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 l x l region we bound the mixing time by$O({\cal}l^4 log {\cal}l)$, which improves on the previous bound of$O({\cal}l^7)$, 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$O(n^3 logn)$upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.