Cutoff for the asymmetric riffle shuffle

DOI10.1214/22-AOP1582zbMATH Open1500.60042arXiv2103.05068OpenAlexW3133839624MaRDI QIDQ2087390

Author name not available (Why is that?)

Publication date: 27 October 2022

Published in: (Search for Journal in Brave)

Abstract: In the Gilbert-Shannon-Reeds shuffle, a deck of

N

cards is cut into two approximately equal parts which are then riffled uniformly at random. Bayer and Diaconis famously showed that this Markov chain undergoes cutoff in total variation after

f r a c 3 l o g (N) 2 l o g (2)

shuffles. We establish cutoff for the more general asymmetric riffle shuffles in which one cuts the deck into differently sized parts before riffling. The value of the cutoff point confirms a conjecture of Lalley from 2000. Some appealing consequences are that asymmetry always slows mixing and that total variation mixing is strictly faster than separation and

L^{i n f t y}

mixing.

Full work available at URL: https://arxiv.org/abs/2103.05068

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