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.

‘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 theory—an essential read for aspiring young researchers.’

Hugo Duminil-Copin - Institut des Hautes Études Scientifiques and Université de Genève

‘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