Poisson approximation for large contours in low-temperature Ising models

Pablo A. Ferrari and Pierre Picco

Physica A: Statistical Mechanics and its Applications, 2000, vol. 279, issue 1, 303-311

Abstract: We consider the contour representation of the infinite volume Ising model at any fixed inverse temperature β>β∗, the solution of ∑θ:θ∋0e−β|θ|=1. Let μ be the infinite-volume “+” measure. Fix V⊂Zd, λ>0 and a (large) N such that calling GN,V the set of contours of length at least N intersecting V, there are in average λ contours in GN,V under μ. We show that the total variation distance between the law of (γ:γ∈GN,V) under μ and a Poisson process is bounded by a constant depending on β and λ times e−(β−β∗)N. 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 Fernández, Ferrari and Garcia.

Keywords: Peierls contours; Animal models; Loss networks; Large contours; Ising model; Poisson approximation; Chen–Stein method (search for similar items in EconPapers)
Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:279:y:2000:i:1:p:303-311

DOI: 10.1016/S0378-4371(99)00536-1

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