Strict monotonicity of weighted inhomogeneous percolation thresholds under covering maps

Abstract

We study Bernoulli bond percolation under weak covering maps for a class of weighted inhomogeneous product measures such that each edge $e$ carries a positive weight $w(e)$ and is open at intensity $λ\geq 0$ with probability $1-\exp(-λw(e))$. We prove a strict monotonicity of the weighted critical intensities $λ_c$ in the covering compatibility setting under bounded degree, uniform ellipticity, tame fibre and disjoint tree-lifting. While for the weighted uniform connection thresholds $λ_{\mathrm{conn}}$, we establish their strict monotonicity for non-injective weak covering maps under bounded degree, covering compatibility, uniform ellipticity, bounded-fibre and disjoint tree-lifting. Moreover, we verify a general-profile theorem on strict monotonicities of the two thresholds $λ_c$ and $λ_{\mathrm{conn}}$ respectively under explicit compact-interval finite-energy, derivative bounds and aforementioned geometric conditions.