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

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	On the size of the largest cluster in 2D critical percolation
2.	Creator	Author's name, affiliation, country	Jacob van den Berg; Centrum Wiskunde & Informatica (CWI); Netherlands
2.	Creator	Author's name, affiliation, country	Rene Conijn; VU University Amsterdam; Netherlands
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Critical percolation; Cluster size
3.	Subject	Subject classification	60K35; 60C05
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2012-12-12
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/2263
10.	Identifier	Digital Object Identifier	10.1214/ECP.v17-2263
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 17
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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