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Partial mixing of semi-random transposition shuffles


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.

Dates and versions

hal-00793202 , version 1 (21-02-2013)



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