A pattern theorem for lattice clusters

1999/02/12 by Neal Madras, Madras, Neal · 3 citations
Mathematics · Physics and Astronomy · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.math/9902161

openalex publication_date 1999/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 asub 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.

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