  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(L^4 log L), which improves on the previous bound
of O(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 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov
chain for linear extensions.
