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The Mathematics of Mixing Things Up

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Abstract

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.

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  1. Departments of Mathematics and Statistics, Stanford University, Stanford, USA

    Persi Diaconis

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Correspondence to Persi Diaconis.

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Research supported in part by National Science Foundation grant DMS 0804324.

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Diaconis, P. The Mathematics of Mixing Things Up. J Stat Phys 144, 445–458 (2011). https://doi.org/10.1007/s10955-011-0284-x

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