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, (z), and the associated mean eigenvector self-overlaps, O (z), at the spectral edge of N N real and complex elliptic Ginibre matrices, as N. Two different regimes of ellipticity are studied: strong non-Hermiticity, keeping the ellipticity parameter fixed and weak non-Hermiticity with 1 as N. At strong non-Hermiticity, we find that both (z) and 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 (z) and 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.