Universal cubic eigenvalue repulsion for random normal matrices

Oas, Gary

doi:10.1103/physreve.55.205

Cited by 23 publications

(34 citation statements)

References 19 publications

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“…It is well known (see e.g. [9,1]), that in terms of the eigenvalues this measure on N (C) is given by the formula dM =…”

Section: The Normal Matrix Model With An Arbitrary Potential Functionmentioning

confidence: 99%

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Etingof

Ma²

2007

SIGMA

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Abstract. Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele-Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is β|z| 2 , this is done by Wiegmann-Zabrodin and Elbau-Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to * -representations of the deformed preprojective algebra of the affine quiver of typeÂ m−1 . We show that this model is equivalent to the usual normal matrix model in the large N limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model.

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“…It is well known (see e.g. [9,1]), that in terms of the eigenvalues this measure on N (C) is given by the formula dM =…”

Section: The Normal Matrix Model With An Arbitrary Potential Functionmentioning

confidence: 99%

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Etingof

Ma²

2007

SIGMA

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“…The difference is that the eigenvalues are complex numbers. The statistical model of normal random matrices was studied in [15,16].…”

Section: Integration Measuresmentioning

confidence: 99%

Matrix Models and Growth Processes: From Viscous Flows to the Quantum Hall Effect

Zabrodin¹

2006

NATO Science Series II: Mathematics, Physics and Chemistry

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We review the recent developments in the theory of normal, normal self-dual and general complex random matrices. The distribution and correlations of the eigenvalues at large scales are investigated in the large N limit. The 1/N expansion of the free energy is also discussed. Our basic tool is a specific Ward identity for correlation functions (the loop equation), which follows from invariance of the partition function under reparametrizations of the complex eigenvalues plane. The method for handling the loop equation requires the technique of boundary value problems in two dimensions and elements of the potential theory. As far as the physical significance of these models is concerned, we discuss, in some detail, the recently revealed applications to diffusion-controlled growth processes (e.g., to the Saffman-Taylor problem) and to the semiclassical behaviour of electronic blobs in the quantum Hall regime.

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“…This logarithmic growth of the number variance is reminiscent of that typical for real eigenvalues of the Hermitian matrices. Another important spectral characteristics which can be simply expressed in terms of the cluster function is the small-distance behavior of the nearest neighbor distance distribution [44,2,40].…”

Section: Gaussian Almost-hermitian Matrices: From Wigner-dyson Tomentioning

confidence: 99%

Almost-Hermitian Random Matrices: Applications to the Theory of Quantum Chaotic Scattering and Beyond

Fyodorov¹

1999

NATO ASI Series

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It is generally accepted that statistics of energy levels in closed chaotic quantum systems is adequately described by the theory of Random Hermitian Matrices. Much less is known about properties of "resonances" -generic features of open quantum systems pertinent for understanding of scattering experiments.In the framework of the Heidelberg approach to quantum chaotic scattering open systems are characterized by an effective non-Hermitian random matrix HamiltonianĤ ef . Complex eigenvalues ofĤ ef are S-matrix poles (resonances). We show how to find the mean density of these poles (Fyodorov and Sommers) and how to use the effective Hamiltonian to calculate autocorrelations of the photodissociation cross section (Fyodorov and Alhassid).In the second part of the paper we review recent results (Fyodorov, Khoruzhenko and Sommers) on non-Hermitian matrices with independent entries in the regime of weak Non-Hermiticity. This regime describes a crossover from Hermitian matrices characterized by Wigner-Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre.

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Universal cubic eigenvalue repulsion for random normal matrices

Cited by 23 publications

References 19 publications

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Matrix Models and Growth Processes: From Viscous Flows to the Quantum Hall Effect

Almost-Hermitian Random Matrices: Applications to the Theory of Quantum Chaotic Scattering and Beyond

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