Electronic Communications in Probability

On the size of the largest cluster in 2D critical percolation

Jacob van den Berg and Rene Conijn

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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.<br />

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 58, 13 pp.

Accepted: 12 December 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability

Critical percolation Cluster size

This work is licensed under a Creative Commons Attribution 3.0 License.


van den Berg, Jacob; Conijn, Rene. On the size of the largest cluster in 2D critical percolation. Electron. Commun. Probab. 17 (2012), paper no. 58, 13 pp. doi:10.1214/ECP.v17-2263. https://projecteuclid.org/euclid.ecp/1465263191

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