Open Access
February 2004 Mixing times of lozenge tiling and card shuffling Markov chains
David Bruce Wilson
Ann. Appl. Probab. 14(1): 274-325 (February 2004). DOI: 10.1214/aoap/1075828054


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.


Download Citation

David Bruce Wilson. "Mixing times of lozenge tiling and card shuffling Markov chains." Ann. Appl. Probab. 14 (1) 274 - 325, February 2004.


Published: February 2004
First available in Project Euclid: 3 February 2004

zbMATH: 1040.60063
MathSciNet: MR2023023
Digital Object Identifier: 10.1214/aoap/1075828054

Primary: 60J10
Secondary: 60C05

Keywords: card shuffling , Cutoff phenomenon , Exclusion process , lattice path , linear extension , lozenge tiling , mixing time , Random walk

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.14 • No. 1 • February 2004
Back to Top