The Annals of Probability

Characterization of cutoff for reversible Markov chains

Riddhipratim Basu, Jonathan Hermon, and Yuval Peres

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

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least $\alpha$, for some $\alpha\in(0,1)$.

We also give general bounds on the total variation distance of a reversible chain at time $t$ in terms of the probability that some “worst” set of stationary measure at least $\alpha$ was not hit by time $t$. As an application of our techniques, we show that a sequence of lazy Markov chains on finite trees exhibits a cutoff iff the product of their spectral gaps and their (lazy) mixing-times tends to $\infty$.

Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 1448-1487.

Dates
Received: December 2014
Revised: November 2015
First available in Project Euclid: 15 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1494835222

Digital Object Identifier
doi:10.1214/16-AOP1090

Mathematical Reviews number (MathSciNet)
MR3650406

Zentralblatt MATH identifier
1374.60129

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

Keywords
Cutoff mixing-time finite reversible Markov chains hitting times trees maximal inequality

Citation

Basu, Riddhipratim; Hermon, Jonathan; Peres, Yuval. Characterization of cutoff for reversible Markov chains. Ann. Probab. 45 (2017), no. 3, 1448--1487. doi:10.1214/16-AOP1090. https://projecteuclid.org/euclid.aop/1494835222


Export citation

References

  • [1] Aldous, D. and Diaconis, P. (1986). Shuffling cards and stopping times. Amer. Math. Monthly 93 333–348.
  • [2] Aldous, D. and Fill, J. A. (2000). Reversible Markov Chains And Random Walks On Graphs. Book in preparation. Available at http://www.stat.berkeley.edu/users/aldous.
  • [3] Aldous, D. J. (1982). Some inequalities for reversible Markov chains. J. Lond. Math. Soc. (2) 2 564–576.
  • [4] Brown, M. (1999). Interlacing eigenvalues in time reversible Markov chains. Math. Oper. Res. 24 847–864.
  • [5] Chen, G.-Y. (2006). The cutoff phenomenon for finite Markov chains Ph.D. thesis, Cornell University.
  • [6] Chen, G.-Y. and Saloff-Coste, L. (2008). The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13 26–78.
  • [7] Chen, G.-Y. and Saloff-Coste, L. (2013). Comparison of cutoffs between lazy walks and Markovian semigroups. J. Appl. Probab. 50 943–959.
  • [8] Chen, G.-Y. and Saloff-Coste, L. (2015). Computing cutoff times of birth and death chains. Electron. J. Probab. 20 1–47.
  • [9] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 1659–1664.
  • [10] Diaconis, P. and Saloff-Coste, L. (2006). Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16 2098–2122.
  • [11] Ding, J., Lubetzky, E. and Peres, Y. (2010). Total variation cutoff in birth-and-death chains. Probab. Theory Related Fields 146 61–85.
  • [12] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge.
  • [13] Fill, J. A. and Lyzinski, V. (2012). Hitting times and interlacing eigenvalues: A stochastic approach using intertwinings. J. Theoret. Probab. 27 954–981.
  • [14] Griffiths, S., Kang, R. J., Oliveira, R. I. and Patel, V. (2012). Tight inequalities among set hitting times in Markov chains. Preprint. Available at arXiv:1209.0039.
  • [15] Hermon, J. (2015). A technical report on hitting times, mixing and cutoff. Preprint. Available at arXiv:1501.01869.
  • [16] Hermon, J. and Peres, Y. (2015). The power of averaging at two consecutive time steps: Proof of a mixing conjecture by Aldous and Fill. Preprint. Available at arXiv:1508.04836.
  • [17] Lancia, C., Nardi, F. R. and Scoppola, B. (2012). Entropy-driven cutoff phenomena. J. Stat. Phys. 149 108–141.
  • [18] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. With a chapter by James G. Propp and David B. Wilson.
  • [19] Lovász, L. and Winkler, P. (1998). Mixing times. In Microsurveys in Discrete Probability (Princeton, NJ, 1997). DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41 85–134. Amer. Math. Soc., Providence, RI.
  • [20] Norris, J., Peres, Y. and Zhai, A. (2015). Surprise probabilities in Markov chains. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms 1759–1773. SIAM, Philadelphia, PA.
  • [21] Oliveira, R. I. (2012). Mixing and hitting times for finite Markov chains. Electron. J. Probab. 17 1–12.
  • [22] Peres, Y. American Institute of Mathematics (AIM) research workshop “Sharp Thresholds for Mixing Times” (Palo Alto, December 2004). Summary available at http://www.Aimath.Org/WWN/mixingtimes.
  • [23] Peres, Y. and Sousi, P. (2013). Total variation cutoff in a tree. Preprint. Available at arXiv:1307.2887.
  • [24] Peres, Y. and Sousi, P. (2015). Mixing times are hitting times of large sets. J. Theoret. Probab. 28 488–519.
  • [25] Starr, N. (1966). Operator limit theorems. Trans. Amer. Math. Soc. 121 90–115.
  • [26] Stein, E. M. (1961). On the maximal ergodic theorem. Proc. Natl. Acad. Sci. USA 47 1894–1897.