A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$

Abstract

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that • for p pc, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound θ(p) ≥ p−pc p(1−pc) . This note presents the argument of [DCT15], which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on Z. 1 Statement of the result Notation. Fix an integer d ≥ 2. We consider the d-dimensional hypercubic lattice (Zd,Ed). Let Λn = {−n, . . . , n}d, and let ∂Λn ∶= Λn ∖ Λn−1 be its vertex-boundary. Throughout this note, S always stands for a finite set of vertices containing the origin. Given such a set, we denote its edge-boundary by ∆S, defined by all the edges {x, y} with x ∈ S and y ∉ S. Consider the Bernoulli bond percolation measure Pp on {0,1}Ed for which each edge of Ed is declared open with probability p and closed otherwise, independently for different edges. Two vertices x and y are connected in S ⊂ V if there exists a path of vertices (vk)0≤k≤K in S such that v0 = x, vK = y, and {vk, vk+1} is open for every 0 ≤ k < K. We denote this event by x S ←→ y. If S = Zd, we drop it from the notation. We set 0 ←→ ∞ (resp. 0 ←→ ∂Λn) if 0 is connected to infinity (resp. 0 is connected to a vertex in ∂Λn).