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Full Description
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.
Contents
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 Measure-Theoretic Probability.
