Random Walks on Infinite Graphs and Groups

Abstract

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.