For sufficiently small ε, the stationary Gaussian Markov random field on Z^d is an explicit factor of an i.i.d. process and admits finitely dependent approximations with exponential total variation decay.
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A class of classical lattice models on Z^d (d≥2) with non-negative integer on-site states and nearest-neighbor interactions exhibits long-range checkerboard order at arbitrarily high temperatures via a purely entropic mechanism, proven using Pirogov-Sinai theory with a Peierls bound.
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Glauber dynamics and coupling-from-the-past for Gaussian fields
For sufficiently small ε, the stationary Gaussian Markov random field on Z^d is an explicit factor of an i.i.d. process and admits finitely dependent approximations with exponential total variation decay.