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Critical Exponents in Percolation via Lattice Animals

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Critical Exponents in Percolation via Lattice Animals
2.	Creator	Author's name, affiliation, country	Alan Hammond; U.C. Berkeley, USA
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	Abstract	We examine the percolation model on $\mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $\beta \in [0,2(d-1))$, let $f(\beta)$ be the asymptotic exponential rate in the number of edges of the number of lattice animals containing the origin which have surface-area-to-volume ratio $\beta$. The function $f$ is bounded above by a function which may be written in an explicit form. For low values of $\beta$ ($\beta \leq 1/p_c - 1$), equality holds, as originally demonstrated by F. Delyon. For higher values ($\beta > 1/p_c - 1$), the inequality is strict. We introduce two critical exponents, one of which describes how quickly $f$ falls away from the explicit form as $\beta$ rises from $1/p_c - 1$, and the second of which describes how large clusters appear in the marginally subcritical regime of the percolation model. We demonstrate that the pair of exponents must satisfy certain inequalities. Other such inequalities yield sufficient conditions for the absence of an infinite cluster at the critical value (c.f. cite{techrep}). The first exponent is related to one of a more conventional nature in the scaling theory of percolation, that of correlation size. In deriving this relation, we find that there are two possible behaviours, depending on the value of the first exponent, for the typical surface-area-to-volume ratio of an unusually large cluster in the marginally subcritical regime.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2005-03-04
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/1131
10.	Identifier	Digital Object Identifier	10.1214/ECP.v10-1131
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 10
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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