Large time out-of-equilibrium dynamics of a manifold in a random potential

We study the out-of-equilibrium dynamics of an elastic manifold in a random potential using mean-field theory. We find two asymptotic time regimes: (i) stationary dynamics; (ii) slow aging dynamics with violation of equilibrium theorems. We obtain an analytical solution valid for all large times with universal scalings of two-time quantities with space. A nonanalytic scaling function crosses over to ultrametricity when the correlations become long range. We propose procedures to test numerically or experimentally the extent to which this scenario holds for a given system.