%0 Journal Article
%T Card shuffling and diophantine approximation
%A Omer Angel
%A Yuval Peres
%A David B. Wilson
%J Mathematics
%D 2007
%I arXiv
%R 10.1214/07-AAP484
%X 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 $\Theta(n^{-3/2})$.
%U http://arxiv.org/abs/0707.2994v3