The Annals of Applied Probability

Mixing time of the card-cyclic-to-random shuffle

Ben Morris, Weiyang Ning, and Yuval Peres

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The card-cyclic-to-random shuffle on $n$ cards is defined as follows: at time $t$ remove the card with label $t$ mod $n$ and randomly reinsert it back into the deck. Pinsky [Probabilistic and combinatorial aspects of the card-cyclic-to-random shuffle (2011). Unpublished manuscript] introduced this shuffle and asked how many steps are needed to mix the deck. He showed $n$ steps do not suffice. Here we show that the mixing time is on the order of $\Theta(n\log n)$.

Article information

Ann. Appl. Probab., Volume 24, Number 5 (2014), 1835-1849.

First available in Project Euclid: 26 June 2014

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Zentralblatt MATH identifier

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

Markov chain mixing time


Morris, Ben; Ning, Weiyang; Peres, Yuval. Mixing time of the card-cyclic-to-random shuffle. Ann. Appl. Probab. 24 (2014), no. 5, 1835--1849. doi:10.1214/13-AAP964.

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