Eigenvector statistics of the product of Ginibre matrices

Burda, Z.; Spisak, B. J.; Vivo, Pierpaolo

doi:10.1103/physreve.95.022134

Cited by 11 publications

(13 citation statements)

References 98 publications

(123 reference statements)

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“…Remarkably, the function O 1 (z) can be efficiently studied within the formalism of free probability [33] which recently allowed to extend the Chalker-Mehlig formulas to a broad class of invariant ensembles beyond the Gaussian case [3,37]. O 1 (z) is also known for a finite size of the complex Ginibre matrix [8,35] and products of small Ginibre matrices [7]. 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: Gin2mentioning

confidence: 99%

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Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Fyodorov

Tarnowski

2020

Ann. Henri Poincaré

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We study the distribution of the eigenvalue condition numbers $$\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}$$ κ i = ( l i ∗ l i ) ( r i ∗ r i ) associated with real eigenvalues $$\lambda _i$$ λ i of partially asymmetric $$N\times N$$ N × N random matrices from the real Elliptic Gaussian ensemble. The large values of $$\kappa _i$$ κ i signal the non-orthogonality of the (bi-orthogonal) set of left $${\mathbf{l}}_i$$ l i and right $${\mathbf{r}}_i$$ 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 density function (JDF) $${{\mathcal {P}}}_N(z,t)$$ P N ( z , t ) of $$t=\kappa _i^2-1$$ t = κ i 2 - 1 and $$\lambda _i$$ λ i taking value z, and investigate its several scaling regimes in the limit $$N\rightarrow \infty $$ N → ∞ . When the degree of asymmetry is fixed as $$N\rightarrow \infty $$ N → ∞ , the number of real eigenvalues is $$\mathcal {O}(\sqrt{N})$$ O ( N ) , and in the bulk of the real spectrum $$t_i=\mathcal {O}(N)$$ t i = O ( N ) , while on approaching the spectral edges the non-orthogonality is weaker: $$t_i=\mathcal {O}(\sqrt{N})$$ t i = O ( N ) . In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as $$N\rightarrow \infty $$ N → ∞ . In such a regime eigenvectors are weakly non-orthogonal, $$t=\mathcal {O}(1)$$ t = O ( 1 ) , and we derive the associated JDF, finding that the characteristic tail $${{\mathcal {P}}}(z,t)\sim t^{-2}$$ 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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Section: Gin2mentioning

confidence: 99%

“…First, we find convenient integral representations, which should allow the use of the Laplace method. For this we start from (39) and, using the integral representation for Hermite polynomials in (7) with both signs, we obtain…”

Section: Asymptotic Analysismentioning

confidence: 99%

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Fyodorov

Tarnowski

2020

Ann. Henri Poincaré

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“…However, when moving to products of m complex Ginibre ensembles the circular law is modified to 1 mπ |x| 2 m −2 on the unit disc. The resulting overlap is then multiplied by this density, with the quadratic power in (4.12) replaced by 2 → 2/m [14].…”

Section: Macroscopic Bulk Limit Of the Overlapsmentioning

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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“…Using a combination of Green's functions and diffusion equations, it was noticed early on that the Dysonian dynamics in this ensemble couples the complex eigenvalues and their eigenvectors in a non-trivial way [10,11]. 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.…”

Section: Introductionmentioning

confidence: 99%

Determinantal Structure and Bulk Universality of Conditional Overlaps in the Complex Ginibre Ensemble

Akemann¹,

Tribe²,

Tsareas³

et al. 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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Eigenvector statistics of the product of Ginibre matrices

Cited by 11 publications

References 98 publications

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Universal eigenvector correlations in quaternionic Ginibre ensembles

Determinantal Structure and Bulk Universality of Conditional Overlaps in the Complex Ginibre Ensemble

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