Scaling limit of the random walk among random traps on Z^d

TL;DR

This paper studies the scaling limit of a random walk in a trap environment on Z^d, showing it converges to a fractional kinetics process under certain conditions.

Contribution

It establishes the quenched subdiffusive scaling limit for the Bouchaud's trap model in high dimensions with heavy-tailed weights.

Findings

The random walk converges to a fractional kinetics process.

The time change converges to a stable subordinator.

Results rely on mixing properties of the environment.

Abstract

Attributing a positive value \tau_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (\tau_x), often known as "Bouchaud's trap model". We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d > 4. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk.