A curiously slowly mixing Markov chain

TL;DR

This paper analyzes a Markov chain called the Burnside process on the hypercube, revealing that its mixing times vary significantly depending on the measurement norm, with rapid mixing in some norms and slow in others.

Contribution

It provides an explicit diagonalization of the Burnside process on the hypercube, demonstrating the differing mixing rates in various norms and for different starting states.

Findings

Fast mixing in  and  norms from all-zero state

Logarithmic mixing time in  norm from any starting point

Polynomial mixing time in  norm for most starting points

Abstract

We study a Markov chain with very different mixing rates depending on how mixing is measured. The chain is the "Burnside process on the hypercube $C_2^n$." Started at the all-zeros state, it mixes in a bounded number of steps, no matter how large $n$ is, in $\ell^1$ and in $\ell^2$. And started at general $x$, it mixes in at most $\log n$ steps in $\ell^1$. But, in $\ell^2$, it takes $\frac{n}{\log n}$ steps for most starting $x$. The $\ell^2$ mixing results follow from an explicit diagonalization of the Markov chain into binomial-coefficient-valued eigenvectors.