A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model

Abstract

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infiniterange models on arbitrary locally finite transitive infinite graphs.For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime β < β c , and the mean-field lower bound P β [0 ←→ ∞] ≥ (β − β c ) β for β > β c . For finite-range models, we also prove that for any β < β c , the probability of an open path from the origin to distance n decays exponentially fast in n.For the Ising model, we prove finiteness of the susceptibility for β < β c , and the mean-field lower boundFor finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for β < β c .The paper is organized in two sections, one devoted to Bernoulli percolation, and one to the Ising model. While both proofs are completely independent, we wish to emphasize the strong analogy between the two strategies.General notation. Let G = (V, E) be a locally finite (vertex-)transitive infinite graph, together with a fixed origin 0 ∈ V . For n ≥ 0, letwhere d(⋅, ⋅) is the graph distance. Consider a set of coupling constants (J x,y ) x,y∈V with J x,y = J y,x ≥ 0 for every x and y in V . We assume that the coupling constants are invariant with respect to some transitively acting group. More precisely, there exists a group Γ of automorphisms acting transitively on V such that J γ(x),γ(y) = J x,y for all γ ∈ Γ. We say that (J x,y ) x,y∈V is finite-range if there exists R > 0 such that J x,y = 0 whenever d(x, y) > R.

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