Riffle Shuffles and Dynamical Systems on the Unit Interval

The preferred method of randomizing a deck of N cards is the riffle shuffle–cut the deck into two stacks, then riffle the two stacks together. A number of mathematical models have been proposed for this process. The model which has received the most attention, and about which the most is known, is the GSR (for Gilbert, Shannon, and Reeds) shuffle. In this model, all permutations with exactly one or two rising sequences are equally likely (a rising sequence for a permutation π is a maximal sequence of consecutive integers i, i + 1, . . . , j such that π(i) < π(i + 1) < π(i + 2) < · · · < π(j) An alternative description of the shuffle is as follows: break the deck at an integer k chosen from the Binomial (N, 12) distribution, then perform an “unbiased” riffle of the two stacks. (In an unbiased riffle of two stacks of sizes A,B, cards are dropped one at a time from the bottoms of the stacks; at any step, if the two stacks have A′, B′ cards, respectively, then the probability that the next card is dropped from the stack with A′ cards is A′/(A′ +B′).) Bayer and Diaconis (1992) gave an intriguing “dynamical” description of the GSR shuffling process. In this representation, the individual cards are represented by points of the unit interval; these points are gotten by taking a sample of size N from the uniform distribution. The assignment of points to cards is such that the original order of the N cards in the deck agrees with the natural order of the corresponding points in the unit interval. The evolution of the deck is determined by the motion of the N points under the action of the doubling map T : x → 2x mod 1: the order of the cards in the deck after n shuffles is determined by the order of the (marked) points in [0,1] after T has been applied n times. Using this representation, Bayer and Diaconis obtained the following exact expression for the probability that the deck is in state π after n shuffles: ( 2n +N − r N )