Universality for Langevin-like spin glass dynamics

TL;DR

This paper proves that the empirical law of Langevin-like spin glass dynamics converges to a universal limit for a broad class of disorder distributions, extending previous Gaussian-specific results to more general cases.

Contribution

It establishes universality of the limiting empirical law for asymmetric spin glass dynamics with non-Gaussian disorder, broadening the scope of prior Gaussian-based findings.

Findings

Empirical law converges almost surely to a universal limit.

Universality holds for disorder with zero mean, unit variance, and exponential tail decay.

Results apply at all temperatures for fixed time intervals.

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 (1995), followed by Guionnet (1997), proved for Gaussian interactions that as the number of particles grows, the short-term empirical law of this dynamics converges a.s. to a non-random law $\mu_\star$ 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 $\mu_\star$.