Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 1 (2017), 51-64.

Mixing times for the rook's walk via path coupling

Cam McLeman, Peter T. Otto, John Rahmani, and Matthew Sutter

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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


The mixing time of a convergent Markov chain measures the number of steps required for the state distribution to be within a prescribed distance of the stationary distribution. In this paper, we illustrate the strength of the probabilistic technique called coupling and its extension, path coupling, to bound the mixing time of Markov chains. The application studied is the rook’s walk on an nd-chessboard, for which the mixing time has recently been studied using the spectral method. Our path-coupling result improves the previously obtained spectral bounds and includes an asymptotically tight upper bound in n for the two-dimensional case.

Article information

Involve, Volume 10, Number 1 (2017), 51-64.

Received: 7 July 2015
Revised: 18 December 2015
Accepted: 19 December 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Markov chains mixing time rook's walk path coupling


McLeman, Cam; Otto, Peter T.; Rahmani, John; Sutter, Matthew. Mixing times for the rook's walk via path coupling. Involve 10 (2017), no. 1, 51--64. doi:10.2140/involve.2017.10.51.

Export citation


  • R. Bubley and M. E. Dyer, “Path coupling: a technique for proving rapid mixing in Markov chains”, pp. 223–231 in Proceedings of the 38th IEEE Symposium on Foundations of Computer Science (Miami, FL, 1997), Institute of Electrical and Electronics Engineers, Los Alamitos, CA, 1997.
  • J. G. Kemeny, J. L. Snell, and A. W. Knapp, Denumerable Markov chains, 2nd ed., Graduate Texts in Mathematics 40, Springer, New York, NY, 1976.
  • S. S. Kim, “Mixing time of a rook's walk”, Undergraduate certificate paper, Princeton University, 2012, hook \posturlhook.
  • D. A. Levin, Y. Peres, and E. L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009.
  • Y. Li and K. Tucker, “Investigating the rook's walk”, Senior thesis, Willamette University, Salem, OR, 2014.
  • T. Lindvall, Lectures on the coupling method, Dover, Mineola, NY, 2002.\hskip -0.4mm