Abstract
The spherical p-spin model is not only a fundamental model in statistical mechanics of disordered system, but has recently gained popularity since many hard problems in machine learning can be mapped on it. Thus the study of the out of equilibrium dynamics in this model is interesting both for the glass physics and for its implications on algorithms solving NP-hard problems. We revisit the long-time limit of the out of equilibrium dynamics of mean-field spherical mixed p-spin models. We consider quenches (gradient descent dynamics) starting from initial conditions thermalized at some temperature in the ergodic phase. We perform numerical integration of the dynamical mean-field equations of the model and we find an unexpected dynamical phase transition. Below an onset temperature, higher than the dynamical transition temperature, the asymptotic energy goes below the "threshold energy" of the dominant marginal minima of the energy function and memory of the initial condition is kept. This behavior, not present in the pure spherical p-spin model, resembles closely the one observed in simulations of glass-forming liquids. We then investigate the nature of the asymptotic dynamics, finding an aging solution that relaxes towards deep marginal minima, evolving on a restricted marginal manifold. Careful analysis, however, rules out simple aging solutions. We compute the constrained complexity in the aim of connecting the asymptotic solution to the energy landscape.
Highlights
Understanding the relation between the dynamical behavior and the underlying energy landscape is a fundamental question in the physics of glassy systems
If we integrate the dynamical equations for a small temperature Tf, we observe that the finite-time energy is continuous, Eðt; TfÞ 1⁄4 Eðt; Tf 1⁄4 0Þ þ oðTfÞ
While the known solution of the relaxation dynamics in the pure p-spin model suggested the existence of a unique dynamical phase transition at TMCT, where ergodicity breaks down, and of a unique threshold energy where the dynamics relaxes below TMCT, our results about the mixed p-spin model reveal a much richer scenario
Summary
Understanding the relation between the dynamical behavior and the underlying energy landscape is a fundamental question in the physics of glassy systems. Its very explicit exact solution provides the complete description of glassy phenomena in the mean field, including the long-time dynamics [14,15], thermodynamics [13], and the structure of metastable states [16] This model makes the RFOT more explicit and makes it easier to produce theoretical predictions [17]. The threshold energy as the value reached after a quench from a completely random point (infinite temperature), we find that in this model, the relaxation dynamics can go below threshold while aging and keeping memory of the short-time evolution This picture is richer than the one that is usually associated with a RFOT, providing support for an improved RFOT already at the mean-field level. In Appendix F, we describe the numerical solution of the dynamical equations
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