Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Abstract

It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that admits a representation in terms of Meijer G-functions. We prove the universality of the local statistics of the squared singular values, namely, the bulk universality given by the sine kernel and the edge universality given by the Airy kernel. The proof is based on the asymptotic analysis for the double contour integral representation of the correlation kernel. Our strategy can be generalized to deal with other models of products of random matrices introduced recently and to establish similar universal results. Two more examples are investigated, one is the product of $M$ Ginibre matrices and the inverse of $K$ Ginibre matrices studied by Forrester, and the other one is the product of $M-1$ Ginibre matrices with one truncated unitary matrix considered by Kuijlaars and Stivigny.

arXiv comment32 pages, 7 figures, revised upon the suggestions of the anonymous referees

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Cited by in corpus (7)

• Recent exact and asymptotic results for products of independent random matrices
• Singular value statistics of matrix products with truncated unitary matrices
• Products of Independent Gaussian Random Matrices
• Local universality in biorthogonal Laguerre ensembles
• Dropping the independence: singular values for products of two coupled random matrices
• Correlation kernels for sums and products of random matrices
• Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices