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.
"Mixing times of lozenge tiling and card shuffling Markov chains." Ann. Appl. Probab. 14 (1) 274 - 325, February 2004. https://doi.org/10.1214/aoap/1075828054