Mean left-right eigenvector self-overlap in the real Ginibre ensemble

We study analytically the Chalker–Mehlig mean diagonal overlap Formula: see text between left and right eigenvectors associated with a complex eigenvalue Formula: see text of Formula: see text matrices in the real Ginibre ensemble (GinOE). We first derive a general finite Formula: see text expression for the mean overlap and then investigate several scaling regimes in the limit Formula: see text. While in the generic spectral bulk and edge of the GinOE the limiting expressions for Formula: see text are found to coincide with the known results for the complex Ginibre ensemble (GinUE), in the region of eigenvalue depletion close to the real axis the asymptotic for the GinOE is considerably different. We also study numerically the distribution of diagonal overlaps and conjecture that it is the same in the bulk and at the edge of both the GinOE and GinUE, but essentially different in the depletion region of the GinOE.