Supercritical sharpness of percolation

TL;DR

This paper establishes that in supercritical percolation on infinite transitive graphs, the probability of the origin being in a large finite cluster decreases exponentially with a graph-specific isoperimetric function.

Contribution

It proves a new exponential decay bound for cluster sizes in supercritical percolation on all infinite transitive graphs, linking decay rate to the graph's isoperimetric properties.

Findings

Exponential decay of large finite cluster probabilities in supercritical percolation.

Decay rate depends on the isoperimetric function of the graph.

Applicable to all infinite transitive graphs.

Abstract

We prove that for supercritical percolation on every infinite transitive graph, the probability that the origin belongs to a finite cluster of size at least $n$ decays exponentially in $\Phi(n)$, where $\Phi$ is the isoperimetric function of the graph.