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

TL;DR

This paper proves conjectures about the expected number and distribution of real eigenvalues of products of truncated orthogonal matrices, revealing different asymptotic behaviors depending on the relation between matrix size and truncation.

Contribution

It establishes the expected number and distribution of real eigenvalues for products of truncated orthogonal matrices, confirming conjectures and extending previous single-matrix results.

Findings

Expected real eigenvalues grow as rac{rac{2mL}{\u03c0}}{ ext{arctanh}( ext{sqrt}(rac{N}{N+L}))} for large N and proportional L.

Limiting distribution of real eigenvalues matches conjectured form.

Expected number of real eigenvalues scales as c_{L,m} rac{rac{1}{ ext{log}(N)}} for fixed L as N rac{rac{1}{ ext{log}(N)}).

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