Shuffling Large Decks of Cards and the Bernoulli-Laplace Urn Model

Abstract

In card games, in casino games with multiple decks of cards and in cryptography, one is sometimes faced with the following problem: How can a human (as opposed to a computer) shuffle a large deck of cards? The procedure we study is to break the deck into several reasonably sized piles, shuffle each thoroughly, recombine the piles, perform a simple deterministic operation, for instance a cut, and repeat. This process can also be seen as a generalised Bernoulli-Laplace urn model. We use coupling arguments and spherical function theory to derive upper and lower bounds on the mixing times of these Markov chains.