Limiting Behavior of Random Gibbs Measures: Metastates in Some Disordered Mean Field Models

We present examples of random mean field spin models for which the size dependence of their Gibbs measures μΛ _n can be rigorously analyzed. We investigate their ‘empirical metastates’ $$1/N \sum\nolimits_{n = 1}^N {\delta _{\mu \Lambda _n } }$$, introduced by Newman and Stein, along the sequence of finite volumes $$\Lambda _n = \{ 1,\,...,\,n\}$$. The empirical metastate is shown not to converge in our examples if the realization of the disorder is fixed. This phenomenon leads us to consider the distributions w.r.t disorder of the empirical metastates for which we show convergence and give explicit limiting expression.