Singular value statistics of matrix products with truncated unitary matrices

Kieburg, Mario; Kuijlaars, Arno B.J.; Stivigny, Dries

doi:10.1093/imrn/rnv242

Singular value statistics of matrix products with truncated unitary matrices

Jan 16, 2015

33 pages

Published in:

Int.Math.Res.Not. 2016 (2016) 11, 3392-3424

Published: Jan 1, 2016

e-Print:

1501.03910 [math.PR]

DOI:

10.1093/imrn/rnv242

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Abstract: (arXiv)

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 random matrices.

Note:

29 pages •
29 pages

matrix model: random
unitarity
perturbation
correlation
statistics
scaling

References(58)

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