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A curiously slowly mixing Markov chain

Persi Diaconis, Andrew Lin, Arun Ram

TL;DR

This paper analyzes the Burnside process on the binary hypercube $C_2^n$, revealing an extreme discrepancy between $\ell^1$ (total-variation) and $\ell^2$ (chi-square) mixing times. It achieves this through an explicit diagonalization of the transition kernel: the nonzero eigenvalues are $\beta_k=\frac{1}{2^{4k}}\binom{2k}{k}^2$ with multiplicities $\binom{n}{2k}$, and corresponding eigenvectors are constructed via lifted discrete Chebyshev polynomials. The authors prove that, for most starting states, the $\ell^2$ mixing time scales as $\Theta\left(\frac{n}{\log n}\right)$, whereas starting from the all-zeros state yields rapid $\ell^2$ convergence; in contrast, $\ell^1$ mixing is bounded by a constant for all starting states. They also demonstrate the lumping structure that reduces the analysis to orbits and discuss extensions to $C_k^n$, highlighting both similarities and substantial challenges beyond the binary case. The work provides sharp qualitative and quantitative insights into how spectral structure and symmetry govern mixing across norms, with implications for related Markov chains and sampling algorithms.

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.

A curiously slowly mixing Markov chain

TL;DR

This paper analyzes the Burnside process on the binary hypercube , revealing an extreme discrepancy between (total-variation) and (chi-square) mixing times. It achieves this through an explicit diagonalization of the transition kernel: the nonzero eigenvalues are with multiplicities , and corresponding eigenvectors are constructed via lifted discrete Chebyshev polynomials. The authors prove that, for most starting states, the mixing time scales as , whereas starting from the all-zeros state yields rapid convergence; in contrast, mixing is bounded by a constant for all starting states. They also demonstrate the lumping structure that reduces the analysis to orbits and discuss extensions to , highlighting both similarities and substantial challenges beyond the binary case. The work provides sharp qualitative and quantitative insights into how spectral structure and symmetry govern mixing across norms, with implications for related Markov chains and sampling algorithms.

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 ." Started at the all-zeros state, it mixes in a bounded number of steps, no matter how large is, in and in . And started at general , it mixes in at most steps in . But, in , it takes steps for most starting . The mixing results follow from an explicit diagonalization of the Markov chain into binomial-coefficient-valued eigenvectors.

Paper Structure

This paper contains 12 sections, 14 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Analytic background
  4. Some uses for the eigenvectors
  5. $\ell^1$ versus $\ell^2$ examples
  6. The Burnside process
  7. Properties of the chain
  8. Eigenvalues and eigenvectors; proof of \ref{['eigenvectors']}
  9. Proof of \ref{['chisquare']}
  10. Related chains
  11. Other Markov chains with similar properties
  12. Generalizing beyond the binary case

Key Result

Theorem 1.1

For the binary Burnside process on $C_2^n$, we have the following:

Figures (1)

  • Figure 1: Alternation count histograms for $100000$ binary strings sampled under $\pi_n(x)$ for $n = 200$ and $n = 2000$. The smooth curve corresponds to the limiting density $\frac{1}{\sqrt{1 - 2x}}$ for a random variable distributed as $2U(1-U)$ for $U$ uniform.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Example 2.3
  • Remark
  • Proposition 2.4
  • proof
  • ...and 28 more