Glauber dynamics of the Sherrington--Kirkpatrick spin-glass model: a dynamical cavity method
Abstract
We present a closed theory, exact in the N→∞ limit, of the equilibrium spin correlations of the Sherrington--Kirkpatrick spin-glass model evolving with Glauber dynamics, above and at the transition temperature Tc. The derivation uses a dynamic cavity approach: in the evolution equation for the selected spin the other spins are represented by a fluctuating field. The influence of the selected spin on this field is included at the linear response level; it is given as a sum of the spin's past flips weighted by the correlations of the unperturbed fluctuating field. The self-consistency condition relates the time-dependent correlations of the selected spin and those of the fluctuating field. The theory predicts a (T−Tc)−2 divergence of the relaxation time upon approaching the transition and an At−1/2 decay of spin correlations at Tc. A numerical solution of the theory gives the amplitude, A=0.964± 0.007. The theoretical predictions are in excellent agreement with Glauber dynamics computer simulations.
Related
- Parisi's solution of the Sherrington--Kirkpatrick model from synchronous Monte Carlo dynamics
- Sampling from the Sherrington-Kirkpatrick Model up to Criticality
- Quenched complexity of marginal states in the Sherrington--Kirkpatrick spin glass
- Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
- Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model
BibTeX
@misc{szamel2026glauber,
title = {{Glauber dynamics of the Sherrington--Kirkpatrick spin-glass model: a dynamical cavity method}},
author = {Grzegorz Szamel},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04078},
eprint = {2611.04078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04078},
}
RIS
TY - UNPB
TI - Glauber dynamics of the Sherrington--Kirkpatrick spin-glass model: a dynamical cavity method
AU - Grzegorz Szamel
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.04078
UR - https://arxiv.org/abs/2611.04078
AB - We present a closed theory, exact in the N→∞ limit, of the equilibrium spin correlations of the Sherrington--Kirkpatrick spin-glass model evolving with Glauber dynamics, above and at the transition temperature T_c. The derivation uses a dynamic cavity approach: in the evolution equation for the selected spin the other spins are represented by a fluctuating field. The influence of the selected spin on this field is included at the linear response level; it is given as a sum of the spin's past flips weighted by the correlations of the unperturbed fluctuating field. The self-consistency condition relates the time-dependent correlations of the selected spin and those of the fluctuating field. The theory predicts a (T−T_c)^−2 divergence of the relaxation time upon approaching the transition and an At^−1/2 decay of spin correlations at T_c. A numerical solution of the theory gives the amplitude, A=0.964± 0.007. The theoretical predictions are in excellent agreement with Glauber dynamics computer simulations.
ER -