An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository papers from renowned and leading experts, many of whom spoke at the workshop 'Groups, Graphs and Random Walks' celebrating the sixtieth birthday of Wolfgang Woess in Cortona, Italy. Topics include: growth and amenability of groups; Schrödinger operators and symbolic dynamics; ergodic theorems; Thompson's group F; Poisson boundaries; probability theory on buildings and groups of Lie type; structure trees for edge cuts in networks; and mathematical crystallography. In what is currently a fast-growing area of mathematics, this book provides an up-to-date and valuable reference for both researchers and graduate students, from which future research activities will undoubtedly stem.
1. Growth of groups and wreath products Laurent Bartholdi; 2. Random walks on some countable groups Alexander Bendikov and Laurent Saloff-Coste; 3. The cost of distinguishing graphs Debra Boutin and Wilfried Imrich; 4. A construction of the measurable Poisson boundary – from discrete to continuous groups Sara Brofferio; 5. Structure trees, networks and almost invariant sets Martin J. Dunwoody; 6. Amenability of trees Behrang Forghani and Keivan Mallahi-Karai; 7. Group-walk random groups Agelos Georgakopoulos; 8. Ends of branching random walks on planar hyperbolic Cayley graphs Lorenz A. Gilch and Sebastian Müller; 9. Amenability and ergodic properties of topological groups – from Bogolyubov onwards Rostislav Grigorchuk and Pierre de la Harpe; 10. Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive substitution Rostislav Grigorchuk, Daniel Lenz and Tatiana Nagnibeda; 11. Thompson's group F is not Liouville Vadim A. Kaimanovich; 12. A proof of the subadditive ergodic theorem Anders Karlsson; 13. Boundaries of Zn-free groups Andrei Malyutin, Tatiana Nagnibeda and Denis Serbin; 14. Buildings, groups of Lie type, and random walks James Parkinson; 15. On some random walks driven by spread-out measures Laurent Saloff-Coste and Tianyi Zheng; 16. Topics in mathematical crystallography Toshikazu Sunada.
Tullio Ceccherini-Silberstein is a professor in the Department of Engineering at Università del Sannio (Benevento), Italy. Maura Salvatori is an associate professor in the Department of Mathematics at Università degli Studi di Milano, Italy. Ecaterina Sava-Huss is a visiting assistant professor in the Department of Mathematics at Cornell University, New York.
Reviews for Groups, Graphs and Random Walks
'All parts are carefully presented, often tending to be self-contained and well documented … historical developments as well as significant fields of application of the theory are described for the enlightenment of the reader.' Jean-Guillaume Eon, Actas Crystallographica, Section A
