Occurrence, repetition and matching of patterns in the low-temperature Ising model

TL;DR

This paper investigates the statistical behavior of pattern occurrences and overlaps in the low-temperature Ising model, establishing exponential, Poisson, and Gumbel laws with precise error bounds.

Contribution

It provides new rigorous results on pattern occurrence laws, including exponential, Poisson, and Gumbel distributions, using disagreement percolation for the low-temperature Ising model.

Findings

Exponential laws with error bounds for occurrence and return times.

Poisson law for large cylindrical event occurrences.

Gumbel law for maximal overlap between independent copies.

Abstract

We continue our study of exponential law for occurrences and returns of patterns in the context of Gibbsian random fields. For the low temperature plus phase of the Ising model, we prove exponential laws with error bounds for occurrence, return, waiting and matching times. Moreover we obtain a Poisson law for the number of occurrences of large cylindrical events and a Gumbel law for the maximal overlap between two independent copies. As a by-product, we derive precise fluctuation results for the logarithm of waiting and return times. The main technical tool we use, in order to control mixing, is disagreement percolation.