WIAS Preprint No. 1241, (2007)

Universality of the REM for dynamics of mean-field spin glasses


  • Ben Arous, Gérard
  • Bovier, Anton
  • Černý, Jiři

2010 Mathematics Subject Classification

  • 82C44 60K35 60G70


  • aging, universality, spin glasses, SK model, random walk




We consider a version of a Glauber dynamics for a $p$-spin Sherrington--Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the $N$-dimensional hypercube. We show that, for any $p geq 3$ and any inverse temperature $beta>0$, there exist constants $g_0>0$, such that for all exponential time scales, $exp(gamma N)$, with $gleq g_0$, the properly rescaled emphclock process (time-change process), converges to an $a$-stable subordinator where $a=g/b^2<1$. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this hbox$alpha$-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of $p$-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case $p=2$) seems to belong to a different universality class.

Appeared in

  • J. Stat. Mech. Theory Exp., 4 (2008) pp. L04003/1--L04003/8.

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