Random Walks on Infinite Graphs and Groups Book

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.