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...
We show that the planar orthogonal polynomials with l logarithmic singularities in the potential are the multiple orthogonal polynomials (Hermite-Padé polynomials) of Type II with l measures. We also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment matrix from the original planar measure.
We study the averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar-distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher–Hartwig singularities in cases where some of the singularities merge together and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the circular orthogonal and symplectic ensembles and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.
We study averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher-Hartwig singularities in cases where some of the singularities merge together, and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the Circular Orthogonal and Symplectic Ensembles, and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.
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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