A lower bound on the two-arms exponent for critical percolation on the lattice

Raphaël Cerf

doi:10.1214/14-AOP940

A lower bound on the two-arms exponent for critical percolation on the lattice

Probability 2015-10-30 v2

Authors: Raphaël Cerf

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Abstract

We consider the standard site percolation model on the $d$ -dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to $1/2$ . We improve slightly this lower bound in any dimension $d\geq2$ . Next, starting only with the hypothesis that $\theta(p)>0$ , without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.

Keywords

critical percolation two-arms exponent lower bound uniqueness of infinite cluster long-range order

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@article{arxiv.1306.3105,
  title  = {A lower bound on the two-arms exponent for critical percolation on the lattice},
  author = {Raphaël Cerf},
  journal= {arXiv preprint arXiv:1306.3105},
  year   = {2015}
}

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Published at http://dx.doi.org/10.1214/14-AOP940 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A lower bound on the two-arms exponent for critical percolation on the lattice

Abstract

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