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

TL;DR

This paper investigates when disordered lattice spin models exhibit Gibbsian or non-Gibbsian measures, linking phase transitions to Gibbsianness, with a focus on the random field Ising model and disordered couplings.

Contribution

It provides general criteria for Gibbsianness and non-Gibbsianness in disordered lattice models, connecting phase transitions with measure properties in product spaces.

Findings

Almost sure Gibbsianness in single-phase regions of the random field Ising model.

Almost sure non-Gibbsianness in multi-phase regions of the same model.

Identifies mechanisms causing Gibbsianness or non-Gibbsianness in disordered spin systems.

Abstract

We consider disordered lattice spin models with finite volume Gibbs measures $\mu_{\L}[\eta](d\s)$. Here $\s$ denotes a lattice spin-variable and $\eta$ a lattice random variable with product distribution $\P$ describing the disorder of the model. We ask: When will the joint measures $\lim_{\L\uparrow\Z^d}\P(d\eta)\mu_{\L}[\eta](d\s)$ 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.