Zero-one laws for binary random fields

Abstract

A set of binary random variables indexed by a lattice torus is considered. Under a mixing hypothesis, the probability of any proposition belonging to the first order logic of colored graphs tends to 0 or 1, as the size of the lattice tends to infinity. For the particular case of the Ising model with bounded pair potential and surface potential tending to $-\infty$, the threshold functions of local propositions are computed, and sufficient conditions for the zero-one law are given.