Riffle shuffles of a deck with repeated cards

TL;DR

This paper investigates how quickly a deck of cards becomes random under riffle shuffles, especially when only certain features like ranks are considered, extending previous models with new formulas and asymptotic analysis.

Contribution

It introduces a general framework for analyzing convergence rates of Markov chains on groups and cosets, with exact formulas and asymptotic solutions for feature-specific shuffling.

Findings

Derived exact formulas for feature-based convergence

Developed new analytic methods for asymptotic analysis

Provided accurate asymptotic results for various deck features

Abstract

We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate.