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

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 $d\ge 2$. 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.

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)

References in corpus (1)

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