Eigenvalues of truncated unitary matrices: disk counting statistics

Ameur, Yacin; Charlier, Christophe; Moreillon, Philippe

doi:10.1007/s00605-023-01920-4

Cited by 5 publications

(2 citation statements)

References 13 publications

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“…[1, 3, 6-8, 20, 32, 35-37, 41], sum of random matrices e.g. [38,46] and also investigated what happens to the distribution of eigenvalues when one would delete columns and rows of the matrix [4,36].…”

Section: Introduction 1state Of the Artmentioning

confidence: 99%

Correlation functions between singular values and eigenvalues

Allard,

Kieburg

2024

Preprint

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Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on j eigenvalues and k singular values that we coin j,k-point correlation measure. We fully derive all j,k-point correlation measures in the simplest cases for matrices of size n = 1 and n = 2 . For n > 2 , we find a general formula for the 1, 1-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a Pólya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble. MSC Classification: 60B20 , 15B52 , 43A90 , 42B10 , 42C05

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Section: Introduction 1state Of the Artmentioning

confidence: 99%

Correlation functions between singular values and eigenvalues

Allard,

Kieburg

2024

Preprint

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“…In [73], the mean, the variance and all higher order cumulants were determined for the GinUE and its extension to rotational invariant potentials W(|z|), the random normal matrix ensembles in the GinUE class. (See also [8] for recent work on the case of the potential W with a hard wall.) In this case, the cumulants are given in terms of poly-logarithms for finite and infinite N. These results were extended to normal symplectic ensembles in the GinSE class for the mean and variance in [1].…”

Section: Introductionmentioning

confidence: 99%

Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble

Akemann,

Byun,

Ebke

et al. 2023

J. Phys. A: Math. Theor.

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In this article, we compute and compare the statistics of the number of eigenvalues in a centred disc of radius R in all three Ginibre ensembles. We determine the mean and variance as functions of R in the vicinity of the origin, where the real and symplectic ensembles exhibit respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit, all three ensembles coincide and display a universal bulk behaviour of O ( R 2 ) for the mean, and O(R) for the variance. We present detailed conjectures for the bulk and edge scaling behaviours of the real Ginibre ensemble, having real and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and conjectures are corroborated by numerical simulations.

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Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

Ameur,

Charlier,

Cronvall

2024

J Stat Phys

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We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant $$\Gamma =2$$ Γ = 2 and that the number n of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order 1/n from the hard edge. At distances much larger than $$1/\sqrt{n}$$ 1 / n , the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the “semi-hard edge”. More precisely, we provide asymptotics for the correlation kernel $$K_{n}(z,w)$$ K n ( z , w ) as $$n\rightarrow \infty $$ n → ∞ in two microscopic regimes (with either $$|z-w| = \mathcal{O}(1/\sqrt{n})$$ | z - w | = O ( 1 / n ) or $$|z-w| = \mathcal{O}(1/n)$$ | z - w | = O ( 1 / n ) ), as well as in three macroscopic regimes (with $$|z-w| \asymp 1$$ | z - w | ≍ 1 ). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.

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Eigenvalues of truncated unitary matrices: disk counting statistics

Abstract: Let T be an $$n\times n$$ n × n truncation of an $$(n+\alpha )\times (n+\alpha )$$ ( n + α ) × ( n + α ) Haar distributed unitary matrix. We consider the disk counti… Show more

Cited by 5 publications

References 13 publications

Correlation functions between singular values and eigenvalues

Correlation functions between singular values and eigenvalues

Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble

Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

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