The behaviour of aging functions in one-dimensional Bouchaud's trap model
- Černý, Jiri
2010 Mathematics Subject Classification
- 82D30 60K37 82C41
- Trap models, aging, Levy processes, singular diffusions
Let tau_x be a collection of i.i.d. positive random variables with distribution in the domain of attraction of alpha-stable law with alpha <1. The symmetric Bouchaud's trap model on Z is a Markov chain X(t) whose transition rates are given by w_xy=(2tau_x)^-1 if x, y are neighbours in Z. We study the behaviour of two correlation functions: P[X(t_w+t)=X(t_w)] and P[X(t')=X(t_w)forall t'in[t_w,t_w+t]]. It is well known that for any of these correlation functions a time-scale t=f(t_w) such that aging occurs can be found. We study these correlation functions on time-scales different from f(t_w), and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of Bouchaud's trap model.