Ergodic Theory Lectures: Nishant Chandgotia

Suppose mu is a probability measure which is shift invariant on {0,1}^{Z^d} and we know that for almost every configuration x in {0,1}^{Z^d} there are connected components of 1s which are infinite. In this talk, we will follow a paper by Burton and Keane (generalising results by Aizenman, Kesten and Newman) to give an elegant proof of the fact that, under fairly general conditions (say full support), the number of connected components of infinite cardinality is at exactly one.