Vol. 10, No. 1, 2017

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Mixing times for the rook's walk via path coupling

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

Vol. 10 (2017), No. 1, 51–64
Abstract

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.

Keywords
Markov chains, mixing time, rook's walk, path coupling
Mathematical Subject Classification 2010
Primary: 60J10
Milestones
Received: 7 July 2015
Revised: 18 December 2015
Accepted: 19 December 2015
Published: 11 October 2016

Communicated by John C. Wierman
Authors
Cam McLeman
Department of Mathematics
University of Michigan–Flint
Flint, MI 48502 United States
Peter T. Otto
Department of Mathematics
Willamette University
Salem, OR 97301
United States
John Rahmani
Department of Mathematics
Virginia Tech
Blacksburg, VA 24061
United States
Matthew Sutter
Department of Mathematics
University of Michigan–Flint
Flint, MI 48502 United States