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2022 On the number of real eigenvalues of a product of truncated orthogonal random matrices
Alex Little, Francesco Mezzadri, Nick Simm
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Electron. J. Probab. 27: 1-32 (2022). DOI: 10.1214/21-EJP732

Abstract

Let O be chosen uniformly at random from the group of (N+L)×(N+L) orthogonal matrices. Denote by O˜ 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 NR(m) of the product matrix O˜1O˜m, where the matrices {O˜j}j=1m are independent copies of O˜. When L grows in proportion to N, we prove that

E(NR(m))=2mLπarctanhNN+L+O(1),N.

We 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(NR(m))cL,mlog(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).

Funding Statement

A. L. would like to gratefully acknowledge the support of the UK Engineering and Physical Sciences Research Council (EPSRC) DTP (grant number EP/N509619/1). F. M. is grateful for support from a University Research Fellowship of the University of Bristol. N. S. gratefully acknowledges support of the Royal Society University Research Fellowship ‘Random matrix theory and log-correlated Gaussian fields’, reference URF\R1\180707.

Acknowledgments

We are grateful to an anonymous referee who pointed out to us references [14, 32] and whose comments improved the presentation of the paper.

Citation

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Alex Little. Francesco Mezzadri. Nick Simm. "On the number of real eigenvalues of a product of truncated orthogonal random matrices." Electron. J. Probab. 27 1 - 32, 2022. https://doi.org/10.1214/21-EJP732

Information

Received: 25 February 2021; Accepted: 9 December 2021; Published: 2022
First available in Project Euclid: 14 January 2022

Digital Object Identifier: 10.1214/21-EJP732

Subjects:
Primary: 15B52 , 60B20

Keywords: Products of random matrices , real eigenvalues , truncated orthogonal matrices

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