Card shuffling and Diophantine approximation

Abstract

The “overlapping-cycles shuffle” mixes a deck of n cards by moving either the nth 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 αn for a fixed real α∈(0, 1). Then for rational α the spectral gap is Θ(n⁻²), while for poorly approximable irrational numbers α, such as the reciprocal of the golden ratio, the spectral gap is Θ(n^−3/2).