A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model

TL;DR

This paper introduces a new proof demonstrating the sharpness of phase transitions in Bernoulli percolation and the Ising model, applicable to various infinite-range models on transitive graphs, with results on susceptibility and correlation decay.

Contribution

It provides a novel, more general proof of phase transition sharpness for both models, extending to infinite-range and arbitrary transitive graphs.

Findings

Finiteness of susceptibility in subcritical regimes.

Mean-field lower bounds for percolation and Ising models.

Exponential decay of connection probabilities and correlations in finite-range models.

Abstract

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $\beta<\beta_c$, and the mean-field lower bound $\mathbb{P}_\beta[0\longleftrightarrow\infty]\ge (\beta-\beta_c)/\beta$ for $\beta>\beta_c$. For finite-range models, we also prove that for any $\beta<\beta_c$, 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 $\beta<\beta_c$, and the mean-field lower bound $\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}$ for $\beta>\beta_c$. For finite-range models, we also prove that the two-point correlations functions decay exponentially fast in the distance for $\beta<\beta_c$.