Dynamics of the spherical Sherrington-Kirkpatrick model and average case complexity for top eigenvectors

Abstract

In this Master’s thesis, we investigate the Langevin dynamics on the spherical Sherrington- Kirkpatrick (SSK) model, a classical mean-field spin glass model. The first contribution of this thesis is the asymptotic limit of energy function of the SSK model, a critical property linked to the model’s equilibrium state. The thermodynamic limit of energy of the system is characterized in terms of a system of integro-differential equations as the size of the system goes to infinity. Then we look at the behavior of the limiting dynamics as the time goes to infinity. This long time behavior of the energy has a phase transition. In the regime of below the critical inverse temperature, the limiting result is zero. In the regime of above the critical inverse temperature, the limiting result is a constant depending on the temperature.

The second contribution of this thesis is that we analyze the complexity of the zerotemperature Langevin dynamics (a.k.a. the gradient descent algorithm) on the SSK model. We establish lower and upper bound for the hitting time, defined as the first time required for the output of the algorithm to achieve a small overlap with the eigenvector corresponding to the smallest eigenvalue of the Wigner matrix.