilities, and the relation between harmonic functions and hitting probabilities. They also introduce some of the most important tools of the trade: the ergodic theorems of Birkhoff and Kingman, Markov operators, Dirichlet forms, and isoperimetric inequalities. Finally, they culminate, in Chapter 7, in one of the subject’s landmark achievements, Varopoulos’ classification of recurrent groups.

Chapters 8–10 are devoted to the study of bounded harmonic functions. Among the highlights of the theory are Blackwell’s characterization of such functions (Chapter 9) and the theorem of Avez, Derriennic, and Kaimanovich & Vershik showing that a random walk has the Liouville property if and only if it has Avez entropy 0 (Chapter 10). In Chapter 11, I discuss group actions and their use in the study of random walks. Here I introduce the important notion of a boundary, and show how one important special case, the Busemann boundary, figures in Karlsson and Ledrappier’s characterization of speed. Chapter 12 lays out the basic facts about Poisson boundaries, including Kaimanovich and Vershik’s entropic characterization of Poisson boundaries.

The last three chapters deal with several more specialized topics: hyperbolic groups (Chapter 13); unbounded harmonic functions (Chapter 14), and (iii) Gromov’s classification of groups of polynomial growth (Chapter 15), a special case of which is used in the proof of Varopoulos’ growth criterion for recurrent groups in Chapter 7. The latter two chapters are independent of Chapters 8–13; Chapter 15 depends logically on the main result of Chapter 14, but could be read separately.

I have included a large number of exercises, especially in the early chapters. These are an integral part of the exposition. Some of the exercises develop interesting examples, and some fill crevices in the theory; many of the results developed in the exercises are used later in the book.

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