Poisson Approximations for the Ising Model

A d-dimensional Ising model on a lattice torus is considered. As the size n of the lattice tends to infinity, a Poisson approximation is given for the distribution of the number of copies in the lattice of any given local configuration, provided the magnetic field a = a(n) tends to −∞ and the pair potential b remains fixed. Using the Stein-Chen method, a bound is given for the total variation error in the ferromagnetic case. AMS SUBJECT CLASSIFICATION: 60F05, 82B20.