Crossover from stationary to aging regime in glassy dynamics

We study the non-equilibrium dynamics of the spherical p-spin models in the scaling regime near the plateau and derive the corresponding scaling functions for the correlators. Our main result is that the matching between different time regimes fixes the aging function in the aging regime to $h(t)=\exp(t^{1-\mu})$. The exponent $\mu$ is related to the one giving the length of the plateau. Interestingly $1-\mu$ is quickly very small when one goes away from the dynamic transition temperature in the glassy phase. This gives new light on the interpretation of experiments and simulations where simple aging was found to be a reasonable but not perfect approximation, which could be attributed to the existence of a small but non-zero stretching exponent.