A non-local Random Walk on the Hypercube

TL;DR

This paper analyzes a non-local random walk on the hypercube where multiple coordinates are flipped simultaneously, establishing mixing times and cutoff phenomena depending on the number of flipped coordinates.

Contribution

It provides new theoretical results on the mixing time and cutoff behavior for a non-local hypercube random walk with multiple coordinate flips.

Findings

Mixing time is of order (n/k) log n.

Cutoff occurs at (n/2k) log n when k=o(n).

The cutoff window is proportional to n/(2k).

Abstract

This paper studies the random walk on the hypercube $(\mathbb{Z}/2\mathbb{Z})^n$ which at each step flips $k$ randomly chosen coordinates. We prove that the mixing time for this walk is of order $\frac{n}{k} \log n$. We also prove that if $k=o(n)$, then the walk exhibits cutoff at $\frac{n}{2k} \log n$ with window $\frac{n}{2k} $.