Percolation and local isoperimetric inequalities

TL;DR

This paper explores how geometric properties of graphs, like polynomial growth and isoperimetric inequalities, influence percolation thresholds and phase transitions in related statistical models.

Contribution

It establishes that graphs with polynomial growth and local isoperimetric inequalities have a percolation threshold less than one, linking geometry to phase transition behavior.

Findings

Graphs with polynomial growth and local isoperimetric inequalities have p_c(G) < 1.

Under bounded degree, such graphs exhibit phase transitions in Ising, Widom-Rowlinson, and beach models.

Results include uniqueness of infinite clusters and estimates on finite component sizes.

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 p_c(G) < 1. This gives a partial answer to a question of Benjamini and Schramm. As a consequence of this result we derive, under the additional condition of bounded degree, 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.