Shuffling large decks of cards and the Bernoulli-Laplace urn model

TL;DR

This paper analyzes a human-like card shuffling method involving dividing, shuffling, recombining, and repeating, modeled as a Bernoulli-Laplace urn process, and derives bounds on how quickly it randomizes the deck.

Contribution

It introduces a novel analysis of a practical shuffling procedure using advanced mathematical tools to estimate mixing times of the associated Markov chain.

Findings

Derived upper bounds on mixing times for the shuffling process.

Applied coupling arguments and spherical function theory in the analysis.

Provided insights into the efficiency of human-like shuffling methods.

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, do some 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 bounds on the mixing times of these Markov chains.