The Mathematics of Mixing Things Up



How long should a Markov chain Monte Carlo algorithm be run? Using examples from statistical physics (Ehrenfest urn, Ising model, hard discs) as well as card shuffling, this tutorial paper gives an overview of a body of mathematical results that can give useful answers to practitioners (viz: seven shuffles suffice for practical purposes). It points to new techniques (path coupling, geometric inequalities, and Harris recurrence). The discovery of phase transitions in mixing times (the cutoff phenomenon) is emphasized.


Markov chains Rates of convergence Cutoff phenomenon 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Departments of Mathematics and StatisticsStanford UniversityStanfordUSA

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