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Harmonic analysis for rank-1 randomised Horn problems

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Abstract

The randomised Horn problem, in both its additive and multiplicative versions, has recently drawn an increasing interest. Especially, closed analytical results have been found for the rank-1 perturbation of sums of Hermitian matrices and products of unitary matrices. We will generalise these results to rank-1 perturbations for products of positive-definite Hermitian matrices and prove the other results in a new unified way. Our ideas work along harmonic analysis for matrix groups via spherical transforms that have been successfully applied in products of random matrices in the past years. In order to achieve the unified derivation of all three cases, we define the spherical transform on the unitary group and prove its invertibility.

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Notes

  1. In probability theory, conventionally a characteristic function is defined as \(\mathbb {E}_X[e^{itX}]\) instead.

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Acknowledgements

The work of PJF and JZ was supported by the Australian Research Council (ARC) through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS). PJF also acknowledges partial support from ARC grant DP170102028, and JZ acknowledges the support of a Melbourne postgraduate award, and an ACEMS top up scholarship. We are grateful for fruitful discussions with Jesper R. Ipsen. We also thank Vadim Gorin for correspondence leading to the penultimate paragraph of the Introduction. And we also thank the referees for valuable comments.

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  1. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Jiyuan Zhang, Mario Kieburg & Peter J. Forrester

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  1. Jiyuan Zhang
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  2. Mario Kieburg
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Correspondence to Jiyuan Zhang.

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Zhang, J., Kieburg, M. & Forrester, P.J. Harmonic analysis for rank-1 randomised Horn problems. Lett Math Phys 111, 98 (2021). https://doi.org/10.1007/s11005-021-01429-7

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