A pattern theorem for lattice clusters |
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Authors: | Neal Madras |
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Affiliation: | (1) Department of Mathematics and Statistics, York University, 4700 Keele St., M3J 1P3 Toronto, Ontario, Canada |
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Abstract: | We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration (pattern) of sites and bonds can occur in large clusters, then for some constantc>0, it occurs at leastcn times in most clusters of sizen. An analogous theorem for self-avoiding walks was proven in 1963 by Kesten [9]. We use the pattern theorem to prove the convergence of limnan+1/an, wherean is the number of clusters of sizen, up to translation. The results also apply to weighted sums, and in particular, we can takean to be the probability that the percolation cluster containing the origin consists of exactlyn sites. Another consequence is strict inequality of connective constants for sublattices and for certain subclasses of clusters.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.The author was visiting the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, while writing this paper. |
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Keywords: | 82B41 60K35 82B43 |
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