Percolation on infinite graphs and isoperimetric inequalities

TL;DR

This paper establishes new criteria for percolation thresholds on infinite graphs using isoperimetric inequalities, unifying results for both amenable and non-amenable graphs, and analyzes finite connectivity decay in these graphs.

Contribution

It introduces a general criterion based on isoperimetric inequalities for percolation thresholds, extending previous results to broader classes of graphs including those without bi-infinite geodesics.

Findings

Graphs with bi-infinite geodesics have exponential decay of finite connectivity near p=1.

Graphs without bi-infinite geodesics can have polynomial decay of finite connectivity even close to p=1.

The criterion unifies percolation behavior across amenable and non-amenable graphs.

Abstract

We consider the Bernoulli bond percolation process (with parameter $p$) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay as $p$ is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for $p$ arbitrarily close to one.