Boundary-connectivity via graph theory

Abstract

We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of Z d \mathbb {Z}^d , where “connectedness” and “boundary” are understood with respect to various graphs on the vertices of Z d \mathbb {Z}^d . These theorems are widely used in statistical physics and related areas of probability. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory instead of topology.