Universality for Products of Random Matrices I: Ginibre and Truncated Unitary Cases

Liu, Dang-Zheng; Wang, Yanhui

doi:10.1093/imrn/rnv247

Cited by 11 publications

(11 citation statements)

References 39 publications

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“…However, the correlation kernel that arises for products is more complicated and has to be expressed in terms of Meijer G-functions. The determinantal machinery allowed them to analyse the point process of eigenvalues in various microscopic regimes for fixed m. A similar theory has been developed to analyse products of truncated unitary random matrices [4,1,35]. Recently these developments have been extended to study the question of double scaling limits as both m and N tend to infinity simultaneously [3,33,34].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the number of real eigenvalues of a product of truncated orthogonal random matrices

Little

Mezzadri

Simm

2022

Electron. J. Probab.

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Let O be chosen uniformly at random from the group of (N + L) × (N + L) orthogonal matrices. Denote by Õ the upper-left N × N corner of O, which we refer to as a truncation of O. In this paper we prove two conjectures of Forrester, Ipsen and Kumar (2020) on the number of real eigenvalues N (m) R of the product matrix Õ1 . . . Õm, where the matrices { Õj} m j=1 are independent copies of Õ. When L grows in proportion to N , we prove thatWe also prove the conjectured form of the limiting real eigenvalue distribution of the product matrix. Finally, we consider the opposite regime where L is fixed with respect to N , known as the regime of weak non-orthogonality. In this case each matrix in the product is very close to an orthogonal matrix. We show that E(N (m) R ) ∼ cL,m log(N ) as N → ∞ and compute the constant cL,m explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and Życzkowski (2010).

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the number of real eigenvalues of a product of truncated orthogonal random matrices

Little

Mezzadri

Simm

2022

Electron. J. Probab.

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show abstract

“…However, the correlation kernel that arises for products is more complicated and has to be expressed in terms of Meijer G-functions. The determinantal machinery allowed them to analyse the point process of eigenvalues in various microscopic regimes for fixed m. A similar theory has been developed to analyse products of truncated unitary random matrices [4,1,33]. Recently these developments have been extended to study the question of double scaling limits as both m and N tend to infinity simultaneously [3,31,32].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the number of real eigenvalues of a product of truncated orthogonal random matrices

Little¹,

Mezzadri²,

Simm³

2021

Preprint

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“…Appendix A: Calculation of the integral (57) In this Appendix, we detail the calculation of the integral given by Eq. (57)…”

Section: Acknowledgmentsmentioning

confidence: 99%

Eigenvector statistics of the product of Ginibre matrices

2017

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We develop a method to calculate left-right eigenvector correlations of the product of m independent N × N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N . We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O = 1 + (m/2)(N − 1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N → ∞ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

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Universality for Products of Random Matrices I: Ginibre and Truncated Unitary Cases

Cited by 11 publications

References 39 publications

On the number of real eigenvalues of a product of truncated orthogonal random matrices

On the number of real eigenvalues of a product of truncated orthogonal random matrices

On the number of real eigenvalues of a product of truncated orthogonal random matrices

Eigenvector statistics of the product of Ginibre matrices

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