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Limit Profiles for Reversible Markov Chains

Evita Nestoridi, Sam Olesker-Taylor

TL;DR

The paper develops TV-approximation lemmas that decompose the distance to equilibrium into a main term and a vanishing error term, enabling explicit limit-profile analysis for reversible Markov chains and Markov processes on homogeneous spaces. It extends Teyssier’s limit-profile framework from conjugacy-invariant walks on groups to general reversible chains and to spaces $X=G/K$ via spherical Fourier analysis and Gelfand-pair theory. The authors apply these tools to three key problems: the random $k$-cycle shuffle on $\mathcal{S}_n$, the Ehrenfest urn diffusion with many urns, and Gibbs samplers with Binomial priors, deriving precise TV limit profiles and refining previous cutoff results. The work leverages spectral decompositions, character ratio bounds (via long first-row partitions), and Krawtchouk-type orthogonality to obtain Poisson- and Gaussian-type limit laws within the cutoff window, with corroborating appendices on hypercube walks and binomial-TV distances. Together, these contributions provide a broad, principled method for obtaining detailed convergence profiles in a range of structured Markov chains and homogeneous-space settings, with potential impact on mixing-time analysis and probabilistic modeling in statistics and statistical physics.

Abstract

In a recent breakthrough, Teyssier [Tey20] introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the $k$-cycle shuffle, improving results of Hough [Hou16] and Berestycki, Schramm and Zeitouni [BSZ11]; the Ehrenfest urn diffusion with many urns, improving results of Ceccherini-Silberstein, Scarabotti and Tolli [CST07]; a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, improving results of Diaconis, Khare and Saloff-Coste [DKS08].

Limit Profiles for Reversible Markov Chains

TL;DR

The paper develops TV-approximation lemmas that decompose the distance to equilibrium into a main term and a vanishing error term, enabling explicit limit-profile analysis for reversible Markov chains and Markov processes on homogeneous spaces. It extends Teyssier’s limit-profile framework from conjugacy-invariant walks on groups to general reversible chains and to spaces via spherical Fourier analysis and Gelfand-pair theory. The authors apply these tools to three key problems: the random -cycle shuffle on , the Ehrenfest urn diffusion with many urns, and Gibbs samplers with Binomial priors, deriving precise TV limit profiles and refining previous cutoff results. The work leverages spectral decompositions, character ratio bounds (via long first-row partitions), and Krawtchouk-type orthogonality to obtain Poisson- and Gaussian-type limit laws within the cutoff window, with corroborating appendices on hypercube walks and binomial-TV distances. Together, these contributions provide a broad, principled method for obtaining detailed convergence profiles in a range of structured Markov chains and homogeneous-space settings, with potential impact on mixing-time analysis and probabilistic modeling in statistics and statistical physics.

Abstract

In a recent breakthrough, Teyssier [Tey20] introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the -cycle shuffle, improving results of Hough [Hou16] and Berestycki, Schramm and Zeitouni [BSZ11]; the Ehrenfest urn diffusion with many urns, improving results of Ceccherini-Silberstein, Scarabotti and Tolli [CST07]; a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, improving results of Diaconis, Khare and Saloff-Coste [DKS08].

Paper Structure

This paper contains 25 sections, 40 theorems, 132 equations.

Table of Contents

  1. Introduction: TV Approximation Lemmas and Limit Profiles
  2. Mixing Times and Limit Profiles
  3. TV Convergence Profile for Random Walks
  4. Reversible Markov Chains
  5. Random Walks on Groups
  6. Conjugacy-Invariant Random Walks
  7. Random Walks on Homogenous Spaces
  8. Corollaries to TV Approximation Lemma for Reversible Markov Chains (\ref{['res:tv-approx:rev']})
  9. Organisation of the Paper
  10. Reversible Markov Chains
  11. Proof of TV-Approximation Lemmas for Reversible Markov Chains
  12. Application to Gibbs Sampler with Binomial Priors
  13. Random $k$-Cycle Walk on the Symmetric Group
  14. Walk Definition and Statement of Result
  15. Statements of Character Ratio Bounds
  16. ...and 10 more sections

Key Result

Lemma A

Consider a reversible, irreducible and aperiodic Markov chain on a finite set $\Omega$ with invariant distribution $\pi$. Denote by $-1 < \lambda_{\lvert \Omega \rvert} \le \ldots \le \lambda_2 < \lambda_1 = 1$ its eigenvalues and by $f_{\lvert \Omega \rvert}, ..., f_1$ its corresponding orthonormal

Theorems & Definitions (51)

  • Lemma A: Reversible Markov Chains
  • Theorem A: Gibbs Sampler
  • Definition
  • Definition A
  • Lemma B: T:limit-profile
  • Theorem B: Random $k$-Cycles
  • Lemma C: Homogeneous Spaces
  • Theorem C: Ehrenfest Urn
  • Corollary A.1
  • Corollary A.2
  • ...and 41 more