相关论文: Strong asymptotics for Jacobi polynomials with var…

We investigate the asymptotic behavior of solutions of Hamilton-Jacobi equations with large drift term in an open subset of two-dimensional Euclidean space. When the drift is given by $\varepsilon^{-1} (H_{x_2}, -H_{x_1})$ of a Hamiltonian…

We give the asymptotic distribution of the zeros of Jacobi-Pi\~neiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio…

We address the problem of the weak asymptotic behavior of zeros of families of generalized hypergeometric polynomials as their degree tends to infinity. The main tool is the representation of such polynomials as a finite free convolution of…

For arbitrary $\beta > 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $\beta$ ensembles. We identify the distribution of these…

Exact integral expressions of the skew orthogonal polynomials involved in Orthogonal (beta=1) and Symplectic (beta=4) random matrix ensembles are obtained: the (even rank) skew orthogonal polynomials are average characteristic polynomials…

We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly…

We study right limits of the Bergman Shift matrix. Our results have applications to ratio asymptotics, weak asymptotic measures, relative asymptotics, and zero counting measures of the orthogonal and orthonormal polynomials.

The real and complex zeros of the parabolic cylinder function $U(a,z)$ are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for $a$ positive or negative and large in…

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles of $N\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of…

We study semi-infinite Jacobi matrices $H=H_{0}+V$ corresponding to trace class perturbations $V$ of the "free" discrete Schr\"odinger operator $H_{0}$. Our goal is to construct various spectral quantities of the operator $H$, such as the…

In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlev\'e transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the…

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,…

A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients has the form $t^{-\alpha}a(t)$,~$\alpha>0$, where $a(t)$ is trigonometric polynomial with an arbitrary set of frequencies.…

In this paper, we study the asymptotic behavior of Jacobi biorthogonal polynomials. A Darboux-type formula is established using the method of steepest descent. In the proof, we construct an appropriate contour to apply the Rodrigues…

Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type…

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic…

In this article, we investigate the statistical distribution and asymptotic behavior of the family of monic integer polynomials of degree $n$ having at least one root in a fixed number field $K$. Although the framework of thin sets implies…

We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_\Delta s^kQ_n(s)\rho(s)ds =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ \Delta $ is a Chebotar\"ev (minimal capacity) contour…

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal…

In this work, we study asymptotic zero distribution of random multi-variable polynomials which are random linear combinations $\sum_{j}a_jP_j(z)$ with i.i.d coefficients relative to a basis of orthonormal polynomials $\{P_j\}_j$ induced by…