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)×(N +L) orthogonal matrices. Denote by Õ the upper-left N × 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 [18] on the number of real eigenvalues) ∼ c L,m log(N ) as N → ∞ 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 [28].

Real eigenvalues of elliptic random matrices

“…(See also [8,35] for the scaling limits of real Ginibre matrices.) For the real eigenvalues of other random matrix models, we refer to [14,17,26,27,29,33] and references therein. We also remark that the statistics of real eigenvalues enjoy an intimate relationship with diverse topics, including the Zakharov-Shabat system [7] and the annihilating Brownian motion [32].…”