Publications (5)1.94 Total impact
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Article: Universality of the REM for dynamics of mean-field spin glasses
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ABSTRACT: 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>2 and any inverse temperature \beta>0, there exist constants g>0, such that for all exponential time scales, $\exp(\gamma N)$, with $\gamma< g$, the properly rescaled clock process (time-change process), converges to an \alpha-stable subordinator where \alpha=\gamma/\beta^2<1. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this \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.07/2007; -
Article: Dynamics of trap models
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ABSTRACT: These notes cover one of the topics of the class given in the Les Houches Summer School ``Mathematical statistical physics'' in July 2005. The lectures tried to give a summary of the recent mathematical results about the long-time behaviour of dynamics of (mean-field) spin-glasses and other disordered media. We have chosen here to restrict the scope of these notes to the dynamics of trap models only, but to cover this topic in somewhat more depth.04/2006; -
Article: The arcsine law as a universal aging scheme for trap models
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ABSTRACT: We give a general proof of aging for trap models using the arcsine law for stable subordinators. This proof is based on abstract conditions on the potential theory of the underlying graph and on the randomness of the trapping landscape. We apply this proof to aging for trap models on large two-dimensional tori and for trap dynamics of the Random Energy Model on a broad range of time scales.04/2006; -
Article: Glauber Dynamics of the Random Energy Model I. Metastable motion on the extreme states.
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ABSTRACT: We investigate the long-time behavior of the Glauber dynamics for the random energy model below the critical temperature. We give very precise estimates on the motion of the process to and between the states of extremal energies. We show that when disregarding time, the consecutive steps of the process on these states are governed by a Markov chain that jumps uniformly on all possible states. The mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point. A rst indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order T is itself of order T . The estimates proven in this paper will furnish crucial input for a follow-up paper where aging is analysed in full detail.Communications in Mathematical Physics 11/2003; 235:379--425. · 1.94 Impact Factor -
Article: Bouchaud's model exhibits two different aging regimes in dimension one
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ABSTRACT: Let E_i be a collection of i.i.d. exponential random variables. Bouchaud's model on Z is a Markov chain X(t) whose transition rates are given by w_{ij}=\nu \exp(-\beta ((1-a)E_i-aE_j)) if i, j are neighbors in Z. We study the behavior of two correlation functions: P[X(t_w+t)=X(t_w)] and P[X(t')=X(t_w) \forall t'\in[t_w,t_w+t]]. We prove the (sub)aging behavior of these functions when \beta >1 and a\in[0,1].11/2002;
