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Zero-one laws for binary random fields

David Coupier·Paul Doukhan·Bernard Ycart
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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.

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David Coupier, Paul Doukhan, Bernard Ycart. 2006-05-18. Zero-one laws for binary random fields. https://arxiv.org/abs/math/0605502

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