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December 2021 Universality for Langevin-like spin glass dynamics
Amir Dembo, Eyal Lubetzky, Ofer Zeitouni
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Ann. Appl. Probab. 31(6): 2864-2880 (December 2021). DOI: 10.1214/21-AAP1665

Abstract

We study dynamics for asymmetric spin glass models, proposed by Hertz et al. and Sompolinsky et al. in the 1980’s in the context of neural networks: particles evolve via a modified Langevin dynamics for the Sherrington–Kirkpatrick model with soft spins, whereby the disorder is i.i.d. standard Gaussian rather than symmetric. Ben Arous and Guionnet (Probab. Theory Related Fields 102 (1995) 455–509), followed by Guionnet (Probab. Theory Related Fields 109 (1997) 183–215), proved for Gaussian interactions that as the number of particles grows, the short-term empirical law of this dynamics converges a.s. to a nonrandom law μ of a “self-consistent single spin dynamics,” as predicted by physicists. Here we obtain universality of this fact: For asymmetric disorder given by i.i.d. variables of zero mean, unit variance and exponential or better tail decay, at every temperature, the empirical law of sample paths of the Langevin-like dynamics in a fixed time interval has the same a.s. limit μ.

Funding Statement

A.D. was supported in part by NSF grant DMS- 1954337 and E.L. was supported in part by NSF Grant DMS-1812095. This research was further supported in part by BSF Grant 2018088.

Acknowledgments

We thank G. Ben Arous and A. Guionnet for a valuable feedback on our preliminary draft and for pointing our attention to the references [4, 15, 17, 26].

Citation

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Amir Dembo. Eyal Lubetzky. Ofer Zeitouni. "Universality for Langevin-like spin glass dynamics." Ann. Appl. Probab. 31 (6) 2864 - 2880, December 2021. https://doi.org/10.1214/21-AAP1665

Information

Received: 1 January 2020; Revised: 1 August 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/21-AAP1665

Subjects:
Primary: 60K35
Secondary: 60F10 , 60H10 , 82C31 , 82C44

Keywords: Interacting random processes , Langevin dynamics , SDEs , Universality

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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