Harmonic Functions and Random Walks on Groups [Hardcover]

A modern introduction into the emerging research field of harmonic functions and random walks on groups.The field of random walks on groups is re-emerging with many new ideas and exciting research. This book contains a comprehensive introduction for researchers new to the field. Despite dealing with cutting-edge research, it is accessible even to new graduate students, with worked examples, exercises, and open problems all included.The field of random walks on groups is re-emerging with many new ideas and exciting research. This book contains a comprehensive introduction for researchers new to the field. Despite dealing with cutting-edge research, it is accessible even to new graduate students, with worked examples, exercises, and open problems all included.Research in recent years has highlighted the deep connections between the algebraic, geometric, and analytic structures of a discrete group. New methods and ideas have resulted in an exciting field, with many opportunities for new researchers. This book is an introduction to the area from a modern vantage point. It incorporates the main basics, such as Kesten's amenability criterion, Coulhon and Saloff-Coste inequality, random walk entropy and bounded harmonic functions, the ChoquetDeny Theorem, the MilnorWolf Theorem, and a complete proof of Gromov's Theorem on polynomial growth groups. The book is especially appropriate for young researchers, and those new to the field, accessible even to graduate students. An abundance of examples, exercises, and solutions encourage self-reflection and the internalization of the concepts introduced. The author also points to open problems and possibilities for further research.Part I. Tools and Theory: 1. Background; 2. Martingales; 3. Markov chains; 4. Networks and discrete analysis; Part II. Results and Applications: 5. Growth, dimension, and heat kernel; 6. Bounded harmonic functions; 7. ChoquetDeny groups; 8. The MilnorWolf theorem; 9. Gromov's theorem; Appendices: A. l£2