Math Graduate Student Seminar

What do infinite clusters in supercritical percolation look like? In this talk we will explore a qualitative answer about how infinite clusters interact in highly nonamenable graphs. By the end of this talk we will be able to understand the following statement: for Bernoulli percolation on (non)unimodular quasi-transitive graphs, almost surely, heavy cluster repulsion holds. That is, for any two heavy clusters C and C', the set of vertices in C within distance one of C' is light. This is a significant step towards resolving a longstanding question posed by Häggström, Peres, and Schonmann, and a generalization of a theorem of Timár, who proved the same result in the unimodular setting. This is joint work with Sasha Bell, Owen Rodgers, Grigory Terlov, and Anush Tserunyan.