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 on ${\mathbb{Z}}^d$. 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 $\theta (p)\ge \frac{p-p_c}{p(1-p_c)}$.

In [DCT], we give a more general proof which covers long-range Bernoulli percolation (and the Ising model) on arbitrary transitive graphs. This article presents the argument of [DCT] in the simpler framework of nearest-neighbour Bernoulli percolation on ${\mathbb{Z}}^d$.