Rigorous Results on Some Simple Spin Glass Models,

A. Bovier, I.A. Kurkova

2003, v.9, №2, 209-242


In this paper we review some recent rigorous results that provide an essentially complete solution of a class of spin glass models introduced by Derrida in the 1980ies. These models are based on Gaussian random processes on $\{-1,1\}^N$ whose covariance is a function of an ultrametric distance on that set. We prove the convergence of the free energy as well as the Gibbs measures in an appropriate sense. These results confirm fully the predictions of the replica method including in situations where continuous replica symmetry breaking takes place.

Keywords: Gaussian processes,generalized random energymodel,continuous hierarchies,spin glasses,Poisson cascades,probability cascades,Ghirlanda-Guerra identities


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