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.