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A characterization of \(L_{2}\) mixing and hypercontractivity via hitting times and maximal inequalities

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.

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Acknowledgements

We are grateful to Shirshendu Ganguly, Gady Kozma, James Lee and Prasad Tetali for useful discussions. Most of the work on this paper was done while the first author was an intern at Microsoft Research, Redmond. The first author would like to thank Microsoft Research for two wonderful summers.

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Correspondence to Jonathan Hermon.

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Hermon, J., Peres, Y. A characterization of \(L_{2}\) mixing and hypercontractivity via hitting times and maximal inequalities. Probab. Theory Relat. Fields 170, 769–800 (2018). https://doi.org/10.1007/s00440-017-0769-x

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  • DOI: https://doi.org/10.1007/s00440-017-0769-x

Keywords

  • Mixing-time
  • Finite reversible Markov chains
  • Maximal inequalities
  • Hitting times
  • Hypercontractivity
  • Log-Sobolov inequalities
  • Relative entropy
  • Robustness of mixing times

Mathematics Subject Classification

  • 60J27
  • 60J10
  • 05C81