Probability on Trees and Networks

Russell Lyons is James H. Rudy Professor of Mathematics at Indiana University, Bloomington. He obtained his PhD at the University of Michigan in 1983. He has written seminal papers concerning probability on trees and random spanning trees in networks. Lyons was a Sloan Foundation Fellow and has been an Invited Speaker at the International Congress of Mathematicians and the Joint Mathematics Meetings. He is a Fellow of the American Mathematical Society. Yuval Peres is a Principal Researcher at Microsoft Research in Redmond, Washington. He obtained his PhD at the Hebrew University, Jerusalem in 1990 and later served on their faculty as well as on the faculty at the University of California, Berkeley. He has written more than 250 research papers in probability, ergodic theory, analysis, and theoretical computer science. He has coauthored books on Brownian motion and Markov chain mixing times. Peres was awarded the Rollo Davidson Prize in 1995, the Loève Prize in 2001, and the David P. Robbins Prize in 2011 and was an Invited Speaker at the 2002 ICM. He is a fellow of the American Mathematical Society and a foreign associate member of the US National Academy of Sciences.

'This long-awaited work focuses on one of the most interesting and important parts of probability theory. Half a century ago, most work on models such as random walks, Ising, percolation and interacting particle systems concentrated on processes defined on the d-dimensional Euclidean lattice. In the intervening years, interest has broadened dramatically to include processes on more general graphs, with trees being a particularly important case. This led to new problems and richer behavior, and as a result, to the development of new techniques. The authors are two of the major developers of this area; their expertise is evident throughout.' Thomas M. Liggett, University of California, Los Angeles

• 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.