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, ρ(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 τ fixed and weak non-Hermiticity with τ → 1 as N → ∞. At strong non-Hermiticity, we find that both ρ(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 ρ(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.