On the out-of-equilibrium order parameter in long-range spin-glasses

We show that the dynamical order parameters can be re-expressed in terms of the distribution of the staggered auto-correlation and response functions. We calculate these distributions for the out of equilibrium dynamics of the Sherrington-Kirkpatrick model at long times. The results suggest that the landscape this model visits at different long times in an out-of-equilibrium relaxation process is, in a sense, self-similar. Furthermore, there is a similarity between the landscape seen out of equilibrium at long times and the equilibrium landscape. The calculation is greatly simplified by making use of the superspace notation in the dynamical approach. This notation also highlights the rather mysterious formal connection between the dynamical and replica approaches. We also perform numerical simulations which show good agreement with the analytical results for the out of equilibrium dynamics.