Exploring the Geometric Model of Riffle Shuffling

Card shuffling is an interesting topic to explore because of its complexity. Initially, card shuffling seems simple because it is ubitquitous. The majority of people know how to shuffle a deck of cards but few consider the math behind it. However, when it comes to analyzing the elements of card shuffling, it incorporates linear algebra, group theory, probability theory, and Markov Chains. When playing card games, people use various techniques to keep others from having an unfair advantage, the most prevalent technique being cutting and randomizing the deck. In this paper, we will investigate how well two models—the standard model and Geometric Model—describe how humans shuffle cards. We will find the optimal region of fast riffle shuffles for a small deck and measure the randomization of a deck using variation distance. To gain a better understanding of the Geometric Model of card shuffling we will also examine the transition matrix that is formed from the probabilities of transitioning from one deck to another. We found the eigenvalues and eigenvectors for the transition matrix for small decks.