How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

Abstract

A truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product P m of m independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of P m form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulae, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity.

Time-inhomogeneous random Markov chains

“…As t grows, the eigenvalues increasingly concentrate on the real line. This concentration phenomenon has also been observed for products of other kinds of random matrices [31,32,33,34,35,36,37]. This paper is organized as follows.…”

“…It is known that, for large N , real Ginibre matrices have a finite fraction of real eigenvalues (asymptotically 2N/π of them [30]). Moreover, a product of t such matrices has 2tN/π real eigenvalues [36], to leading order for large N and small t (this √ t growth is also true for a product of truncated orthogonal matrices [37]). Although we are more interested in small matrices, we have observed that something similar also happens in the context we are addressing.…”

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

“…Using this integrable structure, the probability that all eigenvalues of a product of truncated orthogonal random matrices are real was calculated in [20]. Simplified expressions for the correlation kernel of the Pfaffian point process were obtained in the recent work [19].…”

“…Õm form a Pfaffian point process [28]. In the recent work [19] this has been simplified and explicit formulae for the correlation kernel of the Pfaffian point process were identified. Theorem 2.1 (Forrester,Ipsen and Kumar [19]).…”

“…This paper is structured as follows. In Section 2 we begin by recalling the results of [19] regarding the Pfaffian structure associated with products of truncated orthogonal random matrices. Then we give an overview of the proof of Theorem 1.1, postponing the full details until Section 3.…”