Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane

Balogh, F; Грава, Тамара; Merzi, Dario

doi:10.48550/arxiv.1509.05331

Cited by 2 publications

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Section: Strong Asymptotics Of P N and The Location Of Zerosmentioning

confidence: 59%

“…We remark that the case −1 < c < 0 is essentially treated in [2]. We note that the limiting locus of zeros remains the same for both the positive and negative c (which seems unexpected according to Remark 1.2 in [2]). It turns out that, as the value of c gets bigger, we need higher order corrections in the Riemann-Hilbert analysis.…”

Section: Strong Asymptotics Of P N and The Location Of Zerosmentioning

confidence: 59%

“…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%

“…By a simple rotation of the plane, the above Q covers the case with any nonzero a ∈ C. We are interested in the scaling limit where N and n go to infinity while the ratio, n/N , is fixed to a positive number. Below we will set N = n without losing generality since the orthogonality (2) gives the relation…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Strong Asymptotics Of P N and The Location Of Zerosmentioning

confidence: 59%

Section: Strong Asymptotics Of P N and The Location Of Zerosmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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“…When the (boundary of the) domain G displays some topological transition, it is natural to expect that its mother body measure displays some phase transition as well. In fact, this has already been described in several previous works in the context of random normal matrices [6,7,15,36], although in some cases without explicit mention. However, in our situation it is very interesting that the transition for µ * occurs before any transition for Ω.…”

Section: 2mentioning

confidence: 57%

The mother body phase transition in the normal matrix model

Bleher¹,

Silva²

2016

Preprint

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The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth.In this present paper we consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain Ω that we determine explicitly by finding the rational parametrization of its boundary.We also study in details the mother body problem associated to Ω. It turns out that the mother body measure µ * displays a novel phase transition that we call the mother body phase transition: although ∂Ω evolves analytically, the mother body measure undergoes a "one-cut to three-cut" phase transition.To construct the mother body measure, we define a quadratic differential on the associated spectral curve, and embed µ * into its critical graph. Using deformation techniques for quadratic differentials, we are able to get precise information on µ * . In particular, this allows us to determine the phase diagram for the mother body phase transition explicitly.Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated g-functions significantly more involved, and the critical graph of becomes the key technical tool in this analysis as well.

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Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane

Cited by 2 publications

References 0 publications

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

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

The mother body phase transition in the normal matrix model

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