Geography of local configurations

Abstract

A d-dimensional binary Markov random field on a lattice torus is considered. As the size n of the lattice tends to infinity, potentials a=a(n) and b=b(n) depend on n. Precise bounds for the probability for local configurations to occur in a large ball are given. Under some conditions bearing on 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.