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We show how to combine Fourier analysis with coupling arguments to bound themixing times of a variety of Markov chains. The mixing time is the number ofsteps a Markov chain takes to approach its equilibrium distribution. Oneapplication is to a class of Markov chains introduced by Luby, Randall, andSinclair to generate random tilings of regions by lozenges. For an L X L regionwe bound the mixing time by O(L^4 log L), which improves on the previous boundof O(L^7), and we show the new bound to be essentially tight. In anotherapplication we resolve a few questions raised by Diaconis and Saloff-Coste, bylower bounding the mixing time of various card-shuffling Markov chains. Ourlower bounds are within a constant factor of their upper bounds. When we useour methods to modify a path-coupling analysis of Bubley and Dyer, we obtain anO(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markovchain for linear extensions.Comment: 39 pages, 8 figure

Mixing times of lozenge tiling and card shuffling Markov chains