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

TL;DR

This paper classifies the scaling limits of randomly trapped random walks on multidimensional integer lattices, showing they converge to either Brownian motion or a Fractional Kinetics process depending on the trap structure.

Contribution

It provides a complete classification of the scaling limits for randomly trapped random walks on $ olinebreak b Z^d$, including the case of simple random walks, confirming previous conjectures.

Findings

Scaling limit of clock process is either deterministic or a stable subordinator.

Random walk scaling limits are Brownian motion or Fractional Kinetics process.

Results confirm conjectures for simple random walks on $b Z^d$.

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].