Rigorous results on some simple spin glass models

摘要

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