Real Eigenvalues of Elliptic Random Matrices

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