Journal article

Singular Value Statistics of Matrix Products with Truncated Unitary Matrices

Mario Kieburg, Arno BJ Kuijlaars, Dries Stivigny

INTERNATIONAL MATHEMATICS RESEARCH NOTICES | OXFORD UNIV PRESS | Published : 2016

Abstract

We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also include new finite rank perturbations of the Meijer G-kernels found for products of complex Ginibre..

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University of Melbourne Researchers

Grants

Awarded by KU Leuven


Awarded by Belgian Interuniversity Attraction Pole


Awarded by FWO Flanders


Funding Acknowledgements

M.K. acknowledges partial financial support by the Alexander von Humboldt foundation. A.K. and D.S. are supported by KU Leuven Research Grant OT/12/073 and the Belgian Interuniversity Attraction Pole P07/18. A.K. is also supported by FWO Flanders projects G.0641.11 and G.0934.13.