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Mathematical aspects of mixing times in Markov chains. (English) Zbl 1193.68138

Summary: In the past few years we have seen a surge in the theory of finite Markov chains, by way of new techniques to bounding the convergence to stationarity. This includes functional techniques such as logarithmic Sobolev and Nash inequalities, refined spectral and entropy techniques, and isoperimetric techniques such as the average and blocking conductance and the evolving set methodology. We attempt to give a more or less self-contained treatment of some of these modern techniques, after reviewing several preliminaries. We also review classical and modern lower bounds on mixing times. There have been other important contributions to this theory such as variants on coupling techniques and decomposition methods, which are not included here; our choice was to keep the analytical methods as the theme of this presentation. We illustrate the strength of the main techniques by way of simple examples, a recent result on the Pollard rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.
Also published as Book Version (ISBN: 1-933019-29-8) and E-book Version (ISBN: 978-1-933019-77-2).

MSC:

68Q25 Analysis of algorithms and problem complexity
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J22 Computational methods in Markov chains
90C40 Markov and semi-Markov decision processes
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