## The Annals of Applied Probability

### Mixing times of lozenge tiling and card shuffling Markov chains

David Bruce Wilson

#### Abstract

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall and Sinclair to generate random tilings of regions by lozenges. For an $\ell\times\ell$ region we bound the mixing time by $O(\ell^4\log\ell)$, which improves on the previous bound of $O(\ell^7)$, and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an $O(n^3\log n)$ upper bound on the mixing time of the Karzanov--Khachiyan Markov chain for linear extensions.

#### Article information

Source
Ann. Appl. Probab. Volume 14, Number 1 (2004), 274-325.

Dates
First available in Project Euclid: 3 February 2004

http://projecteuclid.org/euclid.aoap/1075828054

Digital Object Identifier
doi:10.1214/aoap/1075828054

Mathematical Reviews number (MathSciNet)
MR2023023

Zentralblatt MATH identifier
02072701

#### Citation

Wilson, David Bruce. Mixing times of lozenge tiling and card shuffling Markov chains. The Annals of Applied Probability 14 (2004), no. 1, 274--325. doi:10.1214/aoap/1075828054. http://projecteuclid.org/euclid.aoap/1075828054.

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