Strict monotonicity of percolation thresholds under covering maps

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.