Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On sensitivity of uniform mixing times

Jonathan Hermon

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show that the order of the $L_{\infty}$-mixing time of simple random walks on a sequence of uniformly bounded degree graphs of size $n$ may increase by an optimal factor of $\Theta(\log\log n)$ as a result of a bounded perturbation of the edge weights. This answers a question and a conjecture of Kozma.

Résumé

Nous montrons que le temps de mélange pour la distance $L_{\infty}$ d’une marche aléatoire sur une suite de graphe de taille $n$ et de degré uniformément borné peut être multiplié par un facteur d’ordre $\log\log n$ (optimal) en perturbant le poids des arrêtes du graphe de manière uniformément bornée. Ceci résout une question et une conjecture de Kozma.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 234-248.

Dates
Received: 6 July 2016
Revised: 26 September 2016
Accepted: 17 October 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030827

Digital Object Identifier
doi:10.1214/16-AIHP802

Mathematical Reviews number (MathSciNet)
MR3765888

Zentralblatt MATH identifier
06868370

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Sensitivity Mxing-time Sensitivity of mixing times Hitting times

Citation

Hermon, Jonathan. On sensitivity of uniform mixing times. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 234--248. doi:10.1214/16-AIHP802. https://projecteuclid.org/euclid.aihp/1519030827


Export citation

References

  • [1] L. Addario-Berry and M. I. Roberts. Mixing time bounds via bottleneck sequences. J. Stat. Phys. (2017) 1–27.
  • [2] D. Aldous and J. Fill. Reversible Markov chains and random walks on graphs. Unfinished manuscript. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [3] R. Basu, J. Hermon and Y. Peres. Characterization of cutoff for reversible Markov chains. Ann. Probab. 45 (3) (2017) 1448–1487.
  • [4] I. Benjamini. Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theoret. Probab. 4 (3) (1991) 631–637.
  • [5] L. Boczkowski, Y. Peres and P. Sousi. Sensitivity of mixing times in Eulerian digraphs. Preprint, 2016. Available at arXiv:1603.05639.
  • [6] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (3) (1996) 695–750.
  • [7] J. Ding and Y. Peres. Sensitivity of mixing times. Electron. Commun. Probab. 18 (2013) 1–6.
  • [8] S. Goel, R. Montenegro and P. Tetali. Mixing time bounds via the spectral profile. Electron. J. Probab. 11 (1) (2006) 1–26.
  • [9] J. Hermon and Y. Peres. A characterization of $L_{2}$ mixing and hypercontractivity via hitting times and maximal inequalities. Probab. Theory Related Fields (2017) 1–32.
  • [10] J. Hermon and Y. Peres. On sensitivity of mixing times and cutoff. Preprint, 2016. Available at arXiv:1610.04357.
  • [11] G. Kozma. On the precision of the spectral profile. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 321–329.
  • [12] D. Levin, Y. Peres and E. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, 2017.
  • [13] E. Lubetzky and A. Sly. Explicit expanders with cutoff phenomena. Electron. J. Probab. 16 (15) (2011) 419–435.
  • [14] B. Morris and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2) (2005) 245–266.
  • [15] Y. Peres and P. Sousi. Mixing times are hitting times of large sets. J. Theoret. Probab. 28 (2) (2015) 488–519.
  • [16] C. Pittet and L. Saloff-Coste. On the stability of the behavior of random walks on groups. J. Geom. Anal. 10 (4) (2000) 713–737.
  • [17] L. Saloff-Coste. Lectures on Finite Markov Chains. In Lectures on Probability Theory and Statistics 301–413. Springer, Berlin, 1997.