Skip to main content

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


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.

This is a preview of subscription content, access via your institution.


  1. Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs. Unfinished monograph (2002).

  2. Basu, R., Hermon, J., Peres, Y.: Characterization of cutoff for reversible Markov chains. Ann. Probab. arXiv:1409.3250 (2013) (to appear)

  3. Bobkov, S.G., Tetali, P.: Modified logarithmic Sobolev inequalities in discrete settings. J. Theor. Probab. 19(2), 289–336 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  4. Boczkowski, L., Peres, Y., Sousi, P.: Sensitivity of mixing times in Eulerian digraphs. arXiv:1603.05639 (2016)

  5. Cattiaux, P., Guillin, A.: Hitting times, functional inequalities, lyapunov conditions and uniform ergodicity. arXiv:1604.06336 (2016)

  6. Diaconis, P., Saloff-Coste, L.: Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6(3), 695–750 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  7. Ding, J., et al.: Sensitivity of mixing times. Electron. Commun. Probab. 18, 1–6 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  8. Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002)

    Article  MATH  Google Scholar 

  9. Goel, S., Montenegro, R., Tetali, P.: Mixing time bounds via the spectral profile. Electron. J. Probab. 11(1), 1–26 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  10. Hermon, J.: On sensitivity of uniform mixing times. Annales de l’Institut Henri Poincaré Probabilités et Statistiques (2016, to appear). arXiv:1607.01672

  11. Hermon, J., Peres, Y.: The power of averaging at two consecutive time steps: proof of a mixing conjecture by Aldous and Fill. Ann. Insti. Henri Poincaré Prob. Stat. arXiv:1508.04836 (2015) (to appear)

  12. Kozma, G.: On the precision of the spectral profile. Lat. Am. J. Probab. Math. Stat. 3, 321–329 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  14. Mathai, A.M., Rathie, P.N.: Basic Concepts in Information Theory and Statistics: Axiomatic Foundations and Applications. Halsted Press, New York (1975)

    MATH  Google Scholar 

  15. Mossel, E., Oleszkiewicz, K., Sen, A.: On reverse hypercontractivity. Geom. Funct. Anal. 23(3), 1062–1097 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  16. Norris, J., Peres, Y., Zhai, A.: Surprise probabilities in Markov chains. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1759–1773. SIAM (2015)

  17. Peres, Y., Sousi, P.: Mixing times are hitting times of large sets. J. Theor. Probab. 28(2), 488–519 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  18. Saloff-Coste, L.: Lectures on finite Markov chains. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math., vol. 1665, pp. 301–413. Springer, Berlin (1997)

  19. Starr, N.: Operator limit theorems. Trans. Am. Math. Soc. 121, 90–115 (1966)

    MathSciNet  Article  MATH  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jonathan Hermon.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


  • 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