## 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 language | English (US) |
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Pages (from-to) | 157-188 |

Number of pages | 32 |

Journal | Probability Theory and Related Fields |

Volume | 182 |

Issue number | 1-2 |

DOIs | |

State | Published - 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