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

$$\mathbb{E}(N_{\mathbb{R}}^{(m)})=\sqrt{\frac{2mL}{\mathrm{\pi }}}\mathrm{arctanh}\left(\sqrt{\frac{N}{N+L}}\right)+O(1),N\to \mathrm{\infty }.$$

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 $N\to \mathrm{\infty }$ 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).