Topological complexity of random landscapes: the Mezard-Parisi Elastic Manifold (Gerard BEN AROUS, New York University)

10.03.2022 16:15 – 18:00

We will cover the recent progress on the understanding of the topological complexity of functions of many variables. We will first review very quickly the relatively well understood cases of (spherical) spin glasses and Tensor PCA before focusing on the recent work with Paul Bourgade and Ben Mc Kenna on the Mezard-Parisi model of the Elastic Manifold. Our main result will confirm the recent formulas proposed by Yan Fyodorov and Pierre Le Doussal (in 2020).
The Elastic Manifold model is a paradigmatic representative of the class of disordered elastic systems. These are surfaces with rugged shapes resulting from a competition between elastic self-interactions (preferring ordered configurations) on the one hand, and random spatial impurities (preferring disordered configurations) on the other. The elastic manifold model is interesting because it displays a crucial de-pinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the high-dimensional limit by Mézard and Parisi (1992) using the replica method. We study the energy landscape of this model, and compute the (annealed) topological complexity both of total critical points and of local minima, in the Mezard-Parisi high dimensional limit. It gives the phase diagram and identifies the boundary between simple and disordered (glassy) phases. Our approach relies on new exponential asymptotics of determinants of non-invariant random matrices.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-15

entrée libre