TY - JOUR

T1 - An emergent autonomous flow for mean-field spin glasses

AU - MacLaurin, James

N1 - Funding Information:
Much thanks to Colin MacLaurin (U. Queensland) for obtaining some preliminary numerical results that were incorporated into the introduction. Much thanks also to Gerard Ben Arous (NYU), David Shirokoff (NJIT), Victor Matveev (NJIT), Bruno Cessac (INRIA) and Etienne Tanre (INRIA) for interesting discussions and very helpful feedback.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/6

Y1 - 2021/6

N2 - 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.

AB - 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.

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U2 - 10.1007/s00440-021-01040-w

DO - 10.1007/s00440-021-01040-w

M3 - Article

AN - SCOPUS:85105428715

VL - 180

SP - 365

EP - 438

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1-2

ER -