Annals of Applied Probability

Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

David Coupier

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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.

Article information

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

First available in Project Euclid: 21 July 2008

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Poisson approximation Ising model ferromagnetic interaction Stein–Chen method


Coupier, David. Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model. Ann. Appl. Probab. 18 (2008), no. 4, 1326--1350. doi:10.1214/1214/07-AAP487.

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