Electronic Journal of Probability

Intersection and mixing times for reversible chains

Yuval Peres, Thomas Sauerwald, Perla Sousi, and Alexandre Stauffer

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We consider two independent Markov chains on the same finite state space, and study their intersection time, which is the first time that the trajectories of the two chains intersect. We denote by $t_I$ the expectation of the intersection time, maximized over the starting states of the two chains. We show that, for any reversible and lazy chain, the total variation mixing time is $O(t_I)$. When the chain is reversible and transitive, we give an expression for $t_I$ using the eigenvalues of the transition matrix. In this case, we also show that $t_I$ is of order $\sqrt{n \mathbb {E}\!\left [I\right ]} $, where $I$ is the number of intersections of the trajectories of the two chains up to the uniform mixing time, and $n$ is the number of states. For random walks on trees, we show that $t_I$ and the total variation mixing time are of the same order. Finally, for random walks on regular expanders, we show that $t_I$ is of order $\sqrt{n} $.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 12, 16 pp.

Received: 5 February 2016
Accepted: 28 November 2016
First available in Project Euclid: 3 February 2017

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Zentralblatt MATH identifier

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

intersection time random walk mixing time martingale Doob’s maximal inequality

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Peres, Yuval; Sauerwald, Thomas; Sousi, Perla; Stauffer, Alexandre. Intersection and mixing times for reversible chains. Electron. J. Probab. 22 (2017), paper no. 12, 16 pp. doi:10.1214/16-EJP18. https://projecteuclid.org/euclid.ejp/1486090892

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  • [1] David Aldous and James Fill. Reversible Markov Chains and Random Walks on Graphs. In preparation, http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [2] Tugkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing closeness of discrete distributions. J. ACM, 60(1):4, 2013.
  • [3] Itai Benjamini and Ben Morris. The birthday problem and Markov chain Monte Carlo, 2007. arXiv:math/0701390.
  • [4] Artur Czumaj and Christian Sohler. Testing expansion in bounded-degree graphs. Combinatorics, Probability & Computing, 19(5-6):693–709, 2010.
  • [5] Quentin de Mourgues and Thomas Sauerwald. Intersection time, 2013. preprint.
  • [6] P. J. Fitzsimmons and Thomas S. Salisbury. Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist., 25(3):325–350, 1989.
  • [7] Olle Häggström, Yuval Peres, and Jeffrey E. Steif. Dynamical percolation. Ann. Inst. H. Poincaré Probab. Statist., 33(4):497–528, 1997.
  • [8] Satyen Kale and C. Seshadhri. Combinatorial approximation algorithms for maxcut using random walks. In Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 7-9, 2011. Proceedings, pages 367–388, 2011.
  • [9] Satyen Kale and C. Seshadhri. An expansion tester for bounded degree graphs. SIAM J. Comput., 40(3):709–720, 2011.
  • [10] Liran Katzir, Edo Liberty, Oren Somekh, and Ioana A. Cosma. Estimating sizes of social networks via biased sampling. Internet Mathematics, 10(3-4):335–359, 2014.
  • [11] Jeong Han Kim, Ravi Montenegro, Yuval Peres, and Prasad Tetali. A birthday paradox for Markov chains with an optimal bound for collision in the Pollard rho algorithm for discrete logarithm. Ann. Appl. Probab., 20(2):495–521, 2010.
  • [12] Gregory F. Lawler. Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1991.
  • [13] Jean-François Le Gall and Jay Rosen. The range of stable random walks. Ann. Probab., 19(2):650–705, 1991.
  • [14] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson.
  • [15] Russell Lyons and Shayan Oveis Gharan. Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding, 2012. arXiv:1211.0589.
  • [16] Russell Lyons, Yuval Peres, and Oded Schramm. Markov chain intersections and the loop-erased walk. Ann. Inst. H. Poincaré Probab. Statist., 39(5):779–791, 2003.
  • [17] Ben Morris and Yuval Peres. Evolving sets and mixing. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pages 279–286 (electronic). ACM, New York, 2003.
  • [18] Roberto Imbuzeiro Oliveira. Mixing and hitting times for finite Markov chains. Electron. J. Probab., 17:no. 70, 12, 2012.
  • [19] Yuval Peres and Perla Sousi. Mixing times are hitting times of large sets. Journal of Theoretical Probability, pages 1–32, 2013.
  • [20] Thomas S. Salisbury. Energy, and intersections of Markov chains. In Random discrete structures (Minneapolis, MN, 1993), volume 76 of IMA Vol. Math. Appl., pages 213–225. Springer, New York, 1996.