Correlation Functions for β=1 Ensembles of Matrices of Odd Size
Christopher D Sinclair
本文用Tracy和Widom的方法重新导出了N × N矩阵的β=1 Hermitian和实非对称系综的关联函数.
Using the method of Tracy and Widom we rederive the correlation functions for β=1 Hermitian and real asymmetric ensembles of N x N matrices with N odd.
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A method to calculate correlation functions for $β=1$ random matrices of odd size
可以使用Pfweans来执行的用于$β=1$随机矩阵集合的关联函数的计算具有需要根据矩阵大小N的奇偶性进行单独计算的特殊特征。在计算真实高斯矩阵的Ginibre正交包络的相关性时也存在同样的复杂性。事实上，用于计算$β=1$，N奇数，相关性的方法在N奇数的情况下打破了真正的吉尼布尔矩阵，需要一个新的方法来解决这两个问题。在这项工作中采取的新方法是推导出的$β=1$，N奇数相关的限制情况下，他们的N偶对应物，当一个粒子被移向无穷大。这种方法被证明是N个奇数实高斯矩阵的相关性。
1/N phenomenon for some symmetry classes of the odd alternating sign matrices
我们考虑具有某种中心对称的奇数阶交替符号矩阵。也就是说，我们处理的矩阵不变下的半圈，四分之一圈和翻转的两个对角线。在所有这些情况下，在矩阵的中心有两个自然结构。例如，对于在半圈下不变的矩阵，中心元素等于$\pm 1$。最近发现，$A^+_{HT}(2m+1)/A^-_{HT}(2m+1)$ =（m+1）/m。我们推测，类似的非常简单的关系是有效的，在其余两种情况下。
Enumeration of quarter-turn symmetric alternating-sign matrices of odd order
Kuperberg证明了与偶数阶四分之一圈对称交替符号矩阵有关的方冰模型的配分函数是两个相似因子的乘积。本文提出了一个状态与奇阶四分之一圈对称交替符号矩阵成双射的方冰模型，并证明了该模型的配分函数也可以用类似的方法表示.这允许证明，特别是，由罗宾斯有关枚举的四分之一圈对称交替符号矩阵的命题。
Averages over Ginibre's Ensemble of Random Real Matrices
给出了实非对称矩阵高斯系综上乘性类函数系综平均的一种计算方法。这些平均值表示在类函数的反对称矩阵的类反对称内积形成相对于Pfweian。
The Ginibre ensemble of real random matrices and its scaling limits
本文给出了一类非对称实矩阵系综的关联函数的一个封闭形式，它是由与系综相关联的$2 \times 2$矩阵核所构成的反对称矩阵的Pfuman表示的.我们将这一结果应用到真正的Ginibre系综和计算的体积和边缘缩放限制的关联函数的矩阵的大小变得很大。
Edge Universality for Orthogonal Ensembles of Random Matrices
证明了具有实解析势和一个区间极限谱的正交不变矩阵模型局部特征值统计量的边普适性。我们的出发点是\cite{S:08}关于正交系综的再生矩阵核用相应酉系综的标量再生核表示的结果。
Induced Ginibre ensemble of random matrices and quantum operations
介绍了非厄米特随机方阵的Ginibre系综的一个推广。相应的概率测度是由矩形高斯矩阵的集合通过二次化过程引起的。我们推导出这种诱导Ginibre系综的本征值的联合概率密度，并研究了复矩阵和实矩阵的各种谱关联函数，分析了大维数极限下的普遍行为。在此极限下，诱导Ginibre系综的本征值均匀地覆盖复平面上的一个环。真实的诱导Ginibre系综被证明是有用的描述与随机量子操作，输入状态和输出状态的尺寸不同的演化算子的统计特性。
Gaussian diagrammatics from Circular Ensembles of random matrices
该研究发现随机矩阵三个圆系综内的隐藏高斯系综，给出矩计算的新图解规则，明确不同β下矩阵类型与维数设定，并应用于子矩阵迹的矩计算。
Spectral fluctuation properties of constrained unitary ensembles of Gaussian-distributed random matrices
本文研究了随机矩阵约束系综在大矩阵维数限制下的谱涨落性质（定义为矩阵元素个数N（Q）为零的条件，该条件以酉不变形式施加）。我们表明，只要N（Q）是小于一个临界值（在该临界值的随机矩阵的高斯酉系综的二次水平排斥可能会被破坏）的所有光谱波动措施具有相同的形式为高斯酉系综。
We consider non-gaussian ensembles of random normal matrices with the constraint that the ensembles are invariant under unitary transformations. We show that the level density of eigenvalues exhibits disk to ring transition in the complex plane. We also show that the n-eigenvalue correlation and the spacing distribution are universal and identical to that of complex (Gaussian) Ginibre ensemble. Our results are confirmed by Monte Carlo calculations. We verify the universality for dissipative quantum kicked rotor system.
We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kern…
On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
给定任意固定的$N \times N$半正定对角矩阵$G\ge 0$，我们导出了形式为$A=U\sqrt{G}$ }的随机矩阵$A$的复特征值密度的显式公式，其中随机酉矩阵$U$按Haar测度分布在群$\mathrm{U(N)}$上.
One-dimensional stochastic growth and Gaussian ensembles of random matrices
在这篇综述论文中，我们考虑了多核生长（PNG）模型在一个空间维度和它的关系，随机矩阵合奏。对于弯曲和平坦的增长，表面波动的标度函数与来自随机矩阵的某些高斯系综的极限分布函数相一致。这种连接可以通过与PNG模型和随机矩阵集合相关联的点过程来解释，分别通过对多层PNG和多矩阵模型的扩展。我们还解释了其他模型，相当于PNG模型：定向聚合物，最长的增加子序列的问题，杨tableaux，定向渗流模型，扭结-反扭结气体，和Hammersley过程。
Enumerations of half-turn symmetric alternating-sign matrices of odd order
Kuperberg证明了与偶数阶半圈对称交替符号矩阵有关的方冰模型的配分函数是两个相似因子的乘积。本文提出了一个状态为奇阶半圈对称交替符号矩阵双射的方冰模型。模型的配分函数由上述因素表示。找出了以1或-1为中心元素的交替符号矩阵对应的状态对配分函数的贡献，并得到了相应的枚举。
We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using the supersymmetry approach we calculate analytically the moments and the distribution function of the eigenvectors components for a generic matrix $W$. We show that specific choices of $W$ can modify significantly the nature of the eigenvectors changing them from extended to critical to localized. Our analytical results are supported by numerical simulations…
The theory of Pólya ensembles of positive definite random matrices provides structural formulas for the corresponding biorthogonal pair, and correlation kernel, which are well suited to computing the hard edge large $N$ asymptotics. Such an analysis is carried out for products of Laguerre ensembles, the Laguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles and their inverses. The latter includes as a special case the Jacobi unitary ensemble. In each case the hard edge scaled kernel permits an expansion in powers of $1/N$, with the leading term given in a structured form involv…
Correlation Functions of Asymmetric Real Matrices
本文给出了非对称实矩阵系综的关联函数的一个封闭形式，它是由与系综相关联的$2 \times 2$矩阵核所构成的反对称矩阵的Pfuman表示的.我们还推导出封闭形式的矩阵核和关联函数的Ginibre的真正的合奏。
A geometrical triumvirate of real random matrices
本文基于（斜）正交多项式方法，提出了一种计算实随机矩阵系综特征值关联函数的五步法。该方案系统化了现有的方法，并涉及一些新的技术。考虑的合奏是：高斯正交系综（GOE）、实Ginibre系综、部分对称实高斯矩阵的系综、（矩阵$A^{-1}B$的）实球面系综和实反球面系综（由截断的实正交矩阵组成）。特别强调的是支付给处理奇数大小的矩阵所需的方法的变化。各种普遍性的结果进行了讨论，并提出了一个“反球法”的猜想。
On Universality for Orthogonal Ensembles of Random Matrices
本文证明了具有实解析势的正交不变矩阵模型的局部特征值统计量在谱中的普遍性。我们的出发点是矩阵再生核的Tracy-Widom公式。证明的关键思想是将写在正交多项式基上的微分算子矩阵表示为正Toeplitz矩阵和两对角斜对称Toeplitz矩阵的乘积。
We investigate the microcanonical version of the complex induced Ginibre ensemble, by introducing a fixed trace constraint for its second moment. Like for the canonical Ginibre ensemble, its complex eigenvalues can be interpreted as a two-dimensional Coulomb gas, which are now subject to a constraint and a modified, collective confining potential. Despite the lack of determinantal structure in this fixed trace ensemble, we compute all its density correlation functions at finite matrix size and compare to a fixed trace ensemble of normal matrices, representing a different Coulomb gas. Our main …
We study a general ansatz for an odd supersymmetric version of the Kronecker elliptic function, which satisfies the genus one Fay identity. The obtained result is used for construction of the odd supersymmetric analogue for the classical and quantum elliptic $R$-matrices. They are shown to satisfy the classical Yang-Baxter equation and the associative Yang-Baxter equation. The quantum Yang-Baxter is discussed as well. It acquires additional term in the case of supersymmetric $R$-matrices.
We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.
Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ra…
Selberg Integrals, Super hypergeometric functions and Applications to $β$-Ensembles of Random Matrices
研究了一类新的含$n+m$待定项的Selberg型积分，它与变形的Calogero-Sutherland系统有关.我们证明了该积分满足$n+m$非对称线性偏微分方程的完整系统。我们还证明了一个特殊的超几何函数定义的超级杰克多项式是唯一的解决方案的系统。建立了对偶关系、积分公式、Pfaff-Euler变换和Kummer变换等性质。作为一个直接的应用，我们评估的经典随机矩阵理论的$β$ -合奏的特征多项式的比率的期望值。
We consider the ensemble of Real Ginibre matrices with a positive fraction $α>0$ of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and we introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability $p^n_{αn}$ that an $n\times n$ Ginibre matrix has $k=αn$ real eigenvalues and we characterize the spectral measures of these matrices.
Distribution Functions for Random Variables for Ensembles of positive Hermitian Matrices
对于正埃尔米特矩阵的系综，渐近地计算依赖于参数的随机变量的分布函数。分布的傅里叶逆变换被证明是某个算子的弗雷德霍姆行列式，该算子是Wiener-Hopf算子的类似物。渐近公式表明，直到$o(1)$阶项，分布是高斯的。
Ensemble Averages when βis a Square Integer
当L为任意整数且β=L^2和L为奇整数且β = L^2 + 1时，我们给出了厄米系综和循环系综的配分函数和系综平均的一个超精确公式.
Derivation of determinantal structures for random matrix ensembles in a new way
对于对称空间中的随机矩阵系综、循环矩阵系综、手征矩阵系综等有几种处理方法。正交多项式和超对称方法是特别强大的技术。在这里，我们提出了一种新的方法来计算特征多项式的比率的平均值。乍看起来有些矛盾，人们可以将我们的方法称为“没有超对称的超对称”，因为我们使用了来自超对称的结构，而没有实际映射到超空间。我们解决了两种积分，涵盖了广泛的应用随机矩阵集成。对于概率密度分解的特征值，我们发现行列式的结构在一个统一的方式。作为一个新的应用，我们导出了外场作用下厄米矩阵上任意旋转不变概率密度的k点关联函数的表达式。
A real quaternion spherical ensemble of random matrices
人们可以确定一个三方分类的随机矩阵合奏到对应的平面，球和反球的几何普适性类。平面被确定为Ginibre型（iid）矩阵和反球与截断的酉矩阵。本文主要研究与球面相对应的系综：矩阵的形式为$\bY= \bA^{-1} \bB$，其中$\bA$和$\bB$是独立的$N\times N$矩阵，具有iid标准高斯实四元数元素。通过应用类似于那些用于类似的复杂和真正的球形系综，本征值jpdf和关联函数的计算技术。这就完成了使用传统Dyson指数$β=1,2,4$的球形矩阵的探索。我们发现，本征值密度（球面投影后）有一个耗尽的本征值沿环对应的实轴，与此环的反射对称。然而，在大矩阵维数的限制下，该特征值密度接近于相应的复系综的密度，该密度在球体上是均匀的。该结果与球面定律（类似于iid矩阵的圆形定律）一致，该定律指出，对于具有球面结构的矩阵$\bY= \bA^{-1} \bB$，其中$\bA$和$\bB$是独立的，iid矩阵的（立体投影的）本征值密度在球面上趋于均匀。
We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $β>0$, not merely $β=1,2,4$.
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbin…
Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
考虑一个具有独立元素的$N\times N$埃尔米特随机矩阵，不一定是高斯矩阵，即所谓的维格纳矩阵。本文证明了局部间距分布，即谱的某一部分中最近邻本征值之间的距离的分布，在$N\to\infty$的极限下，与GUE的厄米特随机矩阵的分布相同。我们证明了这一猜想的厄米特维格纳矩阵的某个子类。
On Airy solutions of P$_\mathrm{II}$ and the complex cubic ensemble of random matrices, II
本研究借助Airy解与三次势随机矩阵系综分析中正交多项式递推系数的关联，刻画了第二Painlevé方程P$_{\mathrm{II}}$单参数族特解的无极点区域。
Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Circular Beta Ensembles
本文研究了系数随机衰减的CMV矩阵（一种离散的一维Dirac型算子）。在温和的假设下，我们确定的本地特征值统计的自然缩放限制。对于快速减小的系数，特征值具有刚性间距（就像时钟上的数字）;在缓慢减小的情况下，特征值根据泊松过程分布。对于一定的临界衰变率，我们得到了随机矩阵理论的圆β系综。温度β^{-1}表示为耦合常数的平方。
From Alternating Sign Matrices to the Gaussian Unitary Ensemble
本文的目的是证明边界附近的均匀随机交替符号矩阵（相当于具有畴壁边界条件的六顶点模型的配置）的波动由高斯酉包络和GUE角点过程描述。
We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M family of integrable matrices consists of exactly N-M independent commuting N-by-N matrices linear in a real parameter. We first develop a rotationally invariant parametrization of such matrices, previously only constructed in a preferred basis. For example, an arbitrary choice of a vector and two commuting Hermitian matrices defines a type-1 family and vice ver…
Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices
一般由正交多项式的正则变条件下的递推系数来研究随机矩阵酉系综的特征值密度。首先我们直接计算密度的矩。然后，通过研究矩的某些变形，得到了密度满足的一类一阶微分方程（见定理1.2），并通过求解这些微分方程给出了密度。进一步，我们证明了密度在权函数的多项式扰动后是不变的（见定理1.5）。
Spectral density of the non-central correlated Wishart ensembles
随机矩阵理论的Wishart系综在模拟经典和量子混沌系统中遇到的正定矩阵方面很有用。我们考虑构成矩阵A的条目的非零均值，其将相关Wishart矩阵定义为W = AA t，并且将这样的Wishart矩阵的系综称为非中心相关Wishart系综（nc-CWE）。我们推导出Pastur自洽方程，该方程描述了大矩阵维数下nc-CWE的谱密度。
Universal Spectral Correlations in Orthogonal-Unitary and Symplectic-Unitary Crossover Ensembles of Random Matrices
随机矩阵的正交幺正和辛幺正交叉系综在许多场合都是相关的，特别是在量子混沌系统的时间反演对称性破缺的研究中。使用斜正交多项式，我们表明，相同的通用形式的n级关联函数得到的Jacobi家族的交叉合奏，包括拉盖尔和高斯的情况下。对于大矩阵，我们发现通用形式的展开关联函数表示的重新缩放的过渡参数与任意的初始水平密度。
Numerical ranges of non-normal random matrices: elliptic Ginibre and non-Hermitian Wishart ensembles
研究非正交随机矩阵集合的数值范围几何特性，揭示椭圆Ginibre和非厄米特Wishart集合在大系统极限下的形状特征。
We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel-Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel-Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogon…
The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices
给定一个NxN随机矩阵的集合，一个自然的问题是典型矩阵的经验谱测度是否收敛到N --> oo时的极限谱测度。虽然这已经证明了许多薄模式合奏坐在所有实对称矩阵，往往没有很好的封闭形式的表达的限制措施。此外，目前的定理提供了几个图片的合奏之间的过渡。我们考虑元素为i. i. d. r. v.的对称m-块循环矩阵的系综。这些矩阵具有周期为m的环形对角线。我们把m看作是一个“转盘”，我们可以把它从极限特征值密度为高斯的对称循环矩阵的稀疏集合“转”到极限特征值密度为半圆的所有实对称矩阵。当m趋于无穷大时，极限特征值密度f_m显示出视觉上惊人的收敛到半圆，我们证明了这一点。与大多数模式化矩阵系综的研究不同，本文给出了密度的显式封闭形式表达式。证明了f_m是高斯多项式与2 m-2次多项式的乘积，其公式与m × m高斯酉包络（GUE）的公式相同.证据是由时刻。新的功能，使我们能够获得封闭形式的表达式，是转换的中心组合问题的时刻计算到一个等价的代数拓扑计数问题。最后，我们放弃了m个随机变量是不同的假设，推广了m块循环模式。我们证明了极限谱分布的存在，并确定由独立的元素在一个m周期内的模式，不仅取决于每个元素出现的频率，而且元素的排列方式。
Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $β$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $β$-ensemble. As particular cases of the main result we consider the classical $β$-Gaussian, $β$-Wishart and $β$-Jacobi ensembles, for which we derive previously available results as well as new …
Particle Systems with Repulsion Exponent $β$ and Random Matrices
我们考虑一类粒子系统推广的随机矩阵理论的$β$ -合奏。在这些新的合奏中，粒子在接近时会受到$β>0$的排斥，这与$β$合奏中的情况相同。对于大于零的距离，允许相互作用不同于随机本征值。我们发现，本地散装相关的$β$ -合奏，在随机矩阵理论的普遍性，也出现在这些新的合奏。
Clustering of periodic orbits and ensembles of truncated unitary matrices
混沌系统中的周期轨道形成簇，其元素在相空间中近似地穿过相同的点。团簇大小的分布依赖于轨道的长度n和控制轨道作用的接近程度的参数p。我们表明，在面包师的地图集群的大小计数可以变成一个频谱问题的合奏截断酉矩阵。基于对这些系综谱边特征值分布普适性的猜想，我们得到了在n和p都趋于无穷大的情况下簇分布二阶矩的渐近性质.这一结果使我们能够估计平均集群大小作为一个函数的数量遇到周期性轨道。
Recently subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya e…
Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ensemble and the second from a polynomial ensemble of derivative type. This allows us to re-derive and …
The eigenvalue PDF for some well known classes of non-Hermitian random matrices --- the complex Ginibre ensemble for example --- can be interpreted as the Boltzmann factor for one-component plasma systems in two-dimensional domains. We address this theme in a systematic fashion, identifying the plasma system for the Ginibre ensemble of non-Hermitian Gaussian random matrices $G$, the spherical ensemble of the product of an inverse Ginibre matrix and a Ginibre matrix $G_1^{-1} G_2$, and the ensemble formed by truncating unitary matrices, as well as for products of such matrices. We do this when …
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。
When dealing with non-stationary systems, for which many time series are available, it is common to divide time in epochs, i.e. smaller time intervals and deal with short time series in the hope to have some form of approximate stationarity on that time scale. We can then study time evolution by looking at properties as a function of the epochs. This leads to singular correlation matrices and thus poor statistics. In the present paper, we propose an ensemble technique to deal with a large set of short time series without any consideration of non-stationarity. Given a singular data matrix, we r…
The permanent of unitary matrices and their blocks has attracted increasing attention in quantum physics and quantum computation because of connections with the Hong-Ou-Mandel effect and the Boson Sampling problem. In that context, it would be useful to know the distribution of the permanent or other immanants for random matrices, but that seems a difficult problem.We advance this program by calculating the average of the squared modulus of a generic immanant for blocks from random matrices in the unitary group, in the orthogonal group and in the circular orthogonal ensemble. In the case of th…
Singular values of products of random matrices and polynomial ensembles
Akemann，Ipsen和Kieburg最近证明了M个复Ginibre矩阵乘积的平方奇异值按行列式点过程分布。我们介绍了一个多项式系综的概念，并展示了他们的结果可以被解释为多项式系综的变换。我们还证明了M-1个复Ginibre矩阵与一个截断酉矩阵乘积的平方奇异值是一个多项式系综，并导出了与此系综相关的相关核的二重积分表示.我们用它来计算在硬边的缩放极限，这原来是相同的缩放极限Kuijlaars和张发现的平方奇异值的M个复杂的Ginibre矩阵的产品。我们的最终结果是，这些限制核也出现为参数θ> 0的Borodin的双正交系综的标度极限，如果θ或1/θ是一个整数。这进一步支持了猜想，这些内核有一个普遍的字符。
Tridiagonal realization of the anti-symmetric Gaussian $β$-ensemble
反对称高斯酉系综的一个成员的Householder约化给出了一个反对称三对角矩阵，其中所有元素都是独立的。随机变量允许引入一个正参数$β$，相应的随机矩阵的本征值概率密度函数可以明确计算，可以分布$\{q_i\}$，本征向量的第一个组件。给出了三个证明。一个涉及到一个归纳建设的基础上接壤的一个家庭的随机矩阵具有相同的分布作为反对称三对角矩阵。这个证明使用来自Selberg积分理论的Dixon-Anderson积分。第二个证明涉及到显式计算的雅可比矩阵的变化之间的真正反对称三对角矩阵，其特征值和$\{q_i\}$的变量。第三个证明将反对称高斯$β$系综的矩阵映射到实现拉盖尔$β$系综的特定示例的矩阵。除了这些证明，我们注意到随机矩阵的射击特征向量和相关的Prüfer相位的一些简单的性质。
We study several kinds of polynomial ensembles of derivative type which we propose to call Pólya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matric…
In this paper we study the distribution of level crossings for the spectra of linear families A+lambda B, where A and B are square matrices independently chosen from some given Gaussian ensemble and lambda is a complex-valued parameter. We formulate a number of theoretical and numerical results for the classical Gaussian ensembles and some generalisations. Besides, we present intriguing numerical information about the distribution of monodromy in case of linear families for the classical Gaussian ensembles of 3 * 3 matrices.
Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting $PT$-symmetry. Here we show that there is a one-to-one correspondence between complex $PT$-symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. They are related to the split signature versions of the complex and the quaternionic numbers, respectively. We conjecture that these ensembles rep…
Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices
Wigner的一个经典结果是，对于对角线上和对角线上具有独立元素的埃尔米特矩阵，当矩阵大小趋于无穷大时，平均经验特征值分布弱收敛于幂律.在本文中，我们证明了类似的维格纳定理的随机矩阵取自所有无穷小版本的经典对称空间。这是一类模型，其中包括维格纳和戴森研究的模型，以及凝聚态物理中产生的其他七个模型。像维格纳的，我们的结果是普遍的，因为它们只依赖于某些假设的时刻的矩阵项，但不对他们的分布的具体情况。更重要的是，我们允许一定数量的矩阵项之间的依赖性，在精神上最近推广的维格纳定理，由于申克和舒尔茨-巴尔德斯。作为一个副产品，我们得到了一个普遍性的结果，样本协方差阵的相关项目。
For each $α\in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $α$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vert…
Spectral distribution of sparse Gaussian Ensembles of Real Asymmetric Matrices
本文采用复杂度参数方法分析多参数稀疏实非对称矩阵高斯系综，推导了实与复特征值的系综平均谱密度，为多种稀疏实非对称系综提供了统一的谱统计数学框架并揭示了其普遍性。
We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature $β$ in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd charge. Our methods provide a unified framework extending the de Bruijn integral identities from classical $β$-ensembles ($β$ = 1, 2, 4) to multicomponent ensembles, as well as to iterated integra…
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