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

Abstract

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 $\mathcal{I}^a$ having a unique minimizer $Q_{\infty}$. The asymptotic dynamics $Q_{\infty}$ 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 principle, according to a (deterministic) good rate functional $\mathcal{I}^q$ satisfying $\mathcal{I}^q\geq \mathcal{I}^a$, 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 $Q_{\infty}$.