The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 14, Number 1 (2004), 274-325.
Mixing times of lozenge tiling and card shuffling Markov chains
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.
Ann. Appl. Probab. Volume 14, Number 1 (2004), 274-325.
First available in Project Euclid: 3 February 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60C05: Combinatorial probability
Wilson, David Bruce. Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 (2004), no. 1, 274--325. doi:10.1214/aoap/1075828054. https://projecteuclid.org/euclid.aoap/1075828054.