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2012 On the size of the largest cluster in 2D critical percolation
Jacob van den Berg, Rene Conijn
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Electron. Commun. Probab. 17: 1-13 (2012). DOI: 10.1214/ECP.v17-2263

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.<br />

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Jacob van den Berg. Rene Conijn. "On the size of the largest cluster in 2D critical percolation." Electron. Commun. Probab. 17 1 - 13, 2012. https://doi.org/10.1214/ECP.v17-2263

Information

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

zbMATH: 1320.60161
MathSciNet: MR3005731
Digital Object Identifier: 10.1214/ECP.v17-2263

Subjects:
Primary: 60K35
Secondary: 60C05

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