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

TL;DR

This paper investigates pattern occurrences in clusters of random fields on Z^d, establishing exponential bounds, ratio concentration, and limit theorems, with implications for maximal clusters in finite regions.

Contribution

It introduces new exponential bounds for pattern occurrences, proves ratio limit theorems under Markov properties, and connects these results to maximal cluster behavior.

Findings

Pattern occurrence probability decays exponentially with cluster size.

Ratio of pattern counts concentrates around a constant for Markov fields.

Ratio of cluster size probabilities converges as size increases.

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 an 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 tends to infinity. Implications for the maximal cluster in a finite box are discussed.