Asymptotics Of Orthogonal Polynomials Research Articles

Relating random matrix map enumeration to a universal symbol calculus for recurrence operators in terms of Bessel–Appell polynomials

Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random matrices and asymptotics of orthogonal polynomials with exponential weights. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the cellular networks within a broad class. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable unwinding identity in terms of Appell polynomials generated by Bessel functions. Our treatment reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis, Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems, Commun. Pure Appl. Math. 46 (1993) 453–499, as well as directly related to universality results for random matrix spectra.

On zeros, bounds, and asymptotics for orthogonal polynomials on the unit circle

Let $\mu$ be a measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\{\varphi_{n}\}$ be the orthonormal polynomials for $\mu$ and $z_{jn}\}$ their zeros. Let $\mu$ be absolutely continuous in an arc $\Delta$ of the unit circle, with $\mu'$ positive and continuous there. We show that uniform boundedness of the orthonormal polynomials in subarcs $\Gamma$ of $\Delta$ is equivalent to certain asymptotic behaviour of their zeros inside sectors that rest on $\Gamma$. Similarly the uniform limit $\lim_{n\to \infty}|\varphi_{n}(z)|^{2}\mu'(z)=1$ is equivalent to related asymptotics for the zeros in such sectors. Bibliography: 27 titles.

Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

AbstractWe derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.

Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients

We study solutions of three-term recurrence relations whose N-step transfer matrices belong to the uniform Stolz class. In particular, we derive the first order of their uniform asymptotics. For orthonormal polynomials we show more. Namely, we find the constructive formula for the density of their orthogonality measure in terms of Turán determinants and we determine their exact asymptotic behavior. We treat both bounded and unbounded cases in a unified way.

Harmonic Analysis in Phase Space and Finite Weyl\u2013Heisenberg Ensembles

Weyl–Heisenberg ensembles are translation-invariant determinantal point processes on mathbb {R}^{2d} associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl–Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with N points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area N, we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain Omega . We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of Omega , as Omega is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.

On the moments of the characteristic polynomial of a Ginibre random matrix

In this article we study the large $N$ asymptotics of complex moments of the absolute value of the characteristic polynomial of a $N\times N$ complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large $N$ asymptotics of $\mathbb{E}|\det(G_N-z)|^{\gamma}$, where $G_N$ is a $N\times N$ matrix whose entries are i.i.d and distributed as $N^{-1/2}Z$, $Z$ being a standard complex Gaussian, $\mathrm{Re}(\gamma)>-2$, and $|z|<1$. This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher-Hartwig type of singularity: $\det(\int_\mathbb{C} w^{i}\overline{w}^j |w-z|^\gamma e^{-N|w|^{2}}d^2 w)_{i,j=0}^{N-1}$. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight $|w-z|^\gamma e^{-N|w|^2}d^2 w$ along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher-Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two-dimensional set and a Fisher-Hartwig singularity, have been previously unknown.

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper, the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|2αe−(x4+tx2), x ∈ R, where α is a constant larger than −1/2 and t is any real number. They consider this problem in three separate cases: (i) c > −2, (ii) c = −2, and (iii) c < −2, where c:= tN−1/2 is a constant, N = n + α and n is the degree of the polynomial. In the first two cases, the support of the associated equilibrium measure μt is a single interval, whereas in the third case the support of μt consists of two intervals. In each case, globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993).

Stability of ratio asymptotics of orthogonal polynomials under some class of measure perturbations

Stability of ratio asymptotics of orthogonal polynomials under some class of measure perturbations

Asymptotics of Orthogonal Polynomials with Complex Varying Quartic Weight: Global Structure, Critical Point Behavior and the First Painlevé Equation

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials \(\pi _n(z)\) with the quartic exponential weight \(\exp \left[-N\left(\frac{1}{2}z^2 + \frac{1}{4}t z^4\right)\right]\), where \(t\in {\mathbb C}\) and \(N\rightarrow \infty \). We consider in detail the points \(t = -\frac{1}{12}\) and \(t = \frac{1}{15}\), where the recurrence coefficients of the orthogonal polynomial exhibit a behavior that involves special solutions of the first Painleve Riemann–Hilbert problem (RHP). Our principal concern is the description of their behavior in a neighborhood of special points in the \(t\)-plane (accumulating at the indicated values) where the corresponding Painleve function has poles. The nonlinear steepest descent method for the RHP is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing nonlinear Schrodinger equation near the point of gradient catastrophe that the present authors solved in 2013. Our approach is based on the technique developed in that earlier work. We also provide a numerical investigation of the “phase diagrams” in the \(t\)-plane where the recurrence coefficients exhibit different asymptotic behaviors (nonlinear Stokes’ phenomenon).

Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis

We consider the uniform asymptotics of polynomials orthogonal on (0, ∞) with respect to the exponential weight w(x) = x α e −Q(x) ,w here α> − 1a ndQ(x) is a polynomial with positive leading coefficient. In this paper, we have obtained two types of asymptotic expansions in terms of Laguerre polynomials and elementary functions for z in different overlapping regions, respectively. These two regions together cover the whole complex plane. Our approach is based on the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295-368).

Asymptotics for orthogonal polynomials with respect to the Laguerre measure modified by a rational factor

Asymptotics for orthogonal polynomials with respect to the Laguerre measure modified by a rational factor

Asymptotics of orthogonal polynomials and point perturbation on the unit circle

In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p -type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we add a pure point to a gap of the essential spectrum. For the asymptotically constant case, we give an asymptotic formula for the orthonormal polynomials in the gap, prove that the perturbation term converges and show the limit explicitly. Furthermore, we prove that the perturbation is of bounded variation. Then we generalize the method to the asymptotically periodic case and prove similar results. In the last two sections, we show that the bounded variation condition can be removed if a certain symmetry condition is satisfied. Finally, we consider the special case when the Verblunsky coefficients are real with the rate of convergence being c n . We prove that the rate of convergence of the perturbation is in fact O ( c n ) . In particular, the special case c n = 1 / n will serve as a counterexample to the possibility that the convergence of the perturbed Verblunsky coefficients should be exponentially fast when a point is added to a gap.

Asymptotics of orthogonal polynomials via the Riemann-Hilbert approach

In this survey we give a brief introduction to orthogonal polynomials, including a short review of classical asymptotic methods. Then we turn to a discussion of the Riemann-Hilbert formulation of orthogonal polynomials, and the Deift & Zhou method of steepest descent. We illustrate this new approach, and a modified version, with the Hermite polynomials. Other recent progress of this method is also mentioned, including applications to discrete orthogonal polynomials, orthogonal polynomials on curves, multiple orthogonal polynomials, and certain orthogonal polynomials with singular behavior.

Double Scaling Limit of Semiclassical Orthogonal Polynomials in the Center of a Tree and the Painleve 1 Equation

The asymptotics of orthogonal polynomials with complex weight on complex contours was studied in [1] and [5]. It was shown that in the asymptotic limit, the zeros of these polynomials accumulate on tree-like graphs in and generically, the density of zeros vanishes like a square-root at each vertex of these tree graphs. In this article, we study the double scaling limit in which the density vanishes to an order higher than in an interior vertex. This is a new critical phenomenon that has no parallel in the case of orthogonal polynomials on the real line. Yet, surprisingly, the behavior of the orthogonal polynomial in the double scaling limit can be modeled by a solution of the Painleve 1 equation that was used previously [15] in the study of a different double scale limit.

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject. In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann–Hilbert analysis. In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.

Bulk Universality and Related Properties of Hermitian Matrix Models

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.

On the asymptotics of orthogonal polynomials on the curve with a denumerable mass points

We investigate the asymptotic behavior of orthogonal polynomials with respect to a measure of the type \(\sigma =\alpha +\gamma \), where \(\alpha \) is a measure concentrated on a rectifiable Jordan curve and \(\gamma \) is an infinite discrete measure.

Asymptotics of derivatives of orthogonal polynomials on the unit circle

We show that ratio asymptotics of orthogonal polynomials on the circle imply ratio asymptotics for all their derivatives. Moreover, by reworking ideas of Nevai, we show that uniform asymptotics for orthogonal polynomials on an arc of the unit circle imply asymptotics for all their derivatives.

Uniform asymptotics for orthogonal polynomials via the Riemann--Hilbert approach†

In this survey, a brief description is given to the Riemann--Hilbert approach for uniform asymptotics of orthogonal polynomials. The Jacobi polynomials are used to demonstrate the steps. Some essentially new progress of the approach is reported. A short discussion on prospective development is also included. †Dedicated to Professor Wei Lin on the occasion of his 70th birthday.

Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight

We study polynomials that are orthogonal with respect to a varying quartic weight exp(− N(x2/2 + tx4/4)) 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.