Aging of spherical spin glasses

Sompolinski and Zippelius ( 1981) propose the study of dynamical systems wh ose invariant measures are the Gibbs measures for (hard to analyze) statist ical physics models of interest. In the course of doing so, physicists ofte n report of an "aging" phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified spherical version of this mod el. The induced rotational symmetry of the spherical model reduces the dyna mics in question to an N-dimensional coupled system of Ornstein-Uhlenbeck p rocesses whose random drift parameters an the eigenvalues of certain random matrices. We obtain the limiting dynamics for N approaching infinity and b y analyzing its long time behavior, explain what is aging (mathematically s peaking), what causes this phenomenon, and what is its relationship with th e phase transition of the corresponding equilibrium invariant measures.