Limit profiles for reversible Markov chains

Evita Nestoridi, Sam Olesker-Taylor

Research output: Contribution to journalArticlepeer-review

Abstract

In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) 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, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).

Original languageEnglish (US)
Pages (from-to)157-188
Number of pages32
JournalProbability Theory and Related Fields
Volume182
Issue number1-2
DOIs
StatePublished - Feb 2022

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Characters
  • Cutoff
  • Eigenvalues and eigenfunctions of Markov chains
  • Fourier transform
  • Gelfand pairs
  • Homogeneous spaces
  • Limit profiles
  • Random walk on groups
  • Representation theory
  • Spectral representations
  • Spherical functions
  • Symmetric group

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