Kuelske, C.
METASTATES IN DISORDERED MEAN FIELD MODELS:
RANDOM FIELD AND HOPFIELD MODELS
(85K, TeX)
ABSTRACT. We rigorously investigate the size dependence of disordered mean field models
with finite local spin space in terms of metastates. Thereby we provide an
illustration of the framework of metastates for systems of randomly competing
Gibbs measures. In particular we consider the thermodynamic limit of the empirical
metastate $1/N\sum_{n=1}^N \d_{\mu_n(\eta)}$ where $\mu_n(\eta)$ is the Gibbs
measure in the finite volume $\{1,\dots,n\}$ and the frozen disorder variable $\eta$
is fixed. We treat explicitely the Hopfield model with finitely many patterns
and the Curie Weiss Random Field Ising model. In both examples in the phase
transition regime the empirical metastate is dispersed for large $N$. Moreover it
does not converge for a.e. $\eta$ but rather in distribution for whose limits we
give explicit expressions. We also discuss another notion of metastates, due to
Aizenman and Wehr.