he problem of distributing a large number of points uniformly over the surface of a tsphere has not only inspired mathematical researchers, it has the attracted attention ~f:/t/~176 ::t:u;hifi;l:: ::/i~:l :i~t;~ ogous problem is simply that of uniformly distributing points on the circumference of a disk, and equally spaced points provide an obvious answer. So we are faced with this question: What sets of points on the sphere imitate the role of the roots of unity on the unit circle? One way such points can be generated is via optimization with respect to a suitable criterion such as "generalized energy." Although there is a large and growing literature concerning such optimal spherical configurations of N points when N is "small," here we shall focus on this question from an asymptotic perspective (N--~ ~).The discovery of stable carbon-60 molecules (Kroto, et al., 1985)* with atoms arranged in a spherical (soccer ball) pattern has had a considerable influence on current scientific pursuits. The study of this C60 buckminsterfifllerene also has an elegant mathematical component, revealed by F.R.K. Chung, B. Kostant, and S. Sternberg [5]. Now the search is on for much larger stable carbon molecules! Although such molecules are not expected to have a strictly spherical structure (due to bonding constraints), the construction of large stable configurations of spherical points is of interest here, as an initial step in hypothesizing more complicated molecular net structures.In electrostatics, locating identical point charges on the sphere so that they are in equilibrium with respect to a Coulomb potential law is a challenging problem, sometimes referred to as the dual problem for stable molecules.Certainly, uniformly distributing many points on the sphere has important applications to the field of computation. Indeed, quadrature formulas rely on appropriately chosen sampled data-points in order to approximate area integrals by taking averages in these points. Another example arises in the study of computational complexity, where M. Shub and S. Smale [22] encountered the problem of determining spherical points that maximize the product of their mutual distances.These various points of view clearly lead to different extremal conditions imposed on the distribution of N points. Except for some special values of N (e.g., N = 2, 3, 6, 12, 24) these various conditions yield different optimal configurations. However, and this is the main theme of this article, the general pattern for optimal configurations is the same: points (for N large) appear to arrange themselves according to a hexagonal pattern that is slightly perturbed in order to fit on the sphere.To make this more precise, we introduce some notation. We denote by S 2 the unit sphere in the Euclidean space R3: S 2 = {x E R 3 : Jxt = 1}.Lebesgue (area) measure on S 2 is denoted by ~, so that *

We consider polynomials that are orthogonal on [−1, 1] with respect to a modified Jacobi weight (1 − x) α (1 + x) β h(x), with α, β > −1 and h real analytic and stricly positive on [−1, 1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1, 1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1, 1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szegő function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions.

We continue the study of the Hermitian random matrix ensemble with external source 1where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a = 1, since the eigenvalues of M accumulate on two intervals for a > 1, and on one interval for 0 < a < 1. The case a > 1 was treated in part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0 < a < 1. As in part I we apply the Deift/Zhou steepest descent analysis to a 3 × 3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems.

Abstract. We consider the random matrix ensemble with an external source 1defined on n× n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a > 1, we establish the universal behavior of local eigenvalue correlations in the limit n → ∞, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3 × 3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

We consider the double scaling limit in the random matrix ensemble with an external source 1 Z n e −nTr( 1 2 M 2 −AM) dM defined on n × n Hermitian matrices, where A is a diagonal matrix with two eigenvalues ±a of equal multiplicities. The value a = 1 is critical since the eigenvalues of M accumulate as n → ∞ on two intervals for a > 1 and on one interval for 0 < a < 1. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case a = 1 new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Brézin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a 3 × 3-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.

We study unitary random matrix ensembles of the formwhere α > −1/2 and V is such that the limiting mean eigenvalue density for n, N → ∞ and n/N → 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight |x| 2α e −N V (x) . Here the main focus is on the construction of a local parametrix near the origin with ψ-functions associated with a special solution q α of the Painlevé II equation q = sq + 2q 3 − α. We show that q α has no real poles for α > −1/2, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of q α in the double scaling limit.

We study polynomials that are orthogonal with respect to a varying quartic weight exp(−N (x 2 /2+tx 4 /4)) for t < 0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its, and Kitaev, showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painlevé I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem associated with the polynomials. Moreover, we extend the analysis to non-symmetric combinations of contours. Special features in the steepest descent analysis are a modified equililbrium problem and the use of Ψ-functions for the Painlevé I equation in the construction of the local parametrix.