The topology of the elastic manifold

This is joint work with Paul Bourgade and Benjamin McKenna (both Courant Institute, NYU) and relies on the joint recent papers arXiv:2105.05051 and arXiv:2105.05000.

The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations), on the one hand, and elastic self-interactions (preferring ordered configurations), on the other. The elastic manifold model is interesting because it displays a depinning 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 or by Le Doussal-Mueller-Wiese (2007) which relies on functional renormalization group.

We study the topology of the energy landscape of this model in the Mézard-Parisi setting, and compute the (annealed) topological complexity both of total critical points and of local minima. Our main result confirms the recent formulas obtained by Fyodorov and Le Doussal (2020). It  identifies the boundary between simple and glassy phases, as well as the nature of the phase transition. The main argument relies naturally on Random Matrix Theory, through the Kac-Rice formula.

15:00 - 16:00 UTC  Benjamin McKenna

Random determinants beyond invariance

This is partially joint work with Gérard Ben Arous and Paul Bourgade (both Courant Institute, NYU) and relies on the recent joint paper arXiv:2105.05000 and solo paper arXiv:2105.05043.

To study the topology of high-dimensional random functions via the Kac-Rice formula, the core requirement is the analysis of the asymptotic behavior of large random determinants in the exponential scale. For models like the elastic manifold, these random determinants are not invariant under the usual groups of symmetries, as for usual models of spherical spin glasses for instance. Thus their asymptotic evaluation is rather delicate. We give an abstract result for the behavior of large random determinants beyond the invariant case, which cover many other interesting models beyond the elastic manifold. As an example of one such model, we discuss bipartite spherical spin glasses.


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