Real Eigenvalues of Elliptic Random Matrices

Abstract

Abstract We consider the real eigenvalues of an $(N \times N)$ real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $\tau _N\in [0,1]$. In the almost-Hermitian regime where $1-\tau _N=\Theta (N^{-1})$, we obtain the large-$N$ expansion of the mean and the variance of the number of the real eigenvalues. Furthermore, we derive the limiting densities of the real eigenvalues, which interpolate the Wigner semicircle law and the uniform distribution, the restriction of the elliptic law on the real axis. Our proofs are based on the skew-orthogonal polynomial representation of the correlation kernel due to Forrester and Nagao.