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 −ν 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.

We present a thorough numerical analysis of the relaxational dynamics of the Sherrington–Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes N. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition from a spin glass to a ferromagnetic phase. We show that finite-size effects induce the appearance of a distinctive slow regime in the relaxation dynamics, the extension of which depends on the size of the system and also on the strength of the non-disordered perturbation. The long time dynamics are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues of the spike random matrices in the different regimes, sub-critical, critical, and super-critical, confirming some known results and anticipating others, even in the less studied critical regime. We also numerically characterize the finite-size statistics of the gap, which we hope may encourage analytical work which is lacking. Finally, we compute the finite-size scaling of the long time relaxation of the energy, showing the existence of power laws with exponents that depend on the strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap.