A note on mixed matrix moments for the complex Ginibre ensemble

We find stochastic equations governing eigenvalues and eigenvectors of a dynamical complex Ginibre ensemble reaffirming the intertwined role played between both sets of matrix degrees of freedom. We solve the accompanying Smoluchowski-Fokker-Planck equation valid for any initial matrix. We derive evolution equations for the averaged extended characteristic polynomial and for a class of k-point eigenvalue correlation functions. From the latter we obtain a novel formula for the eigenvector correlation function which we inspect for Ginibre and spiric initial conditions and obtain macro-and microscopic limiting laws.

“…agrees with the well-known result [21]. For completeness, we show below both macro-and microscopic limiting laws arising from Eq.…”

Section: Ginibre Casesupporting

confidence: 91%

“…Let us now use the evolution equations (21) to derive relations between eigenvalue and eigenvector correlation functions. We focus on the k-point eigenvalue correlation functions of the form:…”

Section: Relations Between Eigenvalue and Eigenvector Correlation mentioning

confidence: 99%

Full Dysonian dynamics of the complex Ginibre ensemble

Grela¹,

Warchoł²

2018

“…Our formula (1) is valid only in the limit N → ∞. In order to access condition numbers in the Ginibre ensemble in the finite N , we superimpose the results from [18] and [38], derived with the use of different techniques. We obtain the formula for the averaged squared eigenvalue condition number…”

Section: Examplesmentioning

confidence: 99%

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

Belinschi¹,

Nowak²,

Speicher³

et al. 2017

We extend the so-called "single ring theorem" [1], also known as the Haagerup-Larsen theorem [2], by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows to calculate the conditional expectation of the squared eigenvalue condition number. We give examples and we provide cross-check of the analytic prediction by the large scale numerics.

“…The correlation of angles between eigenvectors in the GinUE was analysed in [7]. Starting from moments of the overlap matrix [49,16], a determinantal structure in terms of a kernel was derived in [3] for the conditional overlaps for finite matrix size N , that allowed to take various large-N limits. The question of localisation of eigenvectors in non-Hermitian RMT was answered in [45,40], going beyond Gaussian ensembles, but we will not pursue this direction in our quaternionic Ginibre ensemble (GinSE).…”

Section: Introductionmentioning

confidence: 99%

Universal eigenvector correlations in quaternionic Ginibre ensembles

Akemann

Förster

Kieburg

2020

Non-Hermitian random matrices enjoy non-trivial correlations in the statistics of their eigenvectors. We study the overlap among left and right eigenvectors in Ginibre ensembles with quaternion valued Gaussian matrix elements. This concept was introduced by Chalker and Mehlig in the complex Ginibre ensemble. Using a Schur decomposition, for harmonic potentials we can express the overlap in terms of complex eigenvalues only, coming in conjugate pairs in this symmetry class. Its expectation value leads to a Pfaffian determinant, for which we explicitly compute the matrix elements for the induced Ginibre ensemble with 2α zero eigenvalues, for finite matrix size N . In the macroscopic large-N limit in the bulk of the spectrum we recover the limiting expressions of the complex Ginibre ensemble for the diagonal and off-diagonal overlap, which are thus universal.