UNIVERSALITY FOR LANGEVIN-LIKE SPIN GLASS DYNAMICS

• 成果类型: Article
• 署名作者: Dembo, Amir; Lubetzky, Eyal; Zeitouni, Ofer
• 署名单位: Stanford University; Stanford University; New York University; Weizmann Institute of Science
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1665
• 发表日期: 2021
• 页码: 2864-2880
• 关键词: mean-field theory; large deviations; neural-networks; systems

摘要

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 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).