Little, Alex, Mezzadri, Francesco and Simm, Nick (2022) On the number of real eigenvalues of a product of truncated orthogonal random matrices. Electronic Journal of Probability, 27. a5 1-32. ISSN 1083-6489
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Abstract
Let $O$ be chosen uniformly at random from the group of $(N+L) \times (N+L)$ orthogonal matrices. Denote by $\tilde{O}$ the upper-left $N \times N$ 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 $N^{(m)}_{\mathbb{R}}$ of the product matrix $\tilde{O}_{1}\ldots \tilde{O}_{m}$, where the matrices $\{\tilde{O}_{j}\}_{j=1}^{m}$ are independent copies of $\tilde{O}$. When $L$ grows in proportion to $N$, we prove that \begin{equation*} \mathbb{E}(N^{(m)}_{\mathbb{R}}) = \sqrt{\frac{2m L}{\pi}}\,\mathrm{arctanh}\left(\sqrt{\frac{N}{N+L}}\right) + O(1), \qquad N \to \infty.\end{equation*}
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 \textit{regime of weak non-orthogonality}. In this case each matrix in the product is very close to an orthogonal matrix. We show that $\mathbb{E}(N^{(m)}_{\mathbb{R}}) \sim c_{L,m}\,\log(N)$ as $N \to \infty$ and compute the constant $c_{L,m}$ explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and \.{Z}yczkowski (2010).
Item Type: | Article |
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Schools and Departments: | School of Mathematical and Physical Sciences > Mathematics |
Research Centres and Groups: | Probability and Statistics Research Group |
SWORD Depositor: | Mx Elements Account |
Depositing User: | Mx Elements Account |
Date Deposited: | 10 Dec 2021 08:33 |
Last Modified: | 10 Mar 2022 14:22 |
URI: | http://sro.sussex.ac.uk/id/eprint/103314 |
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