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El. knyga: Harmonic Functions and Random Walks on Groups

(Ben-Gurion University of the Negev, Israel)
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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 Choquet–Deny Theorem, the Milnor–Wolf 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.

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.

Recenzijos

'This is a wonderful introduction to random walks and harmonic functions on finitely generated groups. The focus is the characterization of Choquet-Deny groups. The text offers a balanced treatment of well-chosen topics involving probabilistic and algebraic arguments presented with accuracy and care. The rich list of exercises with solutions will certainly help and entertain the reader.' Laurent Saloff-Coste, Cornell University 'Written by a leading expert in the field, this book explores the fundamental results of this captivating area at the boundary of probability and geometric group theoryan essential read for aspiring young researchers.' Hugo Duminil-Copin, Institut des Hautes Études Scientifiques and Université de Genčve 'This voluminous book is a substantial contribution to the state of the art of random walk theory, which has evolved enormously in the last decades. A broad initial part on the basics is guided by numerous exercises. The core chapters are on the relation between harmonic functions for random walks and the structure of the underlying groups, in particular growth. The final highlight is a modern exposition of Gromov's theorem on polynomial growth and its strong interplay with the topics of the book's title.' Wolfgang Woess, Technische Universität Graz

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A modern introduction into the emerging research field of harmonic functions and random walks on groups.
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. Hilbert space background; B. Entropy; C. Coupling and total
variation; References; Index.
Ariel Yadin is Professor in the Department of Mathematics at Ben-Gurion University of the Negev. His research is focused on the interplay between random walks and the geometry of groups. He has taught a variety of courses on the subject, and has been part of a new wave of investigation into the structure of spaces of unbounded harmonic functions on groups.
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