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Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field

Belius, David and Černý, Jiří and Nakajima, Shuta and Schmidt, Marius. (2021) Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field. Preprints Fachbereich Mathematik, 2021 (15).

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Official URL: https://edoc.unibas.ch/83754/

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Abstract

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold $h_c$ for the magnitude of the external fieldnecessary for trivialization and conjecture $h_c$ to be sharp. The Kac-Rice formula is our main tool. Our work extends [Fyo15], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Stochastik (Belius)
05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Wahrscheinlichkeitstheorie (Cerny)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Belius, David and Černý, Jiří and Nakajima, Shuta and Schmidt, Marius Alexander
Item Type:Preprint
Publisher:Universität Basel
Language:English
edoc DOI:
Last Modified:25 Jun 2021 14:56
Deposited On:25 Jun 2021 14:56

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