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The cutoff phenomenon in finite Markov chains. (English) Zbl 0849.60070

Summary: 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.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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