Mixing time for the random walk on the range of the random walk on tori

TL;DR

This paper investigates the mixing time of a random walk on the subgraph formed by the range of another random walk on a high-dimensional torus, showing it is typically of order N^2.

Contribution

It establishes that the mixing time for the random walk on the range subgraph is of order N^2 for all dimensions d ≥ 3, with high probability.

Findings

Mixing time is of order N^2 for the subgraph.

High probability bound with exponentially small error.

Results hold for all dimensions d ≥ 3.

Abstract

Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\ge3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\ge 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\log N)^2}$.