Communications in Mathematical Physics

, Volume 282, Issue 3, pp 663-695

First online:

Universality of the REM for Dynamics of Mean-Field Spin Glasses

  • Gérard Ben ArousAffiliated withCourant Institute of the Mathematical Sciences, New York University
  • , Anton BovierAffiliated withWeierstrass Institute for Applied Analysis and StochasticsMathematics Institute, Berlin University of Technology
  • , Jiří ČernýAffiliated withDepartment of Mathematics, ETH Zürich Email author 

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We consider a version of 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 all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ β ,p  > 0, such that for all exponential time scales, exp(γ N), with γ < γ β ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β 2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-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.