Fluctuation Distributions of Energy Minima in Complex Landscapes

We discuss the properties of the distributions of energies of minima obtained by gradient descent in complex energy landscapes.Specifically, we study the distribution of energies of minima in the spherical p-spin model and the distribution of jamming threshold packing fractions in jammed particle configurations as archetypal manifestations of disorder-induced complexity. We numerically find universal distributions that resemble the Tracy-Widom distributions often found in problems of random correlated variables, and non-trivial finite-size scaling. Deeper insight into this problem is achieved by realizing the importance of a first-passage process in the eigenvalues of the Hessian to the termination of the steepest descent process, which also manifests the link to problems where the Tracy-Widom distribution is established. This first-passage view of steepest descent dynamics is generic and therefore we expect similar phenomenology in many problems.