## 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.

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

## References

Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs. Unfinished monograph (2002). https://www.stat.berkeley.edu/~aldous/RWG/book.pdf

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

**(to appear)**Bobkov, S.G., Tetali, P.: Modified logarithmic Sobolev inequalities in discrete settings. J. Theor. Probab.

**19**(2), 289–336 (2006)Boczkowski, L., Peres, Y., Sousi, P.: Sensitivity of mixing times in Eulerian digraphs. arXiv:1603.05639 (2016)

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

Diaconis, P., Saloff-Coste, L.: Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab.

**6**(3), 695–750 (1996)Ding, J., et al.: Sensitivity of mixing times. Electron. Commun. Probab.

**18**, 1–6 (2013)Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev.

**70**(3), 419–435 (2002)Goel, S., Montenegro, R., Tetali, P.: Mixing time bounds via the spectral profile. Electron. J. Probab.

**11**(1), 1–26 (2006)Hermon, J.: On sensitivity of uniform mixing times. Annales de l’Institut Henri Poincaré Probabilités et Statistiques (2016, to appear). arXiv:1607.01672

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)**Kozma, G.: On the precision of the spectral profile. Lat. Am. J. Probab. Math. Stat.

**3**, 321–329 (2007)Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)

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

Mossel, E., Oleszkiewicz, K., Sen, A.: On reverse hypercontractivity. Geom. Funct. Anal.

**23**(3), 1062–1097 (2013)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)

Peres, Y., Sousi, P.: Mixing times are hitting times of large sets. J. Theor. Probab.

**28**(2), 488–519 (2015)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)Starr, N.: Operator limit theorems. Trans. Am. Math. Soc.

**121**, 90–115 (1966)

## 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### 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). https://doi.org/10.1007/s00440-017-0769-x

Received:

Revised:

Published:

Issue Date:

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