Universality for Langevin-like spin glass dynamics

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 ${\mathit{\mu }}_{\star }$ 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 ${\mathit{\mu }}_{\star }$.