Overview
- 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
Part of the book series: Graduate Texts in Mathematics (GTM, volume 297)
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About this book
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.
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Keywords
- Random walk
- Random walk textbook
- Random walk on finitely generated group
- Poisson boundaries of random walks
- Carne-Varoploulos inequality
- Isoperimetric inequalities
- Amenable groups
- Dirichlet's Principle
- Markov chains and harmonic functions
- Bounded harmonic functions
- Unbounded harmonic functions
- Groups of polynomial growth
Table of contents (15 chapters)
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Front Matter
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Back Matter
Reviews
“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)
Authors and Affiliations
About the author
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.
Bibliographic Information
Book Title: Random Walks on Infinite Groups
Authors: Steven P. Lalley
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-031-25632-5
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
Hardcover ISBN: 978-3-031-25631-8Published: 09 May 2023
Softcover ISBN: 978-3-031-25634-9Published: 10 May 2024
eBook ISBN: 978-3-031-25632-5Published: 08 May 2023
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XII, 369
Number of Illustrations: 1 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Potential Theory, Group Theory and Generalizations