An emergent autonomous flow for mean-field spin glasses

Abstract

We study the dynamics of symmetric and asymmetric spin-glass models of size N. The analysis is in terms of the double empirical process: this contains both the spins, and the field felt by each spin, at a particular time (without any knowledge of the correlation history). It is demonstrated that in the large N limit, the dynamics of the double empirical process becomes deterministic and autonomous over finite time intervals. This does not contradict the well-known fact that SK spin-glass dynamics is non-Markovian (in the large N limit) because the empirical process has a topology that does not discern correlations in individual spins at different times. In the large N limit, the evolution of the density of the double empirical process approaches a nonlocal autonomous PDE operator \(\Phi _t\). Because the emergent dynamics is autonomous, in future work one will be able to apply PDE techniques to analyze bifurcations in \(\Phi _t\). Preliminary numerical results for the SK Glauber dynamics suggest that the ‘glassy dynamical phase transition’ occurs when a stable fixed point of the flow operator \(\Phi _t\) destabilizes.