Exponential distribution for the occurrence of rare patterns in Gibbsian random fields

Details Exponential distribution for the occurrence of rare patterns in Gibbsian random fields Exponential distribution for the occurrence of rare patterns in Gibbsian random fields

2003-10-22

arxiv.org/abs/math/0310355v1

Mathematics

Probability

To appear in Commun. Math. Phys

Scientific paper

10.1007/s00220-004-1041-7

We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.

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