On the dynamics of trap models in ℤ d

Abstract

We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of positive iid random variables in the basin of attraction of an alpha-stable law, 0<alpha<1. We may think of x as a trap, and tau_x as the depth of the trap at 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 for the embedded 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 planar random walks, and also many one dimensional random walks. We then derive aging results for the process, namely scaling limits for some two-time correlation functions thereof, a strong form of which requires an assumption of transience, stronger than a, b. The above mentioned scaling limit result is an averaged result with respect to tau. Under an additional condition on the size of the intersection of the ranges of two independent copies of the embeddded random walk, roughly saying that it is small compared with the size of the range, we derive a stronger scaling limit result, roughly stating that it holds in probability with respect to tau. With that additional condition, we also strengthen the aging results, from the averaged version mentioned above, to convergence in probability with respect to tau.