TY - JOUR

T1 - Pattern theorems, ratio limit theorems and Gumbel maximal clusters for random fields

AU - Hofstad, van der, R.W.

AU - Kager, W.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

U2 - 10.1007/s10955-007-9435-5

DO - 10.1007/s10955-007-9435-5

M3 - Article

VL - 130

SP - 503

EP - 522

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -