# Cutoff for a stratified random walk on the hypercube

## TL;DR

This paper proves that a specific random walk on the hypercube exhibits a sharp cutoff at time frac{3}{2}nlog n with a window of size n, answering a longstanding question.

## Contribution

It establishes the cutoff time and window for the hypercube random walk, resolving a question posed by Chung and Graham in 1997.

## Key findings

- The Markov chain exhibits cutoff at frac{3}{2}nlog n.
- The cutoff window size is n.
- The result confirms the sharp transition in mixing time.

## Abstract

We consider the random walk on the hypercube which moves by picking an ordered pair $(i,j)$ of distinct coordinates uniformly at random and adding the bit at location $i$ to the bit at location $j$, modulo $2$. We show that this Markov chain has cutoff at time $\frac{3}{2}n\log n$ with window of size $n$, solving a question posed by Chung and Graham (1997).