Cutoff for a One-sided Transposition Shuffle
Abstract
We introduce a new type of card shuffle called one-sided transpositions. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff for the one-sided transposition shuffle at time . We also study a weighted generalisation of the shuffle which, in particular, allows us to recover the well known mixing time of the classical random transposition shuffle.
Keywords
Cite
@article{arxiv.1907.12074,
title = {Cutoff for a One-sided Transposition Shuffle},
author = {Michael E. Bate and Stephen B. Connor and Oliver Matheau-Raven},
journal= {arXiv preprint arXiv:1907.12074},
year = {2020}
}
Comments
24 pages. In this revised paper we prove the existence of a total variation cutoff for the biased one-sided transposition shuffles with bias $\alpha \geq 1$. This completes the proof of the existence of a cutoff for any biased one-sided transposition shuffle with bias $\alpha \in \mathbb{R}$