Zero-one laws for binary random fields

TL;DR

This paper investigates zero-one laws for binary random fields on lattices, establishing conditions under which properties almost surely occur or not as the lattice size grows, with specific results for the Ising model.

Contribution

It introduces zero-one laws for binary random fields under mixing conditions and computes threshold functions for local propositions in the Ising model.

Findings

Probability of first-order properties tends to 0 or 1 as lattice size increases

Threshold functions for local propositions in the Ising model are derived

Sufficient conditions for zero-one laws are established

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.