Geography of local configurations

A d-dimensional ferromagnetic Ising model on a lattice torus is considered. As the size n of the lattice tends to infinity, the magnetic field a = a(n) and the pair potential depend on n. Precise bounds for the probability for local configurations to occur in a large ball are given. Under some conditions bearing on potentials a(n) and b(n), the distance between copies of different local configurations is estimated according to their weights. Finally, a sufficient condition ensuring that a given local configuration occurs everywhere in the lattice is suggested.