A new proof of the sharpness of the phase transition for Bernoulli percolation on

  • Hugo Duminil-Copin

    Université de Genève, Switzerland
  • Vincent Tassion

    Université de Genève, Switzerland
A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$ cover
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Abstract

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation on . More precisely, we show that

  • for , the probability that the origin is connected by an open path to distance decays exponentially fast in .

  • for , the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound .

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 .

Cite this article

Hugo Duminil-Copin, Vincent Tassion, A new proof of the sharpness of the phase transition for Bernoulli percolation on . Enseign. Math. 62 (2016), no. 1/2, pp. 199–206

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