# L’Enseignement Mathématique

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**Volume 62, Issue 1/2, 2016, pp. 199–206**

**DOI: 10.4171/LEM/62-1/2-12**

Published online: 2017-01-30

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

Hugo Duminil-Copin^{[1]}and Vincent Tassion

^{[2]}(1) Université de Genève, Switzerland

(2) Université de Genève, Switzerland

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 $\theta(p) \ge \tfrac{p-p_c}{p(1-p_c)}$.

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 $\mathbb Z^d$.

*Keywords: *Bernoulli percolation, phase transition, sharpness, mean-field lower bound

Duminil-Copin Hugo, Tassion Vincent: A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$. *Enseign. Math.* 62 (2016), 199-206. doi: 10.4171/LEM/62-1/2-12