Coarse geometry and randomness. École d'Été de Probabilités de Saint-Flour XLI -- 2011
The lecture notes contained in the book provide a bird's eye view of recent advances in coarse geometry and randomness. The first part focuses on the infinite vertex transitive graphs, expander graphs, and hyperbolic graphs. Then many important concepts regarding percolation on graphs and random walks are introduced. In particular, local and scaling limits of graphs, critical percolation clusters, and percolation on expanders are dealt with. The Kaimanovich-Vershik entropic criterion for the existence of harmonic functions on Cayley graphs is presented. Some conjectures regarding the question ``Does non-amenability imply non-Liouville (i.e., it admits non-constant bounded harmonic functions) for general bounded degree graphs?'' are also discussed.
scientific article; zbMATH DE number 6243708