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Strong topological trivialization of multi-species spherical spin glasses

Brice Huang & Mark Sellke

Annals of Probability2026https://doi.org/10.1214/25-aop1785article
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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 multidimensional variational problem, obtained by simplifying certain asymptotic determinant formulas from (Probab. Math. Phys. 3 (2022) 731–789; Ann. Inst. Henri Poincaré Probab. Stat. 60 (2024) 636–657). To obtain more precise quenched results, we develop general purpose techniques to avoid subexponential correction factors and show nonexistence of approximate critical points. Many of the results are new, even in the one-species case.

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https://doi.org/https://doi.org/10.1214/25-aop1785

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@article{brice2026,
  title        = {{Strong topological trivialization of multi-species spherical spin glasses}},
  author       = {Brice Huang & Mark Sellke},
  journal      = {Annals of Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aop1785},
}

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