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

Véronique Gayrard, Adéla Švejda|arXiv (Cornell University)|Sep 12, 2013

Stochastic processes and statistical mechanics3 references4 citations

TL;DR

This paper establishes general criteria for the convergence of clock processes in random environments on infinite graphs using a method by Durrett and Resnick, proving that Bouchaud's asymmetric trap model on $\mathbb{Z}^d$ exhibits normal aging for all $d \geq 2$, with two-time correlation functions converging to the arcsine law. The fractional kinetics process is also shown to age.

ABSTRACT

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 the process at the same site at two time points, converge, as the age of the process diverges, to the distribution function of the arcsine law. As a byproduct we prove that the fractional kinetics process ages.

Motivation & Objective

• To develop general convergence criteria for clock processes in random environments on infinite graphs.
• To extend existing results on aging in trap models to the full range of dimensions $d \geq 2$.
• To establish almost sure convergence of clock processes, improving upon previous probability-only results.
• To demonstrate that the fractional kinetics process exhibits aging behavior.
• To unify the theoretical framework for aging and anomalous diffusion via clock process convergence.

Proposed method

• Adapts the Durrett-Resnick method for dependent random variables to derive convergence criteria for clock processes in infinite graphs.
• Analyzes the discrete and continuous-time clock processes defined via inverse holding times and jump rates in a random environment.
• Uses coupling with Brownian motion and exit time estimates to control first-passage and return probabilities.
• Applies the strong Markov property and moment bounds on Green's functions to control tail probabilities of hitting times.
• Employs scale-dependent estimates on $\mathbb{Z}^d$ for $d=2$ and $d \geq 3$, distinguishing between logarithmic and power-law decay.
• Establishes moment bounds on the number of visits to sites via sub-Gaussian and sub-Weibull tail estimates.

Experimental results

Research questions

• RQ1 Under what conditions does the clock process of a Markov jump process on an infinite graph converge to a stable subordinator?
• RQ2 Does Bouchaud’s asymmetric trap model on $\mathbb{Z}^d$ exhibit normal aging for all $d \geq 2$?
• RQ3 Can the convergence of two-time correlation functions to the arcsine law be established almost surely rather than in probability?
• RQ4 Does the fractional kinetics process, derived from the trap model, exhibit aging behavior?
• RQ5 How do the asymptotic behaviors of first-hitting times and site visit counts depend on the dimension $d$?

Key findings

• For all $d \geq 2$, the clock process of Bouchaud’s asymmetric trap model on $\mathbb{Z}^d$ converges almost surely to a stable subordinator after proper rescaling.
• The two-time correlation function for the probability of returning to the same site at two times converges to the arcsine law as the process age diverges.
• The classical aging behavior—characterized by convergence to the arcsine distribution—holds almost surely, not just in probability.
• The fractional kinetics process derived from the model is shown to age, confirming its anomalous diffusive nature.
• Moment bounds on the number of visits to sites are established via Green’s function estimates and exponential tail controls on exit times.
• For $d=2$, logarithmic corrections in the Green’s function and hitting time estimates are essential to control the tail behavior and ensure convergence.