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

Version 2 2023-06-12, 08:13

Version 1 2023-06-10, 01:59

journal contribution

posted on 2023-06-12, 08:13 authored by Alex Little, Francesco Mezzadri, Nicholas SimmNicholas SimmLet $O$ 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).

## Funding

### Random matrix theory and log-correlated Gaussian fields; G2472; ROYAL SOCIETY; URF\R1\180707

## History

## Publication status

- Published

## File Version

- Published version

## Journal

Electronic Journal of Probability## ISSN

1083-6489## Publisher

Institute of Mathematical Statistics## External DOI

## Volume

27## Page range

1-32## Article number

a5## Department affiliated with

- Mathematics Publications

## Research groups affiliated with

- Probability and Statistics Research Group Publications

## Full text available

- Yes

## Peer reviewed?

- Yes

## Legacy Posted Date

2021-12-10## First Open Access (FOA) Date

2022-01-14## First Compliant Deposit (FCD) Date

2021-12-09## Usage metrics

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