Biased random walk on the interlacement set

TL;DR

This paper investigates a biased random walk on the interlacement set in three or more dimensions, revealing that despite transience, the walk's speed is zero and it moves slower than any polynomial rate.

Contribution

It demonstrates that in three dimensions, the biased walk remains transient but has zero limiting speed regardless of bias, highlighting a novel slow movement phenomenon.

Findings

The walk is always transient in dimensions d≥3.

In d=3, the walk's speed is zero for any bias.

The walk moves slower than any polynomial rate in d=3.

Abstract

We study a biased random walk on the interlacement set of $\mathbb{Z}^d$ for $d\geq 3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.