On the size of the largest cluster in 2D critical percolation
Abstract
We consider (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It is well-known that for all 0< a < b the probability that M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that M_n is between an^2 pi(n) and bn^2 pi(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the argument.
Keywords
Cite
@article{arxiv.1208.4014,
title = {On the size of the largest cluster in 2D critical percolation},
author = {Jacob van den Berg and Rene Conijn},
journal= {arXiv preprint arXiv:1208.4014},
year = {2012}
}
Comments
12 pages, 3 figures, minor changes