Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model



The Annals of Applied Probability

Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

David Coupier

Source: Ann. Appl. Probab. Volume 18, Number 4 (2008), 1326-1350.

Abstract

A d-dimensional ferromagnetic Ising model on a lattice torus is considered. As the size of the lattice tends to infinity, two conditions ensuring a Poisson approximation for the distribution of the number of occurrences in the lattice of any given local configuration are suggested. The proof builds on the Stein–Chen method. The rate of the Poisson approximation and the speed of convergence to it are defined and make sense for the model. Thus, the two sufficient conditions are traduced in terms of the magnetic field and the pair potential. In particular, the Poisson approximation holds even if both potentials diverge.

Primary Subjects: 60F05
Secondary Subjects: 82B20
Keywords: Poisson approximation; Ising model; ferromagnetic interaction; Stein–Chen method

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Mathematical Reviews number (MathSciNet): MR2434173

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