## Involve: A Journal of Mathematics

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

### Mixing times for the rook's walk via path coupling

#### 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.

#### Article information

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

Dates
Revised: 18 December 2015
Accepted: 19 December 2015
First available in Project Euclid: 22 November 2017

https://projecteuclid.org/euclid.involve/1511371082

Digital Object Identifier
doi:10.2140/involve.2017.10.51

Mathematical Reviews number (MathSciNet)
MR3561729

Zentralblatt MATH identifier
1350.60069

#### Citation

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. https://projecteuclid.org/euclid.involve/1511371082

#### References

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