Percolation and local isoperimetric inequalities
Augusto Q. Teixeira (IMPA)
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 G, such as growth or dimension, and the process of percolation on G. In particular one could look for conditions under which the critical percolation threshold p_c(G) is non-trivial, that is: p_c(G) is strictly between zero and one. In a very important paper on this subject, Benjamini and Schramm asked whether it is true that for every graph satisfying dim(G) > 1, one has p_c(G) < 1. We will explain this question in detail, explaining what they meant by the dimension of a graph and we will present a result that has recently been obtained in this direction.