Strict monotonicity of percolation thresholds under covering maps

TL;DR

This paper proves that under certain conditions, taking quotients of graphs via covering maps strictly increases their percolation thresholds, extending to the uniqueness parameter with additional assumptions.

Contribution

It establishes the strict monotonicity of percolation thresholds under covering maps for quasi-transitive graphs and extends results to the uniqueness parameter.

Findings

Quotienting a graph increases its percolation critical parameter $p_c$.

Results apply to general covering maps and the uniqueness parameter $p_u$.

Provides a coupling method based on lifting cluster exploration.

Abstract

We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter $p_c$. More precisely, let $\mathcal{G}=(V,E)$ be a quasi-transitive graph with $p_c(\mathcal{G})<1$, and let $G$ be a nontrivial group that acts freely on $V$ by graph automorphisms. Assume that $\mathcal{H}:=\mathcal{G}/G$ is quasi-transitive. Then one has $p_c(\mathcal{G})<p_c(\mathcal{H})$. We provide results beyond this setting: we treat the case of general covering maps and provide a similar result for the uniqueness parameter $p_u$, under an additional assumption of boundedness of the fibres. The proof makes use of a coupling built by lifting the exploration of the cluster, and an exploratory counterpart of Aizenman-Grimmett's essential enhancements.