Universality and extremal aging for dynamics of spin glasses on sub-exponential time scales

We consider Random Hopping Time (RHT) dynamics of the Sherrington - Kirkpatrick (SK) model and p-spin models of spin glasses. For any of these models and for any inverse temperature we prove that, on time scales that are sub-exponential in the dimension, the properly scaled clock process (time-change process) of the dynamics converges to an extremal process. Moreover, on these time scales, the system exhibits aging like behavior which we called extremal aging. In other words, the dynamics of these models ages as the random energy model (REM) does. Hence, by extension, this confirms Bouchaud's REM-like trap model as a universal aging mechanism for a wide range of systems which, for the first time, includes the SK model.