Abstract
We study convergence to equilibrium for a class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix P the mass is essentially concentrated on few entries. Moreover, the entries are exchangeable within each row. This includes various models of random walks on sparse random directed graphs. The models are generally non reversible and the equilibrium distribution is itself unknown. In this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium, with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of P. As an application, we consider the case where the rows of P are i.i.d. random vectors in the domain of attraction of a Poisson–Dirichlet law with index \(\alpha \in (0,1)\). Our main results are based on a detailed analysis of the weight of the trajectory followed by the walker. This approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Addario-Berry, L., Balle, B., Perarnau, G.: Diameter and stationary distribution of random \(r\)-out digraphs. ArXiv e-prints (2015)
Aldous, D.: Random walks on finite groups and rapidly mixing Markov chains. In: Seminar on Probability, XVII. Lecture Notes in Mathematics, vol. 986, pp. 243–297. Springer, Berlin (1983)
Aldous, D., Diaconis, P.: Shuffling cards and stopping times. Am. Math. Mon. 93, 333–348 (1986)
Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs (2002). http://www.stat.berkeley.edu/~aldous/RWG/book.html
Basu, R., Hermon, J., Peres, Y., et al.: Characterization of cutoff for reversible Markov chains. Ann. Probab. 45(3), 1448–1487 (2017)
Ben-Hamou, A., Salez, J., et al.: Cutoff for nonbacktracking random walks on sparse random graphs. Ann. Probab. 45(3), 1752–1770 (2017)
Berestycki, N., Lubetzky, E., Peres, Y., Sly, A.: Random walks on the random graph. arXiv preprint arXiv:1504.01999 (2015)
Bordenave, C., Caputo, P., Chafaï, D., Piras, D.: Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph. Random Matrices Theory Appl. 6, 1750006 (2017)
Bordenave, C., Caputo, P., Salez, J.: Random walk on sparse random digraphs. Probab. Theory Relat. Fields (to appear). arXiv:1508.06600
Boucheron, S., Lugosi, G., Massart, P.: Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford (2013). (With a foreword by Michel Ledoux)
Diaconis, P.: The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93(4), 1659–1664 (1996)
Diaconis, P., Saloff-Coste, L.: Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16(4), 2098–2122 (2006)
Diaconis, P., Shahshahani, M.: Generating a random permutation with random transpositions. Probab. Theory Relat. Fields 57(2), 159–179 (1981)
Diaconis, P., Wood, P.M.: Random doubly stochastic tridiagonal matrices. Random Struct. Algorithms 42(4), 403–437 (2013)
Ding, J., Lubetzky, E., Peres, Y.: Total variation cutoff in birth-and-death chains. Probab. Theory Relat. Fields 146(1–2), 61–85 (2010)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971)
Freedman, D.A.: On tail probabilities for martingales. Ann. Probab. 3, 100–118 (1975)
Hildebrand, M.: A survey of results on random random walks on finite groups. Probab. Surv. 2, 33–63 (2005)
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)
Lubetzky, E., Peres, Y.: Cutoff on all Ramanujan graphs. Geom. Funct. Anal. 26(4), 1190–1216 (2016)
Lubetzky, E., Sly, A.: Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153(3), 475–510 (2010)
Pitman, J., Yor, M.: The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855–900 (1997)
Smith, A.: The cutoff phenomenon for random birth and death chains. Random Struct. Algorithms 50(2), 287–321 (2017)
Wilson, D.B.: Random random walks on \({ Z}^d_2\). Probab. Theory Relat. Fields 108(4), 441–457 (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bordenave, C., Caputo, P. & Salez, J. Cutoff at the “entropic time” for sparse Markov chains. Probab. Theory Relat. Fields 173, 261–292 (2019). https://doi.org/10.1007/s00440-018-0834-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-018-0834-0