Card shuffling and diophantine approximation

TL;DR

This paper analyzes the spectral gap of an overlapping-cycles shuffle, revealing surprising dependence on the ratio of parameters and number theory properties, with implications for mixing times.

Contribution

It determines the spectral gap for a new card shuffling method, showing how it varies with the ratio of parameters and number-theoretic properties.

Findings

Spectral gap is Θ(n^{-2}) for rational ratios.

Spectral gap is Θ(n^{-3/2}) for certain irrational ratios.

Behavior depends on Diophantine approximation properties.

Abstract

The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either the $n$th card or the $(n-k)$th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of $k$ and $n$, has surprising behavior. For example, suppose $k$ is the closest integer to $\alpha n$ for a fixed real $\alpha\in(0,1)$. Then for rational $\alpha$ the spectral gap is $\Theta(n^{-2})$, while for poorly approximable irrational numbers $\alpha$, such as the reciprocal of the golden ratio, the spectral gap is $\Theta(n^{-3/2})$.