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Cutoff phenomenon for the warp-transpose top with random shuffle

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Abstract

Let \(\{G_n\}_1^{\infty }\) be a sequence of non-trivial finite groups. In this paper, we study the properties of a random walk on the complete monomial group \(G_n\wr S_n\) generated by the elements of the form \(({{\,\textrm{e}\,}},\dots ,{{\,\textrm{e}\,}},g;{{\,\textrm{id}\,}})\) and \(({{\,\textrm{e}\,}},\dots ,{{\,\textrm{e}\,}},g^{-1},{{\,\textrm{e}\,}},\dots ,{{\,\textrm{e}\,}},g;(i,n))\) for \(g\in G_n,\;1\le i< n\). We call this the warp-transpose top with random shuffle on \(G_n\wr S_n\). We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is \(O\left( n\log n+\frac{1}{2}n\log (|G_n|-1)\right) \). We show that this shuffle exhibits \(\ell ^2\)-cutoff at \(n\log n+\frac{1}{2}n\log (|G_n|-1)\) and total variation cutoff at \(n\log n\).

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Acknowledgements

I extend sincere thanks to my PhD advisor Arvind Ayyer for all the insightful discussions during the preparation of this paper. I am very grateful to the anonymous referees of the Journal of Algebraic Combinatorics for many constructive suggestions. I would like to thank an anonymous referee of the Algebraic Combinatorics for the valuable comments, which helped improve the total variation upper bound result and simplify the proof of the total variation lower bound. I am grateful to Professor Tullio Ceccherini-Silberstein for his encouragement and inspiring comments. I would also like to thank Guy Blachar, Ashish Mishra, and Shangjie Yang for their discussions. I would like to acknowledge support in part by a UGC Centre for Advanced Study grant.

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Funding was provided by UGC-DAE Consortium for Scientific Research, University Grants Commission.

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  1. Department of Mathematics, Bar-Ilan University, 5290002, Ramat-Gan, Israel

    Subhajit Ghosh

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  1. Subhajit Ghosh
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Correspondence to Subhajit Ghosh.

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The author has no conflict of interest to disclose. This article is a part of author’s PhD dissertation. The extended abstract of this article was accepted in FPSAC 2020 (online).

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Ghosh, S. Cutoff phenomenon for the warp-transpose top with random shuffle. J Algebr Comb 58, 775–809 (2023). https://doi.org/10.1007/s10801-023-01271-1

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