Universality of the REM for dynamics of mean-field spin glasses

We consider a version of Glauber dynamics for a p-spin Sherrington- Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p >= 3 and all inverse temperatures beta > 0, there exists a constant gamma (beta ,p) > 0, such that for all exponential time scales, exp(gamma N), with gamma < gamma(beta ,p), the properly rescaled clock process (time-change process) converges to an alpha-stable subordinator where alpha = gamma/beta(2) < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this alpha-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.