The Annals of Applied Probability

Biased random-to-top shuffling

Johan Jonasson

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Recently Wilson [Ann. Appl. Probab. 14 (2004) 274–325] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson’s technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so-called move-to-front scheme, and one where it depends on its position.

For the move-to-front scheme, a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector’s problem corresponding to the move-to-front scheme at hand.

For the second class, a version of Wilson’s technique for complex-valued eigenvalues/eigenvectors is used. Such variants were presented in [Random Walks and Geometry (2004) 515–532] and [Electron. Comm. Probab. 8 (2003) 77–85]. Here we present another such variant which seems to be the most natural one for this particular class of problems. To find the eigenvalues for the general case of the second class of problems is difficult, so we restrict attention to two special cases. In the first case the card that is moved to the top is picked uniformly at random from the bottom k=k(n)=o(n) cards, and we find the lower bound (n3/(4π2k(k−1)))logn. Via a coupling, an upper bound exceeding this by only a factor 4 is found. This generalizes Wilson’s [Electron. Comm. Probab. 8 (2003) 77–85] result on the Rudvalis shuffle and Goel’s [Ann. Appl. Probab. 16 (2006) 30–55] result on top-to-bottom shuffles. In the second case the card moved to the top is, with probability 1/2, the bottom card and with probability 1/2, the card at position nk. Here the lower bound is again of order (n3/k2)logn, but in this case this does not seem to be tight unless k=O(1). What the correct order of mixing is in this case is an open question. We show that when k=n/2, it is at least Θ(n2).

Article information

Ann. Appl. Probab., Volume 16, Number 2 (2006), 1034-1058.

First available in Project Euclid: 29 June 2006

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Zentralblatt MATH identifier

Primary: 60G99: None of the above, but in this section 60J99: None of the above, but in this section

Mixing time coupling lower bound upper bound Rudvalis shuffle move-to-front scheme coupon-collector’s problem


Jonasson, Johan. Biased random-to-top shuffling. Ann. Appl. Probab. 16 (2006), no. 2, 1034--1058. doi:10.1214/10505160600000097.

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