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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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