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

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