A pattern theorem for lattice clusters

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 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$sub n$ 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.