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The Cutoff Phenomenon in Finite Markov Chains

Persi Diaconis
Proceedings of the National Academy of Sciences of the United States of America
Vol. 93, No. 4 (Feb. 20, 1996), pp. 1659-1664
Stable URL: http://www.jstor.org/stable/38640
Page Count: 6
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The Cutoff Phenomenon in Finite Markov Chains
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Abstract

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

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