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

Akemann, Gernot; Tribe, Roger; Tsareas, Athanasios; Zaboronski, Oleg

doi:10.1142/s201032632050015x

Cited by 14 publications

(29 citation statements)

References 60 publications

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“…They gave convincing arguments for any z in the bulk, a result then proved by Walters and Starr. Explicit formulae have also been recently obtained in [1] for the conditional expectation of diagonal and off-diagonal overlaps with respect to any number of eigenvalues. We include the following statement and its short proof for the sake of completeness.…”

Section: 3mentioning

confidence: 99%

The distribution of overlaps between eigenvectors of Ginibre matrices

Bourgade

Dubach

2019

Probab. Theory Relat. Fields

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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.

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Section: 3mentioning

confidence: 99%

The distribution of overlaps between eigenvectors of Ginibre matrices

Bourgade

Dubach

2019

Probab. Theory Relat. Fields

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“…Thus, the reduced kernel in (3.7) containing a triple sum is not easy to handle in the limit of N → ∞. Fortunately, in [28], an alternative form was derived after a long calculation. It only contains single sums in terms of the exponential polynomials (3.14) and function (3.13).…”

Section: Determinantal Structure Of the Conditional Diagonal Overlapsmentioning

confidence: 99%

“…For the off-diagonal overlap, more details are given in [28]. In Section 4, we focus on the local statistics of the diagonal overlap everywhere in the bulk of the spectrum, extending the results for the origin from [28]. For further results regarding edge statistics, the limiting connection between edge and bulk as well as the asymptotics for large argument separation in the bulk we refer also to [28].…”

Section: Introductionmentioning

confidence: 99%

Universality of Random Matrix Dynamics

Burda¹

2020

Acta Phys. Pol. B

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In these proceedings, we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of k-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on k ≥ 2 complex eigenvalues. As a new result, we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.

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“…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%

“…Notice the difference in normalisation in ON (x1, x2) compared to[41] 3. Here, contact terms are absent in the 2-point function.…”

mentioning

confidence: 98%

Universal eigenvector correlations in quaternionic Ginibre ensembles

Akemann

Förster

Kieburg

2020

J. Phys. A: Math. Theor.

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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.

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

Cited by 14 publications

References 60 publications

The distribution of overlaps between eigenvectors of Ginibre matrices

The distribution of overlaps between eigenvectors of Ginibre matrices

Universality of Random Matrix Dynamics

Universal eigenvector correlations in quaternionic Ginibre ensembles

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