Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

Details Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

2006-10-30

arxiv.org/abs/math/0610939v2

Annals of Applied Probability 2008, Vol. 18, No. 4, 1326-1350

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/1214/07-AAP487 the Annals of Applied Probability (http://www.imstat.org/aap/) by the

Scientific paper

10.1214/1214/07-AAP487

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.

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