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

Kuijlaars, Arno B. J.; López-García, Abey

doi:10.1088/0951-7715/28/2/347

Cited by 18 publications

(18 citation statements)

References 38 publications

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“…They then study the asymptotic distribution of the zeros of such polynomials confirming the predictions of [24]. Similar results were obtained for the more general external potential [38] W (λ) = |λ| 2 − Re(t|λ| k ), k ≥ 2 and |t| sufficiently small so that the eigenvalue distribution of the matrix model has an analytic simply connected support. Cases in which the eigenvalue support has singularities were analyzed in [39] and [6].…”

Section: Introductionsupporting

confidence: 70%

“…The identities (1.19) and (1.20) in Lemma 1.6 are expected to hold in general for a large class of normal matrix models. It has been verified for several other potentials (see, for example [2,24,6,11,38]). We also observe that for the orthogonal polynomials appearing in random matrices, in some cases, the asymptotic distribution of the zeros is supported on the so-called mother body or potential theoretic skeleton of the support D of the eigenvalue distribution.…”

Section: Statement Of Resultsmentioning

confidence: 64%

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Orthogonal Polynomials for a Class of Measures with Discrete Rotational Symmetries in the Complex Plane

2016

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We obtain the strong asymptotics of polynomials p n (λ), λ ∈ C, orthogonal with respect to measures in the complex plane of the formwhere s is a positive integer, t is a complex parameter, and dA stands for the area measure in the plane. This problem has its origin in normal matrix models. We study the asymptotic behavior of p n (λ) in the limit n, N → ∞ in such a way that n/N → T constant. Such asymptotic behavior has two distinguished regimes according to the topology of the limiting support of the eigenvalues distribution of the normal matrix model. If 0 < |t| 2 < T /s, the eigenvalue distribution support is a simply connected compact set of the complex plane, while for |t| 2 > T /s, the eigenvalue distribution support consists of s connected components. Correspondingly, the support of the limiting zero distribution of the orthogonal polynomials consists of a closed contour contained in each connected component. Our asymptotic analysis is obtained by reducing the planar orthogonality conditions of the polynomials to equivalent contour integral orthogonality conditions. The strong asymptotics for the orthogonal polynomials is obtained from the corresponding Riemann-Hilbert problem by the Deift-Zhou nonlinear steepest descent method.

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Section: Introductionsupporting

confidence: 70%

Section: Statement Of Resultsmentioning

confidence: 64%

Orthogonal Polynomials for a Class of Measures with Discrete Rotational Symmetries in the Complex Plane

2016

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“…From this connection, one might wonder if the zero distribution of P n would tend to the averaged distribution of the particles. Though this is the case with the orthogonal polynomials on the real line (that corresponds to the particles confined on the line), in the cases of two-dimensional orthogonal polynomials so far studied [3,2,16,14,15,5], the limiting zero distribution is observed to be concentrated on a small subset of the droplet, on some kind of potential-theoretic skeleton of K. 1 A skeleton of K will refer to a subset of (the polynomial hull of) K with zero area, such that there exists a measure that is supported exactly on the skeleton and that generates the same logarithmic potential in the exterior of (the polynomial hull of) K as the Lebesgue measure supported on K. One characteristic of such skeleton is that it can be discontinuous under the continuous variation of the droplet K. A simple example [10] comes from the sequence of polygons converging to a disk. The skeleton of the polygon, which is the set of rays connecting each vertex to the center, does not converge to the skeleton of the disk, the single point at the center.…”

Section: Introductionmentioning

confidence: 99%

Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian Weight

Lee

Yang

2017

Commun. Math. Phys.

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We consider the orthogonal polynomials, {P n (z)} n=0,1,··· , with respect to the measure |z − a| 2c e −N |z| 2 dA(z) supported over the whole complex plane, where a > 0, N > 0 and c > −1. We look at the scaling limit where n and N tend to infinity while keeping their ratio, n/N , fixed. The support of the limiting zero distribution is given in terms of certain "limiting potential-theoretic skeleton" of the unit disk. We show that, as we vary c, both the skeleton and the asymptotic distribution of the zeros behave discontinuously at c = 0. The smooth interpolation of the discontinuity is obtained by the further scaling of c = e −ηN in terms of the parameter η ∈ [0, ∞). IntroductionConsider the ensemble of n point particles, {z j } n j=1 ⊂ C, distributed according to the probability measure given by 1where Z n is the normalization constant, N > 0 is a (large) parameter, Q : C → R ∪ {∞} is called an external potential and dA is the standard Lebesgue measure on the plane. The statistical behavior of the particles has been studied [1] for a large class of potentials in various contexts including random normal matrices and two-dimensional Coulomb gas. For example, in the scaling limit where n and N tend to infinity while n/N is fixed, it is known [12] that the counting measure of the particles converges weakly,where ∆Q = (∂ 2 x + ∂ 2 y )Q, χ K is the indicator function of the compact set K ⊂ C that we will call a droplet following [12], and the expectation value is taken with respect to the measure in (1).A connection to orthogonal polynomials can be provided by Heine's formula. It says that the averaged characteristic polynomial of the n particles is the (monic) orthogonal polynomial of degree n, i.e., P n (z) = P n,N (z) = E n j=1 (z − z j ) satisfies the orthogonality condition, C P n,N (z)P m,N (z)e −N Q(z) dA(z) = h n,N δ nm (n, m = 0, 1, 2, . . .),where h n,N is a (positive) norming constant. From this connection, one might wonder if the zero distribution of P n would tend to the averaged distribution of the particles. Though this is the case with the orthogonal polynomials on the real line (that corresponds to the particles confined on the line), in the cases of two-dimensional orthogonal polynomials so far studied [3,2,16,14,15,5], the limiting zero distribution is observed to be concentrated on a small subset of the droplet, on some kind of potential-theoretic skeleton of K. 1 A skeleton of K will refer to a subset of (the polynomial hull of) K with zero area, such that there exists a measure that is supported exactly on the skeleton and that generates the same logarithmic potential in the exterior of (the polynomial hull of) K as the Lebesgue measure supported on K. One characteristic of such skeleton is that it can be discontinuous under the continuous variation of the droplet K. A simple example [10] comes from the sequence of polygons converging to a disk. The skeleton of the polygon, which is the set of rays connecting each vertex to the center, does not converge to the skeleton of the disk, the...

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“…The strong asymptotics of p c j,N were extensively studied in [12,13,15,34], see also recent works [35,36] on the case with multiple point charges. We also refer the reader to [18,[28][29][30][31]37] for the strong asymptotics of planar orthogonal polynomials associated with some other classes of potentials.…”

Section: Figure 1 An Illustration Of a Lemniscate Ensemblementioning

confidence: 99%

Lemniscate ensembles with spectral singularity

Byun¹,

Lee²,

Yang³

2021

Preprint

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We consider a family of random normal matrix models whose eigenvalues tend to occupy lemniscate type droplets as the size of the matrix increases. Under the insertion of a point charge, we derive the scaling limit at the singular boundary point, which is expressed in terms of the solution to the model Painlevé IV Riemann-Hilbert problem. For this, we apply a version of the Christoffel-Darboux identity and the strong asymptotics of the associated orthogonal polynomials, where the latter was obtained by Bertola, Elias Rebelo, and Grava.

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

Cited by 18 publications

References 38 publications

Orthogonal Polynomials for a Class of Measures with Discrete Rotational Symmetries in the Complex Plane

Orthogonal Polynomials for a Class of Measures with Discrete Rotational Symmetries in the Complex Plane

Discontinuity in the Asymptotic Behavior of Planar Orthogonal Polynomials Under a Perturbation of the Gaussian Weight

Lemniscate ensembles with spectral singularity

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