The arcsine law as a universal aging scheme for trap models

Scaling limits and aging for asymmetric trap models on the complete graph and K-processes

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.

Trap models with vanishing drift: Scaling limits and aging regimes

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.

电阻度量空间上Bouchaud陷阱模型的老化与次老化

Aging and sub-aging for Bouchaud trap models on resistance metric spaces

该研究针对电阻度量空间上的Bouchaud陷阱模型，证明了电网络在局部Gromov-Hausdorff拓扑下收敛且满足非爆炸条件时模型收敛并呈现老化，局部结构收敛时存在次老化，结果适用于多类低维图，证明基于拓扑度量化与转移密度预紧性两步。

K-processes, scaling limit and aging for the REM-like trap model

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.

Extremal Aging For Trap Models

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…

Aging in the GREM-like trap model

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…

K-processes, scaling limit and aging for the trap model in the complete graph

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.

Spectral characterization of aging: the rem-like trap model

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 dynami…

Parisi树上Bouchaud和Dean陷阱模型在遍历与老化时间尺度下的标度极限

Scaling limits of the Bouchaud and Dean trap model on Parisi's tree in ergodic and aging time scales

本文在Parisi树上研究Bouchaud和Dean陷阱模型，在遍历与老化时间尺度下建立标度极限，并借助级联跳跃演化的连续性定理推导结果。

Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on ${\Bbb Z}^d$

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 …

Anomalous diffusion, Localization, Aging and Sub-aging effects in trap models at very low temperature

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…

On a non-linear Fluctuation Theorem for the aging dynamics of disordered trap models

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…

Non-linear Response of the trap model in the aging regime : Exact results in the strong disorder limit

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 …

Long range trap models on Z and quasistable processes

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…

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

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 β>0, there exist constants g>0, such that for all exponential time scales, $\exp(γN)$, with $γ< g$, 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 time-time correlation function converging to the arcsine law of this α-stabl…