Strong Topological Trivialization of Multi-Species Spherical Spin Glasses
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913856169705472 |
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| author | Huang, Brice Sellke, Mark |
| author_facet | Huang, Brice Sellke, Mark |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09677 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strong Topological Trivialization of Multi-Species Spherical Spin Glasses Huang, Brice Sellke, Mark Probability Disordered Systems and Neural Networks Mathematical Physics Primary 60G15, 60G60, secondary 82D30 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. |
| title | Strong Topological Trivialization of Multi-Species Spherical Spin Glasses |
| topic | Probability Disordered Systems and Neural Networks Mathematical Physics Primary 60G15, 60G60, secondary 82D30 |
| url | https://arxiv.org/abs/2308.09677 |