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Partial mixing of semi-random transposition shuffles

Abstract : We show that for any semi-random transposition shuffle on $n$ cards, the mixing time of any given $k$ cards is at most $n\log k$, provided $k=o((n/\log n)^{1/2})$. In the case of the top-to-random transposition shuffle we show that there is cutoff at this time with a window of size O(n), provided further that $k\to\infty$ as $n\to\infty$ (and no cutoff otherwise). For the random-to-random transposition shuffle we show cutoff at time $(1/2)n\log k$ for the same conditions on $k$. Finally, we analyse the cyclic-to-random transposition shuffle and show partial mixing occurs at time $\le\alpha n\log k$ for some $\alpha$ just larger than 1/2. We prove these results by relating the mixing time of $k$ cards to the mixing of one card. Our results rely heavily on coupling arguments to bound the total variation distance.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-00793202
Contributor : Richard Pymar Connect in order to contact the contributor
Submitted on : Thursday, February 21, 2013 - 6:08:24 PM
Last modification on : Wednesday, October 20, 2021 - 3:18:53 AM

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  • HAL Id : hal-00793202, version 1
  • ARXIV : 1302.2601

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Richard Pymar. Partial mixing of semi-random transposition shuffles. 2013. ⟨hal-00793202⟩

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