A pattern theorem for lattice clusters

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 limn→∞an+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.