Dorottya Beringer: Critical probability and the question of locality for unimodular  random graphs

Speaker: Dorottya Beringer

Title: Critical probability and the question of locality for unimodular  random graphs

Abstract: The critical probability p_c for Bernoulli percolation on a Cayley graph can be defined in several equivalent ways. But they turn out to be possibly not equal in case of extremal unimodular random graphs. We discuss the relation between the various notions in this wider setup. A famous conjecture of Schramm is that p_c(G_n) converges to p_c(G) whenever G_n is a sequence of Cayley graphs converging to a Cayley graph G and limsup p_c(G_n)<1. While this has only been verified for a few cases, one can find more examples where the conjecture holds, among unimodular random graphs. But what is the right notion of p_c from the point of view of this conjecture? We will discuss similar problems and some progress made, in joint work with Pete and Timar.