Strong Topological Trivialization of Multi-Species Spherical Spin Glasses

TL;DR

This paper analyzes the energy landscape of multi-species spherical spin glasses, identifying phase boundaries for critical point trivialization and demonstrating implications for Langevin dynamics mixing times.

Contribution

It establishes the phase boundary for trivialization, links annealed and quenched properties, and develops new techniques for precise landscape analysis in multi-species models.

Findings

Critical points become trivial in certain regimes

Langevin dynamics mixes logarithmically at low temperature

Results extend to the 1-species case with new insights

Abstract

We study the landscapes of multi-species spherical spin glasses. Our results determine the phase boundary for annealed trivialization of the number of critical points, and establish its equivalence with a quenched strong topological trivialization property. Namely in the "trivial" regime, the number of critical points is constant, all are well-conditioned, and all approximate critical points are close to a true critical point. As a consequence, we deduce that Langevin dynamics at sufficiently low temperature has logarithmic mixing time. Our approach begins with the Kac--Rice formula. We characterize the annealed trivialization phase by explicitly solving a suitable multi-dimensional variational problem, obtained by simplifying certain asymptotic determinant formulas from (Ben Arous--Bourgade--McKenna 2023, McKenna 2024). To obtain more precise quenched results, we develop general purpose techniques to avoid sub-exponential correction factors and show non-existence of approximate critical points. Many of the results are new even in the 1-species case.