Probability Seminar: Tom Hutchcroft

In 1987, Aizenman, Kesten, and Newman proved that percolation on Z^d always has at most one infinite cluster a.s. While their proof has mostly been eclipsed by the more general and arguably more elegant proof of Burton and Keane, the Aizenman-Kesten-Newman proof is more quantitative and yields interesting bounds on certain two-arm probabilities. In this talk, I will discuss a new variation on these bounds that holds universally over a large class of graphs, and is stronger even in the case of Z^d. I will then show how this new bound can be applied to derive new results on percolation in various "infinite-dimensional" settings.