Sharpness of phase transition for Voronoi percolation in hyperbolic space

Abstract

In this paper, we consider Voronoi percolation in the hyperbolic space H d (d ≥ 2) and show that the phase transition is sharp. More precisely, we show that for Voronoi percolation with parameter p generated by a homogeneous Poisson point process with intensity λ, there exists p c := p c (λ, d) such that the probability of a monochromatic path from the origin reaching a distance of n decays exponentially fast in n. We also prove the mean-field lower bound P λ,p (0 ↔ ∞) ≥ c(p − p c ) for p > p c .