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

Abstract

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.