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The spherical p-spin model is not only a fundamental model in statisticalmechanics of disordered system, but has recently gained popularity since manyhard problems in machine learning can be mapped on it. Thus the study of theout of equilibrium dynamics in this model is interesting both for the glassphysics and for its implications on algorithms solving NP-hard problems. Werevisit the long-time limit of the out of equilibrium dynamics of mean-fieldspherical mixed p-spin models. We consider quenches (gradient descent dynamics)starting from initial conditions thermalized at some temperature in the ergodicphase. We perform numerical integration of the dynamical mean-field equationsof the model and we find an unexpected dynamical phase transition. Below anonset temperature, higher than the dynamical transition temperature, theasymptotic energy goes below the "threshold energy" of the dominant marginalminima of the energy function and memory of the initial condition is kept. Thisbehavior, not present in the pure spherical p-spin model, resembles closely theone observed in simulations of glass-forming liquids. We then investigate thenature of the asymptotic dynamics, finding an aging solution that relaxestowards deep marginal minima, evolving on a restricted marginal manifold.Careful analysis, however, rules out simple aging solutions. We compute theconstrained complexity in the aim of connecting the asymptotic solution to theenergy landscape.

Rethinking mean-field glassy dynamics and its relation with the energy landscape: the awkward case of the spherical mixed p-spin model