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

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

doi:10.48550/arxiv.1903.09016

Cited by 4 publications

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“…is the standard Wigner semicircle density characterizing real symmetric GOE matrices, and A = (πρ sc (z)a) 2 .…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

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Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

Fyodorov,

Tarnowski

2019

Preprint

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We study the distribution of the eigenvalue condition numbers κ i = (l * i l i )(r * i r i ) associated with real eigenvalues λ i of partially asymmetric N × N random matrices from the Gaussian elliptic ensemble. The large values of κ i signal about the non-orthogonality of (bi-orthogonal) set of left l i and right r i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint probability density (JPD) P N (z, t) of t i = κ 2 i − 1 for λ conditioned to have a value z and investigate its several scaling regimes in the limit N → ∞. When the degree of asymmetry is fixed as N → ∞, the number of real eigenvalues is O( √ N ), and in the bulk of the real spectrum t i = O(N ), while on approaching the spectral edges the non-orthogonality is weaker:In both cases the corresponding JPDs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices, see [20]. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as N → ∞. In such a regime eigenvectors are weakly non-orthogonal, t = O(1), and we derive the associated JPD, finding that the characteristic tail P(z, t) ∼ t −2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

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“…is the standard Wigner semicircle density characterizing real symmetric GOE matrices, and A = (πρ sc (z)a) 2 .…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…It has been recently shown that for complex Ginibre matrices the one and two-point functions conditioned on an arbitrary number of eigenvalues are related to determinantal point processes [2].…”

Section: Introductionmentioning

confidence: 99%

Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

Fyodorov,

Tarnowski

2019

Preprint

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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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“…It has become apparent only quite recently that all three Ginibre ensembles share the same local eigenvalue statistics in the bulk [9,4] and at the edge of the spectrum [44,9]. One may therefore expect that the same holds true for the local eigenvector statistics in the bulk and at the edge, see [3] for the corresponding results in the complex Ginibre ensemble. The edge statistics is particularly relevant for scattering in chaotic cavities, cf.…”

Section: Introductionmentioning

confidence: 85%

“…Recently, there has been much progress in determining the correlations of overlaps of eigenvectors in RMT, see [49,26,27,10,16,7,3,28] for a non-exhaustive list. Many of these focus on the complex and real Ginibre ensemble introduced in [29].…”

Section: Introductionmentioning

confidence: 99%

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 4 publications

References 0 publications

Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

The distribution of overlaps between eigenvectors of Ginibre matrices

Universal eigenvector correlations in quaternionic Ginibre ensembles

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