Percolation and local isoperimetric inequalities

Abstract

In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension d> 1 , then pc(G) < 1. This gives a partial answer to a question of Benjamini and Schramm (Electron Commun Probab 1(8):71–82 1996). As a consequence of this result and Häggström (Adv Appl Probab 32(1):39–66 2000) we derive, that these graphs also undergo a non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and the beach model. Our techniques are also applied to dependent percolation processes with long range correlations. We provide results on the uniqueness of the infinite percolation cluster and quantitative estimates on the size of finite components. Finally we leave some remarks and questions that arise naturally from this work.