Rapid mixing of dealer shuffles and clumpy shuffles
Electronic Communications in Probability
A famous result of Bayer and Diaconis  is that the Gilbert-Shannon-Reeds (GSR) model for the riffle shuffle of n cards mixes in 3 2 log 2 n steps and that for 52 cards about 7 shuffles suffices to mix the deck. In this paper, we study variants of the GSR shuffle that have been proposed to model more realistically how people actually shuffle a deck of cards. The clumpy riffle shuffle and dealer riffle shuffle differ from the GSR model in that when a card is dropped from one hand, the
... hand, the conditional probability that the next card is dropped from the same hand is higher/lower than for the GSR model. It is believed that these shuffles mix slightly slower than the GSR shuffle, but still in order log n steps. However, rigorous results have so far been missing. In this paper we apply the technique of relative entropy and collisions of Morris  , to show that the clumpy shuffle and the dealer shuffle mix in O(log 4 n) steps. 1 over the last decades. A very prominent subclass of mixing time problems is card shuffling, that is, Markov chains on the symmetric group S n of permutations of n items that one can think of as the cards of a deck. Perhaps the most famous of card shuffles is the Gilbert-Shannon-Reads (GSR) model for the riffle shuffle for which Bayer and Diaconis  proved a remarkably exact result; there is a sharp cutoff at 3 2 log 2 n shuffles after which the deck is well mixed and for a standard deck of 52 cards, about 7 shuffles suffices for mixing. Prior to that, Aldous and Diaconis  had proved, via a striking strong uniform time argument, that 2 log 2 n shuffles is an upper bound on the mixing time. The riffle shuffle is, together with the inefficient overhand shuffle which mixes in order n 2 log n steps (see  and ), the most common way in which people actually shuffle a deck of cards. The model for one step of the GSR shuffle is the following. First the deck is cut into two packets of which one goes into your right hand and the other into your left hand. The number of cards that go into your right (or left if you like) hand is a binomial random variable with parameters n and 1/2. Then the cards are dropped from the two hands in such a way that whenever there are A cards left in your right hand and B cards left in your left hand, the probability that the next card is dropped from your right hand is A/(A + B).