Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles

多重正交多项式族的导数类型

Multiple orthogonal polynomial ensembles of derivative type

本文研究了多重正交多项式族的导数类型，证明了随机矩阵乘积和求和的特征值密度属于特定导数类型，并提供了多项式正交性的解决方案。

双正交系综的形变与普适性

Deformations of biorthogonal ensembles and universality

本文研究连续与离散双正交系综的形变在大粒子数极限下的普适性，通过概率生成泛函方法给出温和充分条件，并解释Painlevé型核为正弦与Airy点过程的形变。

Universality in Random Matrix Theory for orthogonal and symplectic ensembles

We give a proof of the Universality Conjecture for orthogonal and symplectic ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. For such weights the associated equilibrium measure is supported on a single interval. The precise statement of our results is given in Theorem 1.1 below. For a proof of the Universality Conjecture for unitary ensembles, for the same class of weights, see [DKMVZ2]. Our starting point is Widom's representation [W] of the orthogonal and symplectic correlation kernels …

Universality for conditional measures of the sine point process

The sine process is a rigid point process on the real line, which means that for almost all configurations $X$, the number of points in an interval $I = [-R,R]$ is determined by the points of $X$ outside of $I$. In addition, the points in $I$ are an orthogonal polynomial ensemble on $I$ with a weight function that is determined by the points in $X \setminus I$. We prove a universality result that in particular implies that the correlation kernel of the orthogonal polynomial ensemble tends to the sine kernel as the length $|I|=2R$ tends to infinity, thereby answering a question posed by A.I. Bu…

The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $θ= \frac{1}{2}$

The Muttalib-Borodin biorthogonal ensemble is a probability density function for $n$ particles on the positive real line that depends on a parameter $θ$ and an external field $V$. For $θ=\frac{1}{2}$ we find the large $n$ behavior of the associated correlation kernel with only few restrictions on $V$. The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a $3\times 3$ Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the orig…

具有实参数 $θ$ 的 Muttalib-Borodin 系综的硬边普适性

Hard edge universality of Muttalib-Borodin ensembles with real parameter $θ$

本文证明了一般势函数下 Muttalib-Borodin 系综硬边极限相关核的普适性，并利用 Deift-Zhou 最速下降法及新参数核构造方法，将结果推广至所有实参数 $θ>0$。

Asymptotics for averages over classical 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 t…

Universality for eigenvalue correlations from the modified Jacobi unitary ensemble

The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the edge in terms of the Bessel kernel. We will prove that this behavior persists for the Modified Jacobi Unitary Ensemble. This generalization of the Jacobi Unitary Ensemble is associated with the modified Jacobi weight w(x)=(1-x)^α(1+x)^βh(x) where the extra factor h is assumed to be real analytic and strictly positive on [-1,1]. We use the connection with the…

The local universality of Muttalib-Borodin ensembles when the parameter $θ$ is the reciprocal of an integer

The Muttalib-Borodin ensemble is a probability density function for $n$ particles on the positive real axis that depends on a parameter $θ$ and a weight $w$. We consider a varying exponential weight that depends on an external field $V$. In a recent article, the large $n$ behavior of the associated correlation kernel at the hard edge was found for $θ=\frac{1}{2}$, where only few restrictions are imposed on $V$. In the current article we generalize the techniques and results of this article to obtain analogous results for $θ=\frac{1}{r}$, where $r$ is a positive integer. The approach is to rela…

Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices

We prove universality at the edge of the spectrum for unitary (beta=2), orthogonal (beta=1) and symplectic (beta=4) ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2, 1.3 below. For a proof of universality in the bulk of the spectrum, for the same class of weights, for unitary ensembles see [DKMVZ2], and for orthogonal and symplectic ensembles see [DG]. Our starting point in the unitary case is [DKMVZ2], and fo…

Local universality in biorthogonal Laguerre ensembles

We consider $n$ particles $0\leq x_1<x_2< \cdots < x_n < +\infty$, distributed according to a probability measure of the form $$ \frac{1}{Z_n}\prod_{1\leq i <j \leq n}(x_j-x_i)\prod_{1\leq i <j \leq n}(x_j^θ-x_i^θ)\prod_{j=1}^nx_j^αe^{-x_j}\ud x_j, ~~ α>-1,~~ θ>0, $$ where $Z_n$ is the normalization constant. This distribution arises in the context of modeling disordered conductors in the metallic regime, and can also be realized as the distribution for squared singular values of certain triangular random matrices. We give a double contour integral formula for the correlation kernel, which all…

Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary

We investigate a two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to β= 2. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a determinantal point process on the complex plane. We investigate the scaling limit, as N \to \infty, of the associated kernel in the neighborhood of a point on the boundary under the assum…

Universality in unitary random matrix ensembles when the soft edge meets the hard edge

Unitary random matrix ensembles Z_{n,N}^{-1} (\det M)^alpha exp(-N Tr V(M)) dM defined on positive definite matrices M, where alpha > -1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when n, N tend to infinity such that n/N - 1 = O(n^{-2/3}). For each value of alpha > -1 we find a one-parameter family of limiting kernels …

Distribution of the first particle in discrete orthogonal polynomial ensembles

We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational. In a number of classical special cases the recurrence procedure is equivalent to the difference and q-Painleve equations of chao-dyn/9507010, [Sakai]. Our approach is based on the formalism of discrete integrable operators and discrete Riemann--Hilbert problems developed in math.CO/9912093, math-ph/0111008.

Average characteristic polynomials for multiple orthogonal polynomial ensembles

Multiple orthogonal polynomials (MOP) are a non-definite version of matrix orthogonal polynomials. They are described by a Riemann-Hilbert matrix Y consisting of four blocks Y_{1,1}, Y_{1,2}, Y_{2,1} and Y_{2,2}. In this paper, we show that det Y_{1,1} (det Y_{2,2}) equals the average characteristic polynomial (average inverse characteristic polynomial, respectively) over the probabilistic ensemble that is associated to the MOP. In this way we generalize classical results for orthogonal polynomials, and also some recent results for MOP of type I and type II. We then extend our results to arbit…

Strong asymptotics for the Pollaczek multiple orthogonal polynomials ensembles

We study the asymptotic properties of a class of multiple orthogonal polynomials with respect to a Nikishin system generated by two measures $(σ_1, σ_2)$ with unbounded supports (${supp}(σ_1) \subset \mathbb{R}_+$, ${supp}(σ_2) \subset \mathbb{R}_-$), and such that the second measure $σ_2$ is discrete. The weak asymptotics for these polynomials was obtained previously by V. Sorokin. We use his result and the Riemann-Hilbert analysis to derive the strong asymptotics of these polynomials and of the reproducing kernel.

An approach to universality using Weyl m-functions

We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel-Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at th…

条件稀释与Laguerre型正交多项式族的乘法统计

Conditional thinning and multiplicative statistics of Laguerre-type orthogonal polynomial ensembles

研究了受测度乘法变形影响的Laguerre型正交多项式族的条件稀释统计特性，推导了其极限核的显式表达式，并揭示了该系统与Bessel点过程相关结构的联系。