Librería Portfolio

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Provides broad and deep coverage of most key aspects of probability on graphs and their interconnections, including the best proofs available of many important results
Detailed end-chapter notes give context and further reading
More than 850 exercises allow readers to develop their skills and apply the key techniques





Table of Contents

1. Some highlights
2. Random walks and electric networks
3. Special networks
4. Uniform spanning trees
5. Branching processes, second moments, and percolation
6. Isoperimetric inequalities
7. Percolation on transitive graphs
8. The mass-transport technique and percolation
9. Infinite electrical networks and Dirichlet functions
10. Uniform spanning forests
11. Minimal spanning forests
12. Limit theorems for Galton-Watson processes
13. Escape rate of random walks and embeddings
14. Random walks on groups and Poisson boundaries
15. Hausdorff dimension
16. Capacity and stochastic processes
17. Random walks on Galton-Watson trees.