The distribution of overlaps between eigenvectors of Ginibre matrices

Paul Bourgade, Guillaume Dubach

Published 2018-01-04Version 1

We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium. We also propose an analogue of (i) for the real Ginibre ensemble, in the bulk of the spectrum (i.e. far from the unit circle and the real axis). Here, the conjectured limiting distribution is given implicitly as an almost sure martingale limit.