Large deviation bounds for the volume of the largest cluster in 2D critical percolation
Probability
2014-04-09 v1
Abstract
Let M_n denote the number of sites in the largest cluster in critical site percolation on the triangular lattice inside a box side length n. We give lower and upper bounds on the probability that M_n / E(M_n) > x of the form exp(- C x^(2/alpha)) for x > 1 and large n with alpha = 5/48 and C > 0. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions d > 2.
Keywords
Cite
@article{arxiv.1404.2118,
title = {Large deviation bounds for the volume of the largest cluster in 2D critical percolation},
author = {Demeter Kiss},
journal= {arXiv preprint arXiv:1404.2118},
year = {2014}
}
Comments
11 pages