Poisson approximation for large-contours in low-temperature Ising models

Details Poisson approximation for large-contours in low-temperature Ising models Poisson approximation for large-contours in low-temperature Ising models

1999-12-16

arxiv.org/abs/math/9912136v1

Physica A: Statistical Mechanics and its Applications Volume 279, Issues 1-4, 1 May 2000, Pages 303-311

Mathematics

Probability

10 pages, to appear in Physica A

Scientific paper

10.1016/S0378-4371(99)00536-1

We consider the contour representation of the infinite volume Ising model at low temperature. Fix a subset V of Z^d, and a (large) N such that calling G_{N,V} the set of contours of length at least N intersecting V, there are in average one contour in G_{N,V} under the infinite volume "plus" measure. We find bounds on the total variation distance between the law of the contours of lenght at least N intersecting V under the "plus" measure and a Poisson process. The proof builds on the Chen-Stein method as presented by Arratia, Goldstein and Gordon. The control of the correlations is obtained through the loss-network space-time representation of contours due to Fernandez, Ferrari and Garcia.

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