(Non-) Gibbsianness and phase transitions in random lattice spin models
- Külske, Christof
2010 Mathematics Subject Classification
- 82B44 82B26 82B20
- Disordered Systems, Gibbs-measures, non-Gibbsianness, Random Field Model, Random Bond Model, Spinglass
We consider disordered lattice spin models with finite volume Gibbs measures μ Λ[η](dσ). Here σ denotes a lattice spin-variable and η a lattice random variable with product distribution ℙ describing the disorder of the model. We ask: When will the joint measures limΛ↑Zs ℙ(dη)μΛ[η](dσ) 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 spin-glasses and ferromagnets, where various mechanisms are responsible for [non-] Gibbsianness.
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