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

TL;DR

This paper offers a new proof demonstrating the sharpness of the phase transition in Bernoulli percolation on , establishing exponential decay below criticality and a mean-field bound above it.

Contribution

It introduces a simplified proof technique for the sharpness of phase transition applicable to nearest-neighbour Bernoulli percolation on , extending ideas from long-range models.

Findings

Exponential decay of connection probability for p<p_c

Mean-field lower bound for (p) when p>p_c

Unified proof approach for different percolation models

Abstract

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that - for $p<p_c$, the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$. - for $p>p_c$, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $\theta(p)\ge\tfrac{p-p_c}{p(1-p_c)}$. This note presents the argument of \cite{DumTas15}, 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 $\mathbb Z^d$.