We review the aging phenomenon in the context of the simplest trap model, Bouchaud's REM-like trap model, from a spectral theoretic point of view. We show that the generator of the dynamics of this model can be diagonalized exactly. Using this result, we derive closed expressions for correlation functions in terms of complex contour integrals that permit an easy investigation into their large time asymptotics in the thermodynamic limit. We also give a ``grand canonical'' representation of the model in terms of the Markov process on a Poisson point process. In this context we analyze the dynamics on various time scales.
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.
We obtain scaling limit results for asymmetric trap models and their infinite volume counterparts, namely asymmetric K processes. Aging results for the latter processes are derived therefrom.
We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and Černý for a large class of graphs.
We discuss the long term behaviour of trap models on the integers with asymptotically vanishing drift, providing scaling limit theorems and ageing results. Depending on the tail behaviour of the traps and the strength of the drift, we identify three different regimes, one of which features a previously unobserved limit process.
We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the REM-like trap model ``at low temperature'', and subsequently derive aging results for those models in this context.
In the seminal work [5], Ben Arous and Černý give a general characterization of aging for trap models in terms of $α$-stable subordinators with $α\in (0,1)$. Some of the important examples that fall into this universality class are Random Hopping Time (RHT) dynamics of Random Energy Model (REM) and $p$-spin models observed on exponential time scales. In this paper, we explain a different aging mechanism in terms of {\it extremal processes} that can be seen as the extension of $α$-stable aging to the case $α=0$. We apply this mechanism to the RHT dynamics of the REM for a wide range of temperat…
The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume $L$-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit $L\to\infty$ of the two-time correlation function of the infinite volume $L$-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that…
We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the trap model in the complete graph, and subsequently derive aging results for those models in this context.
We study in details the dynamics of the one dimensional symmetric trap model, via a real-space renormalization procedure which becomes exact in the limit of zero temperature. In this limit, the diffusion front in each sample consists in two delta peaks, which are completely out of equilibrium with each other. The statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit $T \to 0$. We first compute disorder averages of one-time observables, such as the diffusion front, the thermal width, the localization para…
In this paper the celebrated arcsine aging scheme of G. Ben Arous and J. Cerný is taken up. Using a brand new approach based on point processes and weak convergence techniques, this scheme is implemented in a wide class of Markov processes that can best be described as Glauber dynamics of discrete disordered systems. More specifically, conditions are given for the underlying clock process (a partial sum process that measures the total time elapsed along paths of a given length) to converge to a subordinator, and this subordinator is constructed explicitly. This approach is illustrated on Bouch…
We consider the dynamics of the disordered trap model, which is known to be completely out-of-equilibrium and to present strong localization effects in its aging phase. We are interested into the influence of an external force, when it is applied from the very beginning at $t=0$, or only after a waiting time $t_w$. We obtain a "non-linear Fluctuation Theorem" for the corresponding one-time and two-time diffusion fronts in any given sample, that implies the following consequences : (i) for fixed times, there exists a linear response regime, where the Fluctuation-Dissipation Relation or Einstein…
Using a method developed by Durrett and Resnick [22] we establish general criteria for the convergence of properly rescaled clock processes of random dynamics in random environments on infinite graphs. This complements the results of [26], [19], and [20]: put together these results provide a unified framework for proving convergence of clock processes. As a first application we prove that Bouchaud's asymmetric trap model on ${\Bbb Z}^d$ exhibits a normal aging behavior for all $d\geq 2$. Namely, we show that certain two-time correlation functions, among which the classical probability to find …
We study the dynamics of the one dimensional disordered trap model presenting a broad distribution of trapping times $p(τ) \sim 1/τ^{1+μ}$, when an external force is applied from the very beginning at $t=0$, or only after a waiting time $t_w$, in the linear as well as in the non-linear response regime. Using a real-space renormalization procedure that becomes exact in the limit of strong disorder $μ\to 0$, we obtain explicit results for many observables, such as the diffusion front, the mean position, the thermal width, the localization parameters and the two-particle correlation function. In …
We survey the recent mathematical results about aging in certain simple disordered models. We start by the Bouchaud trap model. We then survey the results obtained for simple models of spin-glass dynamics, like the REM (the Random Energy Model, which is well approximated by the Bouchaud model on the complete graph), then the spherical Sherrington-Kirkpatrick model. We will insist on the differences in phenomenology for different types of aging in different time scales and different models. This talk is based on joint works with A.Bovier, J.Cerny, A.Dembo, V.Gayrard, A.Guionnet, as well as work…
Let $\mathcal X=\{\mathcal X_t:\, t\geq0,\, \mathcal X_0=0\}$ be a mean zero $β$-stable random walk on $\mathbb{Z}$ with inhomogeneous jump rates $\{τ_i^{-1}: i\in\mathbb{Z}\}$, with $β\in(1,2]$ and $\{τ_i: i\in\mathbb{Z}\}$ a family of independent random variables with common marginal distribution in the basin of attraction of an $α$-stable law, $α\in(0,1)$. In this paper we derive results about the long time behavior of this process, in particular its scaling limit, given by a $β$-stable process time-changed by the inverse of another process, involving the local time of the $β$-stable proces…
We study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatures for which such a result can be expected to hold. This establishes that the dynamics ages in the same way as Bouchaud's REM-like trap model, thus extending the universality class of the latter model. The proof relies on a clock process convergence …
We study the aging behavior of the Random Energy Model (REM) evolving under Metropolis dynamics. We prove that a classical two-time correlation function converges almost surely to the arcsine law distribution function that characterizes activated aging, as predicted in the physics literature, in the optimal domain of the time-scale and temperature parameters where this result can be expected to hold. In the course of the proof we establish that a certain continuous time clock process, after proper rescaling, converges almost surely to a stable subordinator, improving upon the result of Cerny a…
We consider Random Hopping Time (RHT) dynamics of the Sherrington - Kirkpatrick (SK) model and p-spin models of spin glasses. For any of these models and for any inverse temperature we prove that, on time scales that are sub-exponential in the dimension, the properly scaled clock process (time-change process) of the dynamics converges to an extremal process. Moreover, on these time scales, the system exhibits aging like behavior which we called extremal aging. In other words, the dynamics of these models ages as the random energy model (REM) does. Hence, by extension, this confirms Bouchaud's …
We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of positive iid random variables in the basin of attraction of an alpha-stable law, 0<alpha<1. We may think of x as a trap, and tau_x as the depth of the trap at x. We are interested in the trap process, namely the process that associates to time t the depth of the currently visited trap. Our first result is the convergence of the law of that process under sui…
Applying the new tools developed in [G1], we investigate the arcsine aging regime of the random hopping time dynamics of the REM. Our results are optimal in several ways. They cover the full time-scale and temperature domain where this phenomenon occurs. On this domain the limiting clock process and associated time correlation function are explicitly constructed. Finally, all convergence statements w.r.t. the law of the random environment are obtained in the strongest sense possible, except perhaps on the very last scales before equilibrium.
