@article{ECP2263,
author = {Jacob van den Berg and Rene Conijn},
title = {On the size of the largest cluster in 2D critical percolation},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {17},
year = {2012},
keywords = {Critical percolation; Cluster size},
abstract = {We consider (near-)critical percolation on the square lattice. Let $\mathcal{M}_{n}$ 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 $\mathcal{M}_{n}$ is smaller than $a n^2 \pi(n)$ and the probability that $\mathcal{M}_{n}$ is larger than $b n^2 \pi(n)$ are bounded away from $0$ as $n \rightarrow \infty$. 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 $\mathcal{M}_{n}$ is {\em between} $a n^2 \pi(n)$ and $b n^2 \pi(n)$. 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.
},
pages = {no. 58, 1-13},
issn = {1083-589X},
doi = {10.1214/ECP.v17-2263},
url = {http://ecp.ejpecp.org/article/view/2263}}