Finite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington-Kirkpatrick model

Damien Barbier, Pedro H. de Freitas Pimenta, Leticia F. Cugliandolo, Daniel A. Stariolo

Published 2021-03-23Version 1

We revisit the gradient descent dynamics of the spherical Sherrington-Kirkpatrick ($p=2$) model with finite number of degrees of freedom. For fully random initial conditions we confirm that the relaxation takes place in three time regimes: a fast algebraic one controlled by the decay of the eigenvalue distribution of the random exchange interaction matrix at its edge in the infinite size limit; a slower algebraic one determined by the distribution of the gap between the two extreme eigenvalues; and a final exponential one determined by the minimal gap sampled in the disorder average. We also analyse the finite size effects on the relaxation from initial states which are almost projected on the saddles of the potential energy landscape, and we show that for deviations from perfect alignment scaling as $N^{-\nu}$ the system escapes the initial configuration in a time-scale scaling as $\ln N$ after which the dynamics no longer "self-averages" with respect to the initial conditions. We prove these statements with a combination of analytic and numerical methods.