## A characterization of L<inf>2</inf> mixing and hypercontractivity via hitting times and maximal inequalities

dc.contributor.author | Hermon, J | |

dc.contributor.author | Peres, Y | |

dc.date.accessioned | 2018-03-16T15:15:03Z | |

dc.date.available | 2018-03-16T15:15:03Z | |

dc.date.issued | 2018 | |

dc.identifier.issn | 0178-8051 | |

dc.identifier.uri | https://www.repository.cam.ac.uk/handle/1810/274054 | |

dc.description.abstract | There are several works characterizing the total-variation mixing time of a reversible Markov chain in term of natural probabilistic concepts such as stopping times and hitting times. In contrast, there is no known analog for the $L_{2}$ mixing time, $\tau_{2}$ (while there are sophisticated analytic tools to bound $ \tau_2$, in general they do not determine $\tau_2$ up to a constant factor and they lack a probabilistic interpretation). In this work we show that $\tau_2$ can be characterized up to a constant factor using hitting times distributions. We also derive a new extremal characterization of the Log-Sobolev constant, $c_{\mathrm{LS}}$, as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $c_{\mathrm{LS}}$ in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $\tau_2$ is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $\tau_2 $ is robust under bounded perturbations of the edge weights. | |

dc.publisher | Springer Science and Business Media LLC | |

dc.title | A characterization of L<inf>2</inf> mixing and hypercontractivity via hitting times and maximal inequalities | |

dc.type | Article | |

prism.endingPage | 800 | |

prism.issueIdentifier | 3-4 | |

prism.publicationDate | 2018 | |

prism.publicationName | Probability Theory and Related Fields | |

prism.startingPage | 769 | |

prism.volume | 170 | |

dc.identifier.doi | 10.17863/CAM.21135 | |

rioxxterms.versionofrecord | 10.1007/s00440-017-0769-x | |

rioxxterms.version | AM | |

rioxxterms.licenseref.uri | http://www.rioxx.net/licenses/all-rights-reserved | |

rioxxterms.licenseref.startdate | 2018-04-01 | |

dc.contributor.orcid | Hermon, Jonathan [0000-0002-2935-3999] | |

dc.identifier.eissn | 1432-2064 | |

dc.publisher.url | http://dx.doi.org/10.1007/s0044 | |

rioxxterms.type | Journal Article/Review | |

cam.issuedOnline | 2017-03-14 | |

rioxxterms.freetoread.startdate | 2018-03-14 |