Pattern theorems, ratio limit theorems and Gumbel maximal clusters for random fields

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    Abstract

    We study occurrences of patterns on clusters of size n in random fields on Z d . We prove that for a given pattern, there is a constant a>0 such that the probability that this pattern occurs at most na times on a cluster of size n is exponentially small. Moreover, for random fields obeying a certain Markov property, we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value. This leads to an elegant and simple proof of the ratio limit theorem for these random fields, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges as n¿8. Implications for the maximal cluster in a finite box are discussed.
    Original languageEnglish
    Pages (from-to)503-522
    JournalJournal of Statistical Physics
    Volume130
    Issue number3
    DOIs
    Publication statusPublished - 2008

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