Analysis of top to bottom-$k$ shuffles

Details Analysis of top to bottom-$k$ shuffles Analysis of top to bottom-$k$ shuffles

2006-03-09

arxiv.org/abs/math/0603209v1

Annals of Applied Probability 2006, Vol. 16, No. 1, 30-55

Mathematics

Probability

Published at http://dx.doi.org/10.1214/10505160500000062 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/10505160500000062

A deck of $n$ cards is shuffled by repeatedly moving the top card to one of the bottom $k_n$ positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as $k_n$ ranges from a constant to $n$. We also consider a symmetric variant of this shuffle in which at each step either the top card is randomly inserted into the bottom $k_n$ positions or a random card from the bottom $k_n$ positions is moved to the top. For this reversible shuffle we derive bounds on the $L^2$ mixing time. Finally, we transfer mixing time estimates for the above shuffles to the lazy top to bottom-$k$ walks that move with probability 1/2 at each step.

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