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Zero-one laws for binary random fields. (English) Zbl 1104.60017

Summary: 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.

MSC:

60F20 Zero-one laws
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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