On the rate of mixing for $p$-shuffles

Abstract

The p-shuffle is a natural generalization of the dovetail shuffle. It is defined as follows. First, the deck is cut into a top stack and a bottom stack so that the distribution of the size of the top stack is Binomial $(N, p)$, where $N$ is the total number of cards in the deck.Then, conditional on the outcome of the cut,the two stacks are “riffled” in such a way that all possible riffles (interleavings) of these two stacks are equally likely. The main result of the paper is an asymptotic $(N \to \infty)$ bound on the number of repetitions needed to “randomize” the deck.