Singular value statistics of matrix products with truncated unitary matrices

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 random matrices.

29 pages

References in corpus (8)

• Orthogonal polynomial ensembles in probability theory
• Product of random projections, Jacobi ensembles and universality problems arising from free probability
• Differential equations for singular values of products of Ginibre random matrices
• Universality conjecture and results for a model of several coupled positive-definite matrices
• Superbosonization formula and its application to random matrix theory
• Non-Hermitean Wishart random matrices (I)
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices
• Transformations of polynomial ensembles

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