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A new proof of the sharpness of the phase transition for Bernoulli percolation on $$\mathbb{Z}^d$$. (English) Zbl 1359.60118
Summary: 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 $$\theta(p) \geq \frac{p-p_c}{p(1-p_c)}$$.
In [the authors, Commun. Math. Phys. 343, No. 2, 725–745 (2016; Zbl 1342.82026)], 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 $$\mathbb Z^d$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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##### References:
 [1] pp.
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