Yan V. Fyodorov, King's College London, UK

Counting Equilibria in Complex Systems via Random Matrices. (10.5h)

1. Introduction: multiple equilibria in complex systems.

2. Kac-Rice formula in 1D: counting real zeroes of random polynomials.

3. The mean number of stationary points for a p-spin spherical spin glass model. Relation to random matrices. Topology trivialization transition.

4. May-Wigner model and real Ginibre ensemble. Nonlinear analogue of May-Wigner instability transition. Mean number of stable equilibria for purely gradient dynamics and Tracy-Widom distribution. Absence of stable equilibria in non-gradient systems.

5. Mean number of equilibria for directed polymers in a random potential. Applications to depinning phenomena.