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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
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Abstract |
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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 -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 for the two-dimensional case. |
Keywords
Markov chains, mixing time, rook's walk, path coupling
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Mathematical Subject Classification 2010
Primary: 60J10
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Milestones
Received: 7 July 2015
Revised: 18 December 2015
Accepted: 19 December 2015
Published: 11 October 2016
Communicated by John C. Wierman
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Authors |
| Cam McLeman | |
| Department of Mathematics University of Michigan–Flint Flint, MI 48502 United States |
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| Peter T. Otto | |
| Department of Mathematics Willamette University Salem, OR 97301 United States |
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| John Rahmani | |
| Department of Mathematics Virginia Tech Blacksburg, VA 24061 United States |
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| Matthew Sutter | |
| Department of Mathematics University of Michigan–Flint Flint, MI 48502 United States |
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