On the dynamics of trap models in Z(d)

Author(s)

Fontes, L. R. G. ; Mathieu, P.

• Document type: Journal article
• Source: PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY; v. 108, p. 31-pg., 2014-06-01.

Abstract

We consider trap models in a"currency sign(d). These are stochastic processes in a random environment as follows. The environment is given by a family tau=(tau(x), xaa"currency sign(d)) of positive independent and identically distributed random variables in the basin of attraction of an alpha-stable law, 0 <alpha < 1. Given tau, our process is a continuous time Markov pure jump process, whose jump chain is a, in principle, generic random walk in a"currency sign(d), da parts per thousand yen1, independent of tau, and tau represents the holding time averages of the continuous time process. We may think of the sites of a"currency sign(d) as traps, and of tau(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 alpha-stable subordinator at the inverse of another alpha-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 da parts per thousand yen2, and also many one-dimensional random walks, but not the simple symmetric case. We then derive ageing 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 ageing results, from the averaged version mentioned above, to convergence in probability with respect to the environment. (AU)

• FAPESP's process: 09/52379-8 - Stochastic modeling of interacting systems
• Grantee: Fabio Prates Machado
• Support Opportunities: Research Projects - Thematic Grants