(Non-) Gibbsianness and Phase Transitions in Random Lattice Spin Models

We consider disordered lattice spin models with finite-volume Gibbs measures µΛ[η](dσ). Here σ denotes a lattice spin variable and η a lattice random variable with product distribution P describing the quenched disorder of the model. We ask: when will the joint measures limΛ↑Zd 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 discontinuity in the quenched Gibbs expectation 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 random bond ferromagnets, where various mechanisms are responsible for [non-] Gibbsianness.