Tantilla striata

Number of Pages: 1Integrative BiologyGeological Science

Tantilla canula

Number of Pages: 1Integrative BiologyGeological Science

Tantilla miyatai

Number of Pages: 1Integrative BiologyGeological Science

The Making of Modern Science: Science, Technology, Medicine, and Modernity, 1789-1914

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Number of Pages: 2Integrative BiologyGeological Science

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Number of Pages: 3Integrative BiologyGeological Science

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Tantilla lempira

Number of Pages: 1Integrative BiologyGeological Science

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(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.Comment: 39 pages, 8 figure