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

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. Key-words: occurrence and repetition of patterns, low temperature Ising model, disagreement percolation, exponential law, Poisson law, Gumbel law, large deviations.