Large deviations in the Langevin dynamics of a short-range spin glass

We consider a Langevin dynamics associated with a d-dimensional Edwards-Anderson model having Gaussian coupling variables, and show that the averaged law of the empirical process satisfies a large-deviation principle according to a good rate functional I^a having a unique minimizer Q_x. The asymptotic dynamics Q_∞ may be characterized as the unique weak solution corresponding to a non-Markovian system of interacting diffusions having an infinite range of interaction. We then establish that the quenched law of the empirical process also obeys a large-deviation process, according to a (deterministic) good rate functional I^q satisfying I^q ≥ I^a, so that, for a typical realization of the disorder variables, the quenched law of the empirical process also converges exponentially fast to a Dirac mass concentrated at Q_∞.