Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = x^α e^-Q(x), where α > -1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach-type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued Riemann--Hilbert problem, due to Fokas, Its, and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.     

First Colonization of a Hard-Edge in Random Matrix Theory

We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model, a phenomenon also known as the “birth of a cut” near a hard-edge. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials.     

Doubly periodic lozenge tilings of a hexagon and matrix valued orthogonal polynomials

We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period 2 in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel is expressed in terms of non‐Hermitian matrix valued orthogonal polynomials (OPs). This model belongs to a class of models for which the existing techniques for studying asymptotics cannot be applied. The novel part of our method consists of establishing a connection between matrix valued and scalar valued OPs. This allows to simplify the double contour formula for the kernel obtained by Duits and Kuijlaars by reducing the size of a Riemann–Hilbert problem. The proof relies on the fact that the matrix valued weight possesses eigenvalues that live on an underlying Riemann surface of genus 0. We consider this connection of independent interest; it is natural to expect that similar ideas can be used for other matrix valued OPs, as long as the corresponding Riemann surface is of genus 0. The rest of the method consists of two parts, and mainly follows the lines of a previous work of Charlier, Duits, Kuijlaars and Lenells. First, we perform a Deift–Zhou steepest descent analysis to obtain asymptotics for the scalar valued OPs. The main difficulty is the study of an equilibrium problem in the complex plane. Second, the asymptotics for the OPs are substituted in the double contour integral and the latter is analyzed using the saddle point method. Our main results are the limiting densities of the lozenges in the disordered flower‐shaped region. However, we stress that the method allows in principle to rigorously compute other meaningful probabilistic quantities in the model.     

随机矩阵函数Kronecker积的谱半径的不等式

证明了随机矩阵函数Kronecker积的谱半径的几个不等式.     

The asymptotics of a Bessel-kernel determinant which arises in Random Matrix Theory

In Random Matrix Theory the local correlations of the Laguerre and Jacobi Unitary Ensemble in the hard edge scaling limit can be described in terms of the Bessel kernel

     

Long-time asymptotics for the Toda lattice in the soliton region

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice for decaying initial data in the soliton region. In addition, we point out how to reduce the problem in the remaining region to the known case without solitons. Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.     

The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u _t −u _txx +2u _x +3uu _x =2u _x u _xx +uu _xxx on the half-line x≥0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann–Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation. Dedicated to Gennadi Henkin in great admiration.     

On the Diaconis-Shahshahani Method in Random Matrix Theory

If Γ is a random variable with values in a compact matrix group K, then the traces Tr(Γ^j) (j ∊ N) are real or complex valued random variables. As a crucial step in their approach to random matrix eigenvalues, Diaconis and Shahshahani computed the joint moments of any fixed number of these traces if Γ is distributed according to Haar measure and if K is one of U_n, O_n or Sp_n, where n is large enough. In the orthogonal and symplectic cases, their proof is based on work of Ram on the characters of Brauer algebras. The present paper contains an alternative proof of these moment formulae. It invokes classical invariant theory (specifically, the tensor forms of the First Fundamental Theorems in the sense of Weyl) to reduce the computation of matrix integrals to a counting problem, which can be solved by elementary means.     

Factorization of a class of symbols with outer functions

In this paper the Wiener–Hopf (or Riemann–Hilbert) factorization of a class of symbols important in applications is studied. The symbols in this class involve outer functions that appear in applications such as diffraction by strip gratings and infinite-dimensional integrable systems. The method proposed is based on the reduction of a vector Riemann–Hilbert to a scalar problem on an appropriate Riemann surface. Two examples are given leading to the Riemann sphere and to an elliptic curve.     

Matrix Valued Orthogonal Polynomials Arising from the Complex Projective Space

The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer parameters to obtain an immediate extension beyond these cases.     

Discrete Entropies of Orthogonal Polynomials

Given a nontrivial Borel measure on ℝ, let p _n be the corresponding orthonormal polynomial of degree n whose zeros are λ _j ⁽ⁿ⁾, j=1,…,n. Then for each j=1,…,n,

Hilbert空间中的一类随机算子方程

本文提出了随机强制算子与随机强单调算子的新概念。研究了在Hilbert空问中一类随机算子方程的随机解。同时推广了著名的Krasnoselkii定理。     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Universality limits in the bulk for varying measures

Universality limits are a central topic in the theory of random matrices. We establish universality limits in the bulk of the spectrum for varying measures, using the theory of entire functions of exponential type. In particular, we consider measures that are of the form in the region where universality is desired. Wn does not need to be analytic, nor possess more than one derivative—and then only in the region where universality is desired. We deduce universality in the bulk for a large class of weights of the form , for example, when W=e−Q where Q is convex and Q^′ satisfies a Lipschitz condition of some positive order. We also deduce universality for a class of fixed exponential weights on a real interval.     

基于随机矩阵理论决定多元GARCH模型最佳维度研究

基于随机矩阵理论(RMT)的降维技术能够通过去除噪声和只保留有用“信息”,而对相关矩阵估计中用来描述相关的主成分或因子的最佳使用数量做出确定.本文认为利用RMT对相关矩阵估计的降维操作来实现RMT对多元GARCH模型的有效降维是可能的.为说明基于RMT的降维技术用于多元GARCH模型的有效性,本文建立了两类将基于RMT的相关矩阵估计和波动率结合在一起的多元GARCH模型:滑动相关多元GARCH模型(SC-GARCH模型)和改进的O-GARCH模型(IO-GARCH模型).理论分析表明,这两类模型具有降维的相关结构,易于估计,并且利用RMT能确定出它们的理论最佳维度.实证研究中,本文建立了上海证券市场100只股票收益率的两类多元GARCH模型,并在马克维茨证券组合理论的框架下,考察了它们的协方差矩阵预测效果.结果表明这两类模型的预测效果很好.通过两类模型各个维度预测效果的比较可以看出.RMT能够为多元GARCH的降维提供有效的依据并且较准确地确定多元GARCH模型的最佳维度.理论和实证分析结果表明,基于RMT的降维技术是解决多元GARCH模型“维数灾祸”问题的有效手段.     

有关随机矩阵领域最新研究动态与进展的综述报告

本文综述了随机矩阵领域的某些国内外最新前沿课题与进展,以及它们对应的主要研究方法和手段,作者还列出了此领域某些有待解决的问题。