Randomly trapped random walks on ${\mathbb{Z}}^d$

Abstract

We give a complete classification of scaling limits of randomly trapped random walks and associated clock processes on ${\mathbb{Z}}^d$, $d\ge 2$. Namely, under the hypothesis that the discrete skeleton of the randomly trapped random walk has a slowly varying return probability, we show that the scaling limit of its clock process is either deterministic linearly growing or a stable subordinator. In the case when the discrete skeleton is a simple random walk on ${\mathbb{Z}}^d$, this implies that the scaling limit of the randomly trapped random walk is either Brownian motion or the Fractional Kinetics process, as conjectured in [BCCR13].

revised version, 24 pages