Sharpness of Bernoulli percolation via couplings

Abstract

In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice ${\mathbb{Z}}^d$). We prove the following estimate, where ${\theta }_n(p)$ is the probability that there is a path of $p$-open edges from $0$ to the sphere of radius $n$: \[ \forall p\in [0,1],\forall m,n \ge 1, \quad θ_{2n} (p-2θ_m(p))\le C\frac{θ_n(p)}{2^{n/m}}. \] This result implies that ${\theta }_n(p)$ decays exponentially fast in the subcritical phase. It also implies the mean-field lower bound in the supercritical phase. We thus provide a new proof of the sharpness of the phase transition for Bernoulli percolation. Contrary to the previous proofs of sharpness, we do not rely on any differential formula. The main novelty is a stochastic domination result which is inspired by [Russo, 1982]. We also discuss a consequence of our result for percolation in high dimensions, where it can be seen as a near-critical sharpness estimate.

10 pages, very minor changes introduced