Speakers

Title:    Heavy repulsion of clusters in Bernoulli percolation.

Abstract:    Quasi-transitive graphs are naturally equipped with a relative vertex-weight function, which is identically 1 in the unimodular case. In this talk, I will present progress on a question of Häggström, Peres, and Schonmann asking whether two infinite Bernoulli percolation clusters can be within distance 1 of each other infinitely often. More precisely, we prove that, almost surely, for any two clusters C and C', the set of vertices in C that lie within distance 1 of C' is of finite total weight. Our proof combines Timár’s approach, which settled the question in the unimodular case, with recent results on the mcp setting from measured group theory. This is joint work with Sasha Bell, Tasmin Chu, Owen Rodgers, and Anush Tserunyan.