Mean Eigenvector Self-Overlap in the Real and Complex Elliptic Ginibre Ensembles at Strong and Weak Non-Hermiticity

We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in \(N\times N\) non-Hermitian random Gaussian matrices. In a well-known work by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as \(N\rightarrow \infty \) (Chalker and Mehlig in Phys Rev Lett 81(16):3367–3370, 1998). In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles, which are characterised by correlations between off-diagonal entries controlled by a parameter \(\tau \in [0,1]\), with \(\tau =1\) corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite N, for any eigenvalue off the real axis. We further investigate several scaling regimes as \(N\rightarrow \infty \), both in the limit of strong non-Hermiticity keeping a fixed \(\tau \in [0,1)\) and in the weak non-Hermiticity limit, with \(\tau \) approaching unity in such a way that \(N(1-\tau )\) remains finite.