Strict monotonicity of percolation thresholds under covering maps

We prove that, under reasonable conditions, strictly quotienting a graph strictly increases the value of \(p_c\). This answers a question of Benjamini and Schramm. We establish the same statement for the uniqueness parameter \(p_u\) under the additional assumption that we quotient by a finite group. The second statement is false if we take the group to be arbitrary.

Annals of Probability, 47(6):4116–4136, 2019