WIAS Preprint No. 260, (1996)

Metastates in disordered mean field models: Random field and Hopfield models


  • Külske, Christof

2010 Mathematics Subject Classification

  • 60K40 82B44 60G57


  • Disordered Systems, Size Dependence, Random Gibbs States, Metastates, Mean Field Models, Hopfield Model, Random Field Model


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 ∑Nn=1 δμn(η) where μn(η) is the Gibbs measure in the finite volume {1,...,n} and the frozen disorder variable η 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. η but rather in distribution for whose limits we give explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.

Appeared in

  • J. Statist. Phys., 88 (1997), pp. 1257-1293

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