The distribution of overlaps between eigenvectors of Ginibre matrices

Abstract

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.

Symmetries of the quaternionic Ginibre ensemble

“…Proof. The computation is analogous to what was done in [5] in the complex Ginibre case, using the beta-gamma algebra. Theorem 3.4 and Lemma 2.7 give us the following chain of equalities in distribution, where we used that γ (k)…”

Universal eigenvector correlations in quaternionic Ginibre ensembles

“…Rigorous results have been obtained for the distribution of diagonal and off-diagonal overlaps and their correlations for the complex Ginibre ensemble (GinUE) in [10] using probabilistic, and in [26,27] using supersymmetric and orthogonal polynomial techniques, respectively. The latter also included partial results on the real eigenvalues of the real Ginibre ensemble (GinOE).…”

“…The latter also included partial results on the real eigenvalues of the real Ginibre ensemble (GinOE). The authors of [10] numerically assessed the universality of their results in some examples. The correlation of angles between eigenvectors in the GinUE was analysed in [7].…”

On the determinantal structure of conditional overlaps for the complex Ginibre ensemble

“…These techniques were further developed including Feynman diagrams [12,13], free probability [14] or stochastic differential equations [15] and applied to different ensembles including products of elliptic Ginibre matrices [12]. These, as well as truncated unitary and spherical ensembles, were analysed in [16] using probabilistic means, after an earlier breakthrough for these methods in [17], see also [18] for the correlations between angles of eigenvectors. The quaternionic Ginibre ensemble appeared more recently from a probabilistic angle [19,20] as well as for finite-N in [21], using the heuristic tools of [22].…”

“…A common underlying question is that of universality of the newly found eigenvectors correlations. While much of this remains open, numerical checks [17] strongly suggest some universality, and we refer to [13] for a comprehensive list of various ensembles in the global bulk regime, pointing at parallels and differences. A further indication is the recently found universality of complex bulk and edge eigenvalue correlations away from the real line, uniting all three Ginibre ensembles [26,27].…”