A pattern theorem for lattice clusters

TL;DR

This paper proves a pattern theorem for lattice clusters, showing that certain local configurations occur frequently in large clusters, with implications for percolation probabilities and connective constants.

Contribution

It establishes a general pattern theorem for lattice clusters, extending previous results and applying to weighted sums and percolation probabilities.

Findings

Local patterns occur at least proportionally often in large clusters.

Results apply to weighted sums and percolation cluster probabilities.

Proves strict inequalities of connective constants for sublattices.

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.