# A characterization of $$L_{2}$$ L 2 mixing and hypercontractivity via hitting times and maximal inequalities

A characterization of $$L_{2}$$ L 2 mixing and hypercontractivity via hitting times and... 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 $$L_{2}$$ L 2 mixing time, $$\tau _{2}$$ τ 2 (while there are sophisticated analytic tools to bound $$\tau _2$$ τ 2 , in general they do not determine $$\tau _2$$ τ 2 up to a constant factor and they lack a probabilistic interpretation). In this work we show that $$\tau _2$$ τ 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, $$c_{{\mathrm {LS}}}$$ c LS , as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $$c_{{\mathrm {LS}}}$$ c LS in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $$\tau _2$$ τ 2 is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $$\tau _2$$ τ 2 is robust under bounded perturbations of the edge weights. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Probability Theory and Related Fields Springer Journals

# A characterization of $$L_{2}$$ L 2 mixing and hypercontractivity via hitting times and maximal inequalities

, Volume 170 (4) – Mar 14, 2017
32 pages
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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathematical Finance/Insurance; Operations Research/Decision Theory
ISSN
0178-8051
eISSN
1432-2064
D.O.I.
10.1007/s00440-017-0769-x
Publisher site
See Article on Publisher Site

### 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 $$L_{2}$$ L 2 mixing time, $$\tau _{2}$$ τ 2 (while there are sophisticated analytic tools to bound $$\tau _2$$ τ 2 , in general they do not determine $$\tau _2$$ τ 2 up to a constant factor and they lack a probabilistic interpretation). In this work we show that $$\tau _2$$ τ 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, $$c_{{\mathrm {LS}}}$$ c LS , as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $$c_{{\mathrm {LS}}}$$ c LS in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $$\tau _2$$ τ 2 is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $$\tau _2$$ τ 2 is robust under bounded perturbations of the edge weights.

### Journal

Probability Theory and Related FieldsSpringer Journals

Published: Mar 14, 2017

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