Kuelske C.
(Non-) Gibbsianness and phase transitions
in random lattice spin models
(242K, PS)

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.