A remark on Jacobi ensemble

Operator level limit of the circular Jacobi $β$-ensemble

We prove an operator level limit for the circular Jacobi $β$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal $β$-ensemble.

Rectangular Young tableaux and the Jacobi ensemble

It has been shown by Pittel and Romik that the random surface associated with a large rectangular Young tableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle. We show that in the corner, these fluctuations are gaussian wheras, away from the corner and when the rectangle is a square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with the Jacobi ensemble.

Approximation of Beta-Jacobi ensembles by Beta-Laguerre ensembles

Let $λ$ and $μ$ be beta-Jacobi and beta-Laguerre ensembles with joint density function $f_{β, m, a_1, a_2}$ and $f_{β, m, a_1}$, respectively. Here $β>0$ and $a_1, a_2$ and $m$ satisfying . $a_1, a_2>\fracβ{2}(m-1).$ In this paper, we consider the distance between $2(a_1+a_2)λ$ and $μ$ in terms of total variation distance and Kullback-Leibler distance. Following the idea in \cite{JM2017}, we are able to prove that both the two distances go to zero once $a_1m=o(a_2)$ and not so if $\lim_{a_2\to\infty}a_1m/a_2=σ>0.$

具有不同缩放的 beta-Jacobi 系综的非标准极限定理与大偏差

Nonstandard limit theorems and large deviation for beta -Jacobi ensembles with a different scaling

本文研究了参数为 $p_1, p_2\geq n$ 的 $β$-Jacobi 系综，通过谱测度方法证明了重缩放经验测度弱收敛到修正的 Watcher 律，并给出了相应重缩放谱测度的中心极限定理与大偏差。

$β$-Jacobi系综极值特征值的重要性采样

Importance Sampling for the Extremal Eigenvalue of $β$-Jacobi ensemble

本文针对超高维$β$-Jacobi系综极值特征值的尾部概率稀有事件，提出了精确近似与高效估计量，并通过数值模拟验证了该方法的高效性与准确性。

Some asymptotic properties of the spectrum of the Jacobi ensemble

For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | λ_i - λ_j |^β\prod^n_{i=1} (2 - λ_i)^a (2 + λ_i)^b I_{(-2,2)} (λ_i) $$ $(a, b > -1, β> 0) $ a strong uniform approximation by the roots of the Jacobi polynomials is derived if the parameters $a, b,$ $β$ depend on $n$ and $n \to \infty$. Roughly speaking, the eigenvalues can be uniformly approximated by roots of Jacobi polynomials with parameters $((2a+2)/β-1, (2b+2)/β-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$. These results are used to investigate the asymptotic properties of…

Global Fluctuations for Linear Statistics of β-Jacobi Ensembles

We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(λ_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $λ_1, ..., λ_n$ being the eigenvalues of a (general) $β$-Jacobi ensemble, for which tridiagonal models were given by Killip and Nenciu as well as Edelman and Sutton. The fluctuation from the mean ($\sum_{i=1}^n f(λ_i) - \Exp \sum_{i=1}^n f(λ_i)$) is given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation …

The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles

An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix may be regarded as the limit of Gaussian beta ensemble (G$β$E) matrices as the matrix size $N$ tends to infinity with the constraint that $N β$ is a constant.

Jacobi 系综随机矩阵的极端特征值

Extreme eigenvalues of random matrices from Jacobi ensembles

本文研究 Jacobi β-Ensembles 大矩阵的极端特征值分布，在 β 任意、α₁ 为整数时给出最小特征值的两项渐近公式，并推广至最大特征值及特殊情形。

Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles

We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If $n$ is the size of the sample, $r\leq n$ the number of variates and $X_{n,r}$ such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of $\det X_{n,r}$ into a product of $r$ independent gamma or beta random variables. For $n$ fixed, we study the evolution as $r$ grows, and then take the limit of large $r$ and $n$ with $…

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulae for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

Limit Theorems for Beta-Jacobi Ensembles

For a beta-Jacobi ensemble determined by parameters a_1, a_2 and n, under the restriction that the three parameters go to infinity with n and a_1 being of small orders of a_2, we obtain both the bulk and the edge scaling limits. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the Central Limit Theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues.

$β$-Jacobi 系综极值的尾部概率的渐近有效估计量

Asymptotically efficient estimators for tail probabilities of extremals of $β$-Jacobi ensembles

本文研究 $β$-Jacobi 系综极值的尾部概率，通过对系综应用拉伸与平移变换确保极值满足大偏差，构造出依赖于大偏差速率函数的估计量，刻画了不同假设下极值的大偏差与中偏差行为。

随机雅可比矩阵的谱测度与β集合在高温下的狄利克雷过程关系

The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes

本文研究了高温下β集合随机雅可比矩阵的谱测度，证明其收敛于狄利克雷过程并提供显式谱测度的例子。

Laguerre和Jacobi系综的改进集中性

Improved concentration of Laguerre and Jacobi ensembles

本文研究了Laguerre和Jacobi系综在参数发散渐近极限下的集中性，通过改进误差分析获得了更优的集中界，并进一步优化了Jacobi系综第一和第二矩的集中界。

Product of random projections, Jacobi ensembles and universality problems arising from free probability

We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues.

关于Jacobi正交 Ensemble 特征值刚性的研究

On The Eigenvalue Rigidity of the Jacobi Unitary Ensemble

本文研究了Jacobi正交 Ensemble 特征值的刚性性质，通过构造随机测度并证明其收敛到高斯乘法混沌测度，得出最优全局刚性估计。

On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles

Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.

Remarks on a constrained optimization problem for the Ginibre ensemble

We study the limiting distribution of the eigenvalues of the Ginibre ensemble conditioned on the event that a certain proportion lie in a given region of the complex plane. Using an equivalent formulation as an obstacle problem, we describe the optimal distribution and some of its monotonicity properties.

正交与辛随机矩阵的特征多项式、Jacobi 系综与 L-函数

Characteristic Polynomials of Orthogonal and Symplectic Random Matrices, Jacobi Ensembles & L-functions

本文研究辛随机矩阵特征多项式及其二阶导数的联合矩，得到其渐近及 Painlevé 方程表示的首阶系数，并给出 Dirichlet L-函数族的猜想性渐近，同时计算了加性 Jacobi 统计量的渐近。最后处理酉群平均的类比联合矩，获得显式结果。

关于零温度下Jacobi族极值本征值的极限

On the extremal eigenvalues of Jacobi ensembles at zero temperature

本文研究了零温度下Jacobi族极值本征值的极限，通过Bessel函数推导了冻结情况下的协方差极限，对应于硬边分析和有限β的结果。

Fast sampling from $β$-ensembles

We study sampling algorithms for $β$-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler fo…

Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence

Let $A$ and $B$ be independent, central Wishart matrices in $p$ variables with common covariance and having $m$ and $n$ degrees of freedom, respectively. The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that $m$ and $n$ grow in proportion to $p$. We show that after centering and scaling, the distribution is approximated to second-order, $O(p^{-2/3})$, by the Tracy--Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory …

Limit theorems for Jacobi ensembles with large parameters

Consider Jacobi random matrix ensembles with the distributions $$c_{k_1,k_2,k_3}\prod_{1\leq i< j \leq N}\left(x_j-x_i\right)^{k_3}\prod_{i=1}^N \left(1-x_i\right)^{\frac{k_1+k_2}{2}-\frac{1}{2}}\left(1+x_i\right)^{\frac{k_2}{2}-\frac{1}{2}} dx$$ of the eigenvalues on the alcoves $$A:=\{x\in\mathbb R^N| \> -1\leq x_1\le ...\le x_N\leq 1\}.$$ For $(k_1,k_2,k_3)=κ\cdot (a,b,1)$ with $a,b>0$ fixed, we derive a central limit theorem for the distributions above for $κ\to\infty$. The drift and the inverse of the limit covariance matrix are expressed in terms of the zeros of classical Jacobi polynomi…

Local Eigenvalue Density for General MANOVA Matrices

We consider random n\times n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to \log n factors). This result is the analogue of the local …

Convergence and an explicit formula for the joint moments of the Circular Jacobi $β$-Ensemble characteristic polynomial

The problem of convergence of the joint moments, which depend on two parameters $s$ and $h$, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes. Recently, Forrester considered the analogous problem for the Circular $β$-Ensemble (C$β$E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer $s$ and complex $h$. In this paper we consider this problem for a generalisation of the C$β$…

随机杨图与雅可比酉系综

Random Young diagrams and Jacobi Unitary Ensemble

本文研究由外幂表示分解诱导的随机杨图，证明其在特定极限下收敛到雅可比酉系综特征值的大N极限律，并计算相关特征标、讨论曲面计数联系，提出若干关联子猜想。

Mod-Gaussian convergence for random determinants and random characteristic polynomials

The aim of this paper is to give a precise asymptotic description of some eigenvalue statistics stemming from random matrix theory. More precisely, we consider random determinants of the GUE, Laguerre, Uniform Gram and Jacobi beta ensembles and random characteristic polynomials evaluated at 1 for matrices in the Circular and Circular Jacobi beta ensembles. We use the framework of mod-Gaussian convergence to provide quantitative estimates of their logarithmic behavior, as the size of the ensemble grows to infinity. We establish central limit theorems, Berry-Esseen bounds, moderate deviations an…

General beta Jacobi corners process and the Gaussian Free Field

We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the beta goes to infinity limit.

两类 $\beta$-Jacobi 集成的最小特征值分布

Smallest eigenvalue distributions for two classes of $β$-Jacobi ensembles

计算了两类 $\beta$-Jacobi 集成的最小特征值精确分布和极限分布，这些分布由多元超几何 ${}_2F_{1}^{2/\beta}$ 函数给出，并分析了 $\beta\in 2\mathbb{N}_{+}$ 和 $\beta=1$ 时的渐近行为。

Universality of mesoscopic fluctuations for orthogonal polynomial ensembles

We prove that the fluctuations of mesocopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a Central Limit Theorem for…

Circular Jacobi Ensembles and deformed Verblunsky coefficients

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{δ,β}^{(n)} \prod_{1\leq k<l\leq n}| e^{\iiθ_k}-e^{\iiθ_l}|^β\prod_{j=1}^{n}(1-e^{-\iiθ_j})^δ (1-e^{\iiθ_j})^{\overlineδ} $$ with $\Re δ> -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(α_0, ..., α_{n-1})$. We introduce here a deformation $(γ_0, >..., γ_{n-1})$ of these coeff…

Moderate deviations for spectral measures of random matrix ensembles

In this paper we consider the (weighted) spectral measure $μ_n$ of a $n\times n$ random matrix, distributed according to a classical Gaussian, Laguerre or Jacobi ensemble, and show a moderate deviation principle for the standardised signed measure $\sqrt{n/a_n}(μ_n -σ)$. The centering measure $σ$ is the weak limit of the empirical eigenvalue distribution and the rate function is given in terms of the $L^2$-norm of the density with respect to $σ$. The proof involves the tridiagonal representations of the ensembles which provide us with a sequence of independent random variables and a link to or…

正交多项式系综的介观边缘普适性

Mesoscopic Edge Universality of Orthogonal Polynomial Ensembles

本文研究正交多项式系综边缘的介观涨落，证明在递推系数缓慢变化时线性统计量满足普适的中心极限定理，并给出截断雅可比矩阵预解式的估计，适用于雅可比、拉盖尔、高斯酉及随机平铺离散系综。

关于与β系综相关的某些矩阵的平均态密度及在Toda晶格中的应用

On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice

本文研究高温区域β系综相关三对角随机矩阵，证明经验谱分布收敛并显式计算平均态密度，应用于Toda晶格Lax矩阵。

Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in multivariate statistics) and smallest (e.g.~condition numbers of linear systems) eigenvalues. We identify three ranges of parameter values for which the gap probability determining these distributions is a finite sum with respect to particular bases, and moreover make use of a certain differential-difference system fundamental in the theory of the Selberg int…

Laguerre 与 Jacobi β 系综中边缘特征值分布的计算

Computation of marginal eigenvalue distributions in the Laguerre and Jacobi $β$ ensembles

本文针对 Laguerre 和 Jacobi β 系综，提出递归程序精确计算边缘特征值分布，并揭示其在量子输运中 Landauer 电导分布等问题的应用价值。

Jacobi 过程的 Wigner 型与 Marchenko-Pastur 型极限

Wigner- and Marchenko-Pastur-type limits for Jacobi processes

本文研究 BC 型根系驱动的 Jacobi 过程，利用随机微分方程推导出当粒子数趋于无穷时经验分布的 Wigner 半圆律和 Marchenko-Pastur 律的几乎必然极限，并推广了 Jacobi 系综及多项式零点的相关平稳极限结果。

关于Hele-Shaw β-系综单点密度的注记

Remarks on the one-point density of Hele-Shaw $β$-ensembles

本文证明了受Hele-Shaw型外势约束的二维库仑气体在β≥1/2时单点密度族的等度连续性，从而在微观尺度上得到Lipschitz连续的子序列极限密度，并比较了单点密度与热平衡密度。

固定$q$的$q$-Krawtchouk系综的边缘极限

The Fixed-$q$ Edge Limit of the $q$-Krawtchouk Ensemble

本文研究$q$-Krawtchouk系综在固定$q$下的右边缘极限，通过有限多项式对偶性证明相关核收敛，并揭示极限算子的非唯一性及谱原子选择机制。

Gaussian fluctuations for βEnsembles

We study the Circular and Jacobi $β$-Ensembles and prove Gaussian fluctuations for the number of points in one or more intervals in the macroscopic scaling limit.

Ablowitz-Ladik 格点与 Schur 流的大偏差

Large Deviations for Ablowitz-Ladik lattice, and the Schur flow

本文研究 Ablowitz-Ladik 格点与 Schur 流的广义吉布斯系综，推导经验测度的大偏差原理，并证明其几乎必然收敛，极限由圆与 Jacobi beta 系综的平衡测度刻画。