Large deviations for Langevin spin glass dynamics
The authors study the thermodynamic limit (i.e., the number of spins tends to infinity) of an asymmetrical spin glass dynamics defined by a random mean field interaction. This type of Langevin dynamics was introduced by \textit{Sompolinsky} and \textit{Zippelius} [Phys. Rev. 25, 6860 (1983)] for a continuous version of the Sherrington-Kirkpatrick spin glass model. In contrast to the physics paper, the dependency in the coupling matrix, given by the symmetry, is dropped as a simplification. Results: For high temperatures a full large deviation principle for the annealed law (i.e., averaged over the disorder in the coupling matrix) of the empirical measure on the path space is proved. The corresponding rate function attains its minimum at a unique probability measure \(Q\) on the path space, which is the law of a non-Markovian process. From this the convergence of the quenched law (i.e., for almost all realisations of the coupling matrix) of the empirical measure to \(\delta_Q\) is deduced. The study of the quenched behavior of a single spin is started by the investigation of a dynamics for a replicated system of interacting spins.
scientific article; zbMATH DE number 800491
scientific article