Recent progress on the Mezard-Parisi Elastic Manifold model

This is joint work with Paul Bourgade and Benjamin McKenna (Courant Institute, NYU).

In this talk, we celebrate Giorgio Parisi’s 2021 Nobel Prize and come back to an important model Giorgio Parisi introduced with Marc Mezard in 1992.

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. Our main result confirms the recent formulas proposed by Yan Fyodorov and Pierre Le Doussal (in 2020). It gives the phase diagram and identifies the boundary between simple and disordered (glassy) phases.

Our approach relies on new exponential asymptotics of random determinants, for non-invariant random matrices.

If time permits, I will also explain what the next steps are, or should be, in order to reach the long term goal of understanding the de-pinning transition statically and dynamically.