Synopses & Reviews
This eminent work focuses on the interplay between the behavior of random walks and discrete structure theory. Wolfgang Woess considers Markov chains whose state space is equipped with the structure of an infinite, locally-finite graph, or of a finitely generated group. He assumes the transition probabilities are adapted to the underlying structure in some way that must be specified precisely in each case. He also explores the impact the particular type of structure has on various aspects of the behavior of the random walk. In addition, the author shows how random walks are useful tools for classifying, or at least describing, the structure of graphs and groups.
"The organization of the book is well-thought-out...The reviewer has a very high opition of this book" Bulletin of the American Mathematical Society
"a very valuable addition to the literture on this fascinating and important subject." Mathematical Review
The main theme of this book is the interplay between random walks and discrete structure theory.
Table of Contents
Part I. The Type Problem: 1. Basic facts; 2. Recurrence and transience of infinite networks; 3. Applications to random walks; 4. Isoperimetric inequalities; 5. Transient subtrees, and the classification of the recurrent quasi transitive graphs; 6. More on recurrence; Part II. The Spectral Radius: 7. Superharmonic functions and r-recurrence; 8. The spectral radius; 9. Computing the Green function; 10. Spectral radius and strong isoperimetric inequality; 11. A lower bound for simple random walk; 12. Spectral radius and amenability; Part III. The Asymptotic Behaviour of Transition Probabilities: 13. The local central limit theorem on the grid; 14. Growth, isoperimetric inequalities, and the asymptotic type of random walk; 15. The asymptotic type of random walk on amenable groups; 16. Simple random walk on the Sierpinski graphs; 17. Local limit theorems on free products; 18. Intermezzo; 19. Free groups and homogenous trees; Part IV. An Introduction to Topological Boundary Theory: 20. Probabilistic approach to the Dirichlet problem, and a class of compactifications; 21. Ends of graphs and the Dirichlet problem; 22. Hyperbolic groups and graphs; 23. The Dirichlet problem for circle packing graphs; 24. The construction of the Martin boundary; 25. Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth; 27. The Martin boundary of hyperbolic graphs; 28. Cartesian products.