Density of Eigenvalues of Random Normal Matrices

Elbau, Peter; Felder, Giovanni

doi:10.1007/s00220-005-1372-z

Cited by 55 publications

(99 citation statements)

References 9 publications

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“…A natural approach is to use a cut-off, as proposed by Elbau and Felder [14,15]. This approach consists of restricting the model to those normal matrices with spectrum confined in a fixed two-dimensional bounded domain containing the droplet Ω.…”

Section: Normal Matrix Model and Laplacian Growthmentioning

confidence: 99%

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The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree d + 1. The polynomials that we investigate are multiple orthogonal with respect to a system of d analytic weights defined on a symmetric (d + 1)-star centered at the origin. In the first part we analyze in detail a vector equilibrium problem involving a system of d interacting measures (µ1, . . . , µ d ) supported on star-like sets in the plane. We show that in the subcritical regime, the first component µ * 1 of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of µ * 1 . The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann-Hilbert problem of size (d + 1) × (d + 1). The vector equilibrium problem and the Riemann-Hilbert problem that we investigate are generalizations of those studied recently by Bleher-Kuijlaars in the case d = 2.

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Section: Normal Matrix Model and Laplacian Growthmentioning

confidence: 99%

“…The following relations hold: 15) and, in case d is odd, for 17) and for 19) and in case d is odd and for z ∈ S 21) with the convention g d+1 ≡ 0.…”

Section: The ϕ-Functionsmentioning

confidence: 99%

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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“…These concern relations between the asymptotics of the zeros of orthogonal polynomials and the associated equilibrium measures that have previously been studied by Elbau [5,6] for a certain class measures with bounded support. Their validity for the class of measures considered here is supported by numerical calculations.…”

Section: Zeros Of Orthogonal Polynomials and Quadrature Domainsmentioning

confidence: 99%

“…2) The expected density of eigenvalues (or one-point function) of random Hermitian matrices 6) drawn from the probability density…”

Section: Zeros Of Orthogonal Polynomials and Quadrature Domainsmentioning

confidence: 99%

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

Balogh

Harnad

2008

Complex Anal. Oper. Theory

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This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form w(z) = exp(−|z| 2 + U µ (z)), where U µ (z) is the logarithmic potential of a compactly supported positive measure µ. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials U µ (z). It is also shown that the 2 × 2 matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights w(z) supported on a compact domain.

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“…This means that in the limit N → ∞, we should expect the measures δ z for optimal configurations to converge to the equilibrium measure with potential function W . This indeed turns out to be the case, as shown by the following theorem, proved in [3].…”

Section: Asymptotic Eigenvalue Distribution In the Normal Matrix Modelmentioning

confidence: 62%

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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Density of Eigenvalues of Random Normal Matrices

Cited by 55 publications

References 9 publications

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

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

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