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arxiv: math/9912136 · v1 · pith:NC54KMIUnew · submitted 1999-12-16 · 🧮 math.PR · math-ph· math.MP

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

classification 🧮 math.PR math-phmath.MP
keywords contourscontourinfiniteintersectingisingleastmeasureplus
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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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