Cutoff for a One-sided Transposition Shuffle

TL;DR

This paper introduces a new one-sided transposition shuffle, derives its eigenvalues using Young tableaux, and proves a cutoff at time n log n, also analyzing a weighted version related to classical transpositions.

Contribution

It provides an explicit eigenvalue formula for the new shuffle and establishes a total-variation cutoff, extending understanding of mixing times in card shuffles.

Findings

Eigenvalues characterized via Young tableaux

Cutoff at time n log n for the shuffle

Weighted generalization recovers classical results

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 $n\log n$. 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.