Random Walks on Infinite Groups

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.

First textbook devoted solely to random walks on infinite, nonabelian groups Integrated treatment of measure-theoretic probability and random walk theory First textbook to treat Kleiner's approach to Gromov's classification theorem for groups of polynomial growth

Auteur

Steven P. Lalley is professor Emeritus at the Department of Statistics at the University of Chicago. His research includes probability and random processes, in particular: stochastic interacting systems, random walk, percolation, branching processes, combinatorial probability, ergodic theory, and connections between probability and geometry.

Résumé

"This book is about symmetric random walks on finitely generated infinite groups and consists of fifteen chapters followed by an appendix on measure and probability theories. It also offers good accounts on the theories of Markov chains valued in countable spaces and discrete-time martingales." (Nizar Demni, Mathematical Reviews, May 8, 2024)

Contenu
1 First Steps.- 2 The Ergodic Theorem.- 3 Subadditivity and its Ramifications.- 4 The Carne-Varopoulos Inequality.- 5 Isoperimetric Inequalities and Amenability.- 6 Markov Chains and Harmonic Functions.- 7 Dirichlet's Principle and the Recurrence Type Theorem.- 8 Martingales.- 9 Bounded Harmonic Functions.- 10 Entropy.- 11 Compact Group Actions and Boundaries.- 12 Poisson Boundaries.- 13 Hyperbolic Groups.- 14 Unbounded Harmonic Functions.- 15 Groups of Polynomial Growth.- Appendix A: A 57-Minute Course in MeasureTheoretic Probability. <p