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Poisson approximation for large contours in low-temperature Ising models

Author

Listed:
  • Ferrari, Pablo A.
  • Picco, Pierre

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.

Suggested Citation

  • Ferrari, Pablo A. & Picco, Pierre, 2000. "Poisson approximation for large contours in low-temperature Ising models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 279(1), pages 303-311.
  • Handle: RePEc:eee:phsmap:v:279:y:2000:i:1:p:303-311
    DOI: 10.1016/S0378-4371(99)00536-1
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