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August 2021 Cutoff for a one-sided transposition shuffle
Michael E. Bate, Stephen B. Connor, Oliver Matheau-Raven
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Ann. Appl. Probab. 31(4): 1746-1773 (August 2021). DOI: 10.1214/20-AAP1632

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 nonconstant 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 nlogn. 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.

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Michael E. Bate. Stephen B. Connor. Oliver Matheau-Raven. "Cutoff for a one-sided transposition shuffle." Ann. Appl. Probab. 31 (4) 1746 - 1773, August 2021. https://doi.org/10.1214/20-AAP1632

Information

Received: 1 July 2019; Revised: 1 June 2020; Published: August 2021
First available in Project Euclid: 15 September 2021

Digital Object Identifier: 10.1214/20-AAP1632

Subjects:
Primary: 60J10
Secondary: 20C30

Keywords: coupon collecting , Cutoff phenomenon , mixing time , representation theory , Young tableaux

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 4 • August 2021
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