Spectral density of complex eigenvalues and associated mean eigenvector self-overlaps at the edge of elliptic Ginibre ensembles

We consider the density of complex eigenvalues, $\rho (z)$, and the associated mean eigenvector self-overlaps, $\mathcal{O}(z)$, at the spectral edge of $N\times N$ real and complex elliptic Ginibre matrices, as $N\to \mathrm{\infty }$. Two different regimes of ellipticity are studied: strong non-Hermiticity, keeping the ellipticity parameter $\tau$ fixed and weak non-Hermiticity with $\tau \to 1$ as $N\to \mathrm{\infty }$. At strong non-Hermiticity, we find that both $\rho (z)$ and $\mathcal{O}(z)$ have the same leading order behaviour across the elliptic Ginibre ensembles, establishing the expected universality. In the limit of weak non-Hermiticity, we find different results for $\rho (z)$ and $\mathcal{O}(z)$ across the two ensembles. This paper is the final of three papers that we have presented addressing the mean self-overlap of eigenvectors in these ensembles.