Aging in two-dimensional Bouchaud's model

Arous, Gérard Ben; Černý, Jǐŕı; Mountford, Thomas

doi:10.1007/s00440-004-0408-1

Cited by 46 publications

(67 citation statements)

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“…The scaling limit is a singular diffusion now known under the name of FIN. In higher dimension d ≥ 2, the scaling limit is known to be the so called Fractional Kinetics process: d-dimensional Brownian motion time changed by the inverse of an independent stable subordinator as proved in [16], [9] and [7], the strategy being similar to [8]. Besides a number of estimates on the Green kernel of simple symmetric random walk in Z d , the proof in the d = 2 case involves rather sophisticated renormalization technics.…”

Section: Remarkmentioning

confidence: 99%

“…As far as trap models on Z d are concerned, excluding the asymmetric case (where π depends on τ in a specific way, as mentioned above; see [1], [17], [6], [25]), only the case of the simple symmetric random walk was investigated so far. It corresponds to π being the uniform law on the nearest neighbors of the origin.…”

Section: Remarkmentioning

confidence: 99%

“…Let us start by enumerating the full range of X 6) in chronological order (in the natural way, i.e., given x, y ∈ R, we have x < y iff X hits x before it does y); let ψ : N → N be the map…”

Section: Remark 11mentioning

confidence: 99%

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On the dynamics of trap models in ℤ^d

Fontes

Mathieu

2013

Proceedings of the London Mathematical Society

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We consider trap models in Z d . These are stochastic processes in a random environment as follows. The environment is given by a family τ = (τ x , x ∈ Z d ) of positive iid random variables in the basin of attraction of an α-stable law, 0 < α < 1. Given τ , our process is a continuous time Markov pure jump process, whose jump chain is a in principle generic random walk in Z d , d ≥ 1, independent of τ , and τ represents the holding time averages of the continuous time process. We may think of the sites of Z d as traps, and of τ x as the depth of trap 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 suitable scaling. The limit process is given by the jumps of a certain α-stable subordinator at the inverse of another α-stable subordinator, correlated with the first subordinator. For that result, the requirements on the underlying random walk are a) the validity of a law of large numbers for its range, and b) the slow variation at infinity of the tail of the distribution of its time of return to the origin: they include all transient random walks as well as all random walks in d ≥ 2, and also many one dimensional random walks, but not the simple symmetric case. We then derive aging results for our process, namely scaling limits for some two-time correlation functions of the process; a strong form of those results requires an assumption of transience, stronger than a, b above. The scaling limit result mentioned above is an averaged result with respect to the environment. Under an additional condition on the size of the intersection of the ranges of two independent copies of the underlying random walk, roughly saying that it is small compared with size of the range, we derive a stronger scaling limit result, roughly stating that it holds in probability with respect to the environment. With that additional condition, we also strengthen the aging results, from the averaged version mentioned above, to convergence in probability with respect to the environment.

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Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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On the dynamics of trap models in ℤ^d

Fontes

Mathieu

2013

Proceedings of the London Mathematical Society

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“…Let us make a comment about the meaning of the conditional probabilities P(A 0 |l(τ −n 1/(1+α) , 0) = y) appearing in (7). One can use any regular version of the conditional probabilities of (l(τ −n 1/(1+α) , u)) u≥0 given l(τ −n 1/(1+α) , 0) to define the function y → P(A 0 |l(τ −n 1/(1+α) , 0) = y) up to a set of ν 0 -measure zero.…”

Section: Strategy Of the Proofmentioning

confidence: 99%

“…the BTM has a behavior completely different from the one-dimensional case, as shown by Ben Arous andČerný in [5], and by Ben Arous,Černý and Mountford in [7] (see also [17]). In these papers it is shown that the scaling limit of the BTM on For an spectral characterization of aging see [10].…”

Section: Introductionmentioning

confidence: 99%

Sub-Gaussian bound for the one-dimensional Bouchaud trap model

Cabezas¹

2015

Braz. J. Probab. Stat.

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Abstract. We establish a sub-Gaussian lower bound for the transition kernel of the onedimensional, symmetric Bouchaud trap model, which provides a positive answer to the behavior predicted by Bertin and Bouchaud in [8]. The proof rests on the Ray-Knight description of the local time of a one-dimensional Brownian motion. Using the same ideas we also prove the corresponding result for the FIN singular diffusion.

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The arcsine law as a universal aging scheme for trap models

Arous

Černý

2006

Comm Pure Appl Math

Self Cite

100

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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.

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Aging in two-dimensional Bouchaud's model

Cited by 46 publications

References 10 publications

On the dynamics of trap models in ℤ^d

On the dynamics of trap models in ℤ^d

Sub-Gaussian bound for the one-dimensional Bouchaud trap model

The arcsine law as a universal aging scheme for trap models

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Aging in two-dimensional Bouchaud's model

Cited by 46 publications

References 10 publications

On the dynamics of trap models in ℤ d

On the dynamics of trap models in ℤ d

Sub-Gaussian bound for the one-dimensional Bouchaud trap model

The arcsine law as a universal aging scheme for trap models

Contact Info

Product

Resources

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On the dynamics of trap models in ℤ^d

On the dynamics of trap models in ℤ^d