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Fluctuation Moments Induced by Conjugation with Asymptotically Liberating Random Matrix Ensembles

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Abstract

Independent Haar-unitary random matrices and independent Haar-orthogonal random matrices are known to be asymptotically liberating ensembles, and they give rise to asymptotic free independence when used for conjugation of constant matrices. G. Anderson and B. Farrel showed that a certain family of discrete random unitary matrices can actually be used to the same end. In this paper, we investigate fluctuation moments and higher-order moments induced on constant matrices by conjugation with asymptotically liberating ensembles. We show for the first time that the fluctuation moments associated with second-order free independence can be obtained from conjugation with an ensemble consisting of signed permutation matrices and the discrete Fourier transform matrix. We also determine fluctuation moments induced by various related ensembles where we do not get known expressions but others related to traffic free independence.

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Acknowledgements

The author would like to thank James A. Mingo for invaluable discussions during the preparatiion of this paper. The author also thanks the anonymous referees for their insightful suggestions and comments, all of which improved the presentation of this material. Research was partially supported by The Mexican National Council of Science and Technology (CONACYT) ref. 579659/410386.

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  1. Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, ON, K7L 3N6, Canada

    Josue Vazquez-Becerra

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  1. Josue Vazquez-Becerra
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Correspondence to Josue Vazquez-Becerra.

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Research was partially supported by The Mexican National Council of Science and Technology (CONACYT) ref. 579659/410386.

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Vazquez-Becerra, J. Fluctuation Moments Induced by Conjugation with Asymptotically Liberating Random Matrix Ensembles. J Theor Probab 36, 1972–2039 (2023). https://doi.org/10.1007/s10959-023-01246-9

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