A pattern theorem for lattice clusters
概率论
2009-09-25 v1
摘要
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 it occurs at least cN times in most clusters of size n, for some constant c>0. An analogous theorem for self-avoiding walks was proven in 1963 by Kesten. The results also apply to weighted sums, and in particular we can take a to be the probability that the percolation cluster containing the origin consists of exactly n sites. Another consequence is strict inequality of connective constants for sublattices and for certain subclasses of clusters.
引用
@article{arxiv.math/9902161,
title = {A pattern theorem for lattice clusters},
author = {Neal Madras},
journal= {arXiv preprint arXiv:math/9902161},
year = {2009}
}