Poisson approximation for large contours in low-temperature Ising models

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.