An emergent autonomous flow for mean-field spin glasses

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Abstract

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 Φ t. Because the emergent dynamics is autonomous, in future work one will be able to apply PDE techniques to analyze bifurcations in Φ 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 Φ t destabilizes.

Original languageEnglish (US)
Pages (from-to)365-438
Number of pages74
JournalProbability Theory and Related Fields
Volume180
Issue number1-2
DOIs
StatePublished - Jun 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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