Limit Profiles for Reversible Markov Chains
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2020
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| _version_ | 1866908362645438464 |
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| author | Nestoridi, Evita Olesker-Taylor, Sam |
| author_facet | Nestoridi, Evita Olesker-Taylor, Sam |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_13437 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Limit Profiles for Reversible Markov Chains Nestoridi, Evita Olesker-Taylor, Sam Probability Combinatorics Group Theory Representation Theory 20C15, 20C30, 43A30, 43A65, 43A90, 60B15, 60J10, 60J20 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]. |
| title | Limit Profiles for Reversible Markov Chains |
| topic | Probability Combinatorics Group Theory Representation Theory 20C15, 20C30, 43A30, 43A65, 43A90, 60B15, 60J10, 60J20 |
| url | https://arxiv.org/abs/2005.13437 |