Random Walks on Infinite Graphs and Groups

The main theme of this book is the interplay between random walks and discrete structure theory.

Wolfgang Woess (Author)

9780521552929, Cambridge University Press

Hardback, published 13 February 2000

348 pages, 12 b/w illus.
23.7 x 15.8 x 2.4 cm, 0.61 kg

Review of the hardback: '… will be essential reading for all researchers in stochastic processes and related topics.' European Maths Society Journal

The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.

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.

Subject Areas: Calculus & mathematical analysis [PBK]