  We discuss the properties of the distributions of energies of minima obtained
by gradient descent in complex energy landscapes. We find strikingly similar
phenomenology across several prototypical models. We particularly focus on the
distribution of energies of minima in the analytically well-understood
p-spin-interaction spin glass model. We numerically find non-Gaussian
distributions that resemble the Tracy-Widom distributions often found in
problems of random correlated variables, and non-trivial finite-size scaling.
Based on this, we propose a picture of gradient descent dynamics that
highlights the importance of a first-passage process in the eigenvalues of the
Hessian. This picture provides a concrete link to problems in which the
Tracy-Widom distribution is established. Aspects of this first-passage view of
gradient-descent dynamics are generic for non-convex complex landscapes,
rationalizing the commonality that we find across models.
