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arXiv · math-ph/9904024

(Non-) Gibbsianness and phase transitions in random lattice spin models

C. Kuelske
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Abstract

We consider disordered lattice spin models with finite volume Gibbs measures $μ_Ł[η](d\s)$. Here $\s$ denotes a lattice spin-variable and $η$ a lattice random variable with product distribution $¶$ describing the disorder of the model. We ask: When will the joint measures $\lim_{Ł\uparrow\Z^d}¶(dη)μ_Ł[η](d\s)$ be [non-] Gibbsian measures on the product of spin-space and disorder-space? We obtain general criteria for both Gibbsianness and non-Gibbsianness providing an interesting link between phase transitions at a fixed random configuration and Gibbsianness in product space: Loosely speaking, a phase transition can lead to non-Gibbsianness, (only) if it can be observed on the spin-observable conjugate to the independent disorder variables. Our main specific example is the random field Ising model in any dimension for which we show almost sure- [almost sure non-] Gibbsianness for the single- [multi-] phase region. We also discuss models with disordered couplings, including spinglasses and ferromagnets, where various mechanisms are responsible for [non-] Gibbsianness.

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C. Kuelske. 1999-04-26. (Non-) Gibbsianness and phase transitions in random lattice spin models. https://arxiv.org/abs/math-ph/9904024

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