STRICT MONOTONICITY OF PERCOLATION THRESHOLDS UNDER COVERING MAPS

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 G = (V, E) be a quasi-transitive graph with p(c) (G) < 1, and let G be a nontrivial group that acts freely on V by graph automorphisms. Assume that H := G/G is quasi-transitive. Then one has p(c) (G) < P-c (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.