An emergent autonomous flow for mean-field spin glasses

• 成果类型: Article
• 署名作者: MacLaurin, James
• 署名单位: New Jersey Institute of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01040-w
• 发表日期: 2021
• 页码: 365-438
• 关键词: order parameters; propagation; chaos; MODEL; EQUATIONS; DYNAMICS

摘要

We study the dynamics of symmetric and asymmetric spin-glass models of size N. The analysis is in terms of the double empirical process: this contains both the spins, and the field felt by each spin, at a particular time (without any knowledge of the correlation history). It is demonstrated that in the large N limit, the dynamics of the double empirical process becomes deterministic and autonomous over finite time intervals. This does not contradict the well-known fact that SK spin-glass dynamics is non-Markovian (in the large N limit) because the empirical process has a topology that does not discern correlations in individual spins at different times. In the large N limit, the evolution of the density of the double empirical process approaches a nonlocal autonomous PDE operator Phi(t). Because the emergent dynamics is autonomous, in future work one will be able to apply PDE techniques to analyze bifurcations in Phi(t). Preliminary numerical results for the SK Glauber dynamics suggest that the 'glassy dynamical phase transition' occurs when a stable fixed point of the flow operator Phi(t) destabilizes.