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Linear Speed Large Deviations for Percolation Clusters

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Linear Speed Large Deviations for Percolation Clusters
2.	Creator	Author's name, affiliation, country	Yevgeniy Kovchegov; UCLA Mathematics Department, USA
2.	Creator	Author's name, affiliation, country	Scott Roger Sheffield; Microsoft Research
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	Abstract	Let $C_n$ be the origin-containing cluster in subcritical percolation on the lattice $\frac{1}{n} \mathbb Z^d$, viewed as a random variable in the space $\Omega$ of compact, connected, origin-containing subsets of $\mathbb R^d$, endowed with the Hausdorff metric $\delta$. When $d \geq 2$, and $\Gamma$ is any open subset of $\Omega$, we prove that $$\lim_{n \rightarrow \infty}\frac{1}{n} \log P(C_n \in \Gamma) = -\inf_{S \in \Gamma} \lambda(S)$$ where $\lambda(S)$ is the one-dimensional Hausdorff measure of $S$ defined using the correlation norm: $$||u|| := \lim_{n \rightarrow \infty} - \frac{1}{n} \log P (u_n \in C_n )$$ where $u_n$ is $u$ rounded to the nearest element of $\frac{1}{n}\mathbb Z^d$. Given points $a^1, \ldots, a^k \in \mathbb R^d$, there are finitely many correlation-norm Steiner trees spanning these points and the origin. We show that if the $C_n$ are each conditioned to contain the points $a^1_n, \ldots, a^k_n$, then the probability that $C_n$ fails to approximate one of these trees tends to zero exponentially in $n$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2003-12-27
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/1098
10.	Identifier	Digital Object Identifier	10.1214/ECP.v8-1098
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 8
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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