Gradient descent dynamics in the mixed $p$-spin spherical model: finite size simulation and comparison with mean-field integration

TL;DR

This study uses numerical simulations to analyze gradient descent dynamics in a mixed p-spin spherical spin glass model, confirming mean-field predictions and revealing temperature-dependent inherent structures, thus strengthening the analogy with supercooled liquids.

Contribution

It provides the first finite-size simulation validation of mean-field dynamical equations for mixed p-spin models, highlighting the existence of an onset temperature affecting inherent structures.

Findings

Dynamics agree with mean-field equations in large systems

Existence of an onset initial temperature within the liquid phase

Inherent structure energies depend on initial temperature below this onset

Abstract

We perform numerical simulations of a long-range spherical spin glass with two and three body interaction terms. We study the gradient descent dynamics and the inherent structures found after a quench from initial conditions, well thermalized at temperature $T_{in}$. In large systems, the dynamics strictly agrees with the integration of the mean-field dynamical equations. In particular, we confirm the existence of an onset initial temperature, within the liquid phase, below which the energy of the inherent structures undoubtedly depends on $T_{in}$. This behavior is in contrast with that of pure models, where there is a 'threshold energy' that attracts all the initial configurations in the liquid. Our results strengthen the analogy between mean-field spin glass models and supercooled liquids.