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

TL;DR

This paper investigates the finite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington-Kirkpatrick model, revealing multiple relaxation regimes and how initial state deviations influence dynamics.

Contribution

It provides a detailed analysis of the relaxation dynamics in finite systems, highlighting the transition from self-averaging to non-self-averaging behavior based on initial condition deviations.

Findings

Relaxation occurs in three distinct time regimes.

Finite size effects cause a transition from self-averaging to non-self-averaging.

Initial deviations scale as N^{- u} lead to logarithmic escape times.

Abstract

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 first 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 faster 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 scaling as $N^{-\nu}$ from perfect alignment 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.