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

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 $N_{\mathbb{R}}^{(m)}$ of the product matrix ${\tilde{O}}_1\dots {\tilde{O}}_m$, 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 $$\mathbb{E}(N_{\mathbb{R}}^{(m)})=\sqrt{\frac{2mL}{\pi }}\mathrm{arctanh}\left(\sqrt{\frac{N}{N+L}}\right)+O(1),N\to \infty .$$ 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 $\mathbb{E}(N_{\mathbb{R}}^{(m)})\sim c_{L,m}\log (N)$ 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 Życzkowski (2010).

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• Real spectra of large real asymmetric random matrices
• Real eigenvalues of elliptic random matrices
• The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble