Title: Percolation and isoperimetric inequalities.
Abstract: In this talk we will discuss some relations between percolation on a given graph G and its geometry.
There are several interesting questions relating various properties of a graph, such as growth or dimension, and the process of percolation on it. In particular we will look for conditions under which its critical percolation threshold is non-trivial, that is: p_c(G) is strictly between zero and one. In a very influential paper on this subject, Benjamini and Schramm asked whether it was true that for every graph satisfying dim(G) > 1, one has p_c(G) < 1. We will explain this question in detail and present some recent results that have been obtained in this direction. This talk is based on a joint work with Augusto Teixeira (IMPA, Rio de Janeiro, Brazil).