Luen-Chau Li
In this work, we introduce matrix-valued diffusion processes which describe the non-equilibrium situation of the matrix models for the beta-Hermite and the beta-Laguerre ensembles. We also study the corresponding spectral measure process and empirical/singular value process with regard to their limit laws.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
New expressions for Laguerre and Hermite polynomials are shown. They are based on operator algebras commonly used in quantum mechanics.
We study the global spectrum fluctuations for $β$-Hermite and $β$-Laguerre ensembles via the tridiagonal matrix models introduced in \cite{dumitriu02}, and prove that the fluctuations describe a Gaussian process on monomials. We extend our results to slightly larger classes of random matrices.
In this paper we examine the zero and first order eigenvalue fluctuations for the $β$-Hermite and $β$-Laguerre ensembles, using the matrix models we described in \cite{dumitriu02}, in the limit as $β\to \infty$. We find that the fluctuations are described by Gaussians of variance $O(1/β)$, centered at the roots of a corresponding Hermite (Laguerre) polynomial. We also show that the approximation is very good, even for small values of $β$, by plotting exact level densities versus sum of Gaussians approximations.
Hermite, Higher order Hermite, Laguerre type polynomials and Burgers like equations
本文推导了与Hermite及Laguerre多项式相关的Burgers类方程，展示了这些多项式作为热方程解所对应的非线性偏微分方程形式。
We give integral representations for multiple Hermite and multiple Hermite polynomials of both type I and II. We also show how these are connected with double integral representations of certain kernels from random matrix theory.
The standard block orthogonal (SBO) polynomials $P_{i;n}(x), 0\le i\le n$ are real polynomials of degree $n$ which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than $i$. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these polynomials when the first scalar product corresponds to Hermite (resp. Laguerre) polynomials. These new sets of polynomials, we call Hermite (resp. Laguerre) SBO polynomials, provide a basis of…
We study a q-generalization of the classical Laguerre/Hermite orthogonal polynomials. Explicit results include: the recursive coefficients, matrix elements of generators for the Heisenberg algebra, and the Hankel determinants. The power of quadratic relation is illustrated by comparing two ways of calculating recursive coefficients. Finally, we derive a q-deformed version of the Toda equations for both q-Laguerre/Hermite ensembles.
We define a q-deformation of the Dirac operator, inspired by the one dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace operator. This allows to construct q-deformed Schroedinger equations in higher dimensions. The equivalence of these Schroedinger equations with those defined on q-Euclidean space in quantum variables is shown. We also define the m-dimensional q-Clifford-Hermite polynomials and show their connection with the q-Laguerre polynomials. These polynomials are orthogonal…
Completeness of the Klein-Gordon oscillator eigenfunctions via Hermite and Laguerre polynomials
本文证明了Klein-Gordon振子在一维和三维空间中的本征函数的完备性，利用Hermite和广义Laguerre多项式的标准性质，以及三维中球谐函数的完备性，展示了Klein-Gordon场的标量性质使其论证比Dirac振子的类似证明更简单，无需考虑非对角项抵消。
We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of…
We consider Hermite and Laguerre $β$-ensembles of large $N\times N$ random matrices. For all $β$ even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized (multivariate) classical orthogonal polynomials. The corrections to the bulk density are oscillatory terms that depends on $β$. At the edges, the density can be expressed as a multiple integral of the Konstevich type which constitutes…
In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the…
We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of special functions. We show that many of the properties of the resulting functions, including functional equations and reciprocity laws, are direct consequences of transformation formulae of hypergeometric functions. Interest in these results is reinforced by the fact that polynomial or other factors of the Mellin transforms have zeros only on the critical line Re s = 1/2. We additionally present a simple-zero Proposition for the Mellin transform of the wavefunction of the D-dime…
Soft edge limit of the Laguerre beta-ensemble at the lower edge
研究拉盖尔β-系综在下边缘的软边缘极限，通过不同条件下的方法证明其在特定参数变化时收敛到\(\text{Airy}_{\beta}\)过程，完善边缘缩放极限情况，还能证明相关算子层面极限及给出不同条件下的论证。
In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be …
We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre) $N\times N$ hermitian matrices. We study the effect of finite rank perturbations of these ensembles by a source term. The rank $r$ of the perturbation corresponds to the number of non-null eigenvalues of the source matrix. In the perturbed ensembles, the correlation functions can be written in terms of kernels. We show that for all $N$, the difference between the perturbed and the unperturbed kernels is a degenerate kernel of size $r$ which depends on multiple Hermite or Laguerre functions. We also compute asymptotic formulas f…
$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $β$-Hermite, $β$-Laguerre, and $β$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $β\to\infty$ in the $β$-Hermite and $β$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices $Σ_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed $Σ_N^{-1}$ with formulas for the eigenvalues and eigenvectors of …
In a previous paper we deformed Hermite polynomials to three associated polynomials .Here we apply the same deformation to Laguerre polynomials .
Connecting Exceptional Orthogonal Polynomials of Different Kind
本文将经典 Jacobi、Laguerre、Hermite 正交多项式间的渐近极限与二次变换推广到同余维的例外正交多项式，建立不同种类 $X_m$ 多项式间的系统联系。
Through the von Neumann interaction followed by post-selection, we can extract not only the eigenvalue of an observable of the measured system but also the weak value. In this post-selected von Neumann measurement, the initial pointer state of the measuring device is assumed to be a fundamental Gaussian wave function. By considering the optical implementation of the post-selected von Neumann measurement, higher-order Gaussian modes can be used. In this paper, we consider the Hermite--Gaussian (HG) and Laguerre--Gaussian (LG) modes as pointer states and calculate the average shift of the pointe…
We give a proof of the Universality Conjecture for orthogonal (beta=1) and symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels K_{n,beta}, correlation and cluster functions, gap probabilities and the distributions of the largest and smallest eigenvalues. Corresponding results for unitary (beta=2) Laguerre-type ensembles have been proved by the fourth autho…
In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are well-defined, analyze their complexity, and illustrate their performance in practice. Finally, we also present a few of the possible applications of this library.
We call "Dyson process" any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n by n Hermitian matrices, and the eigenvalues describe n curves. Given sets X_1,...,X_m the probability that for each k no curve passes through X_k at time τ_k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of …
Bessel processes associated with the root systems $A_{N-1}$ and $B_N$ describe interacting particle systems with $N$ particles on $\mathbb R$; they form dynamic versions of the classical $β$-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to $β$-Cauchy ensembles in both cases with explicit distributions. For these distributions we derive central limit theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend to infinity. The results are closely related to corresponding known freezing result…
This paper constructs tridiagonal random matrix models for general ($β>0$) $β$-Hermite (Gaussian) and $β$-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for $β= 1,2,4$. Furthermore, in the cases of the $β$-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.
In this paper we study entanglement of the reduced density matrix of a bipartite quantum system in a random pure state. It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace Laguerre ensemble of $N\times N$ random matrices. We showed that for finite $N$ the smallest eigenvalue distribution may be expressed in terms of Jack polynomials. Furthermore, based on the exact results, we found, a limiting distribution, when the smallest eigenvalue is suitably scaled with $N$ followed by a large $N$ limit. Our results turn out to be the same as the s…
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。
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。
The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…
We introduce a one-parameter deformation of the Wishart-Laguerre or chiral ensembles of positive definite random matrices with Dyson index beta=1,2 and 4. Our generalised model has a fat-tailed distribution while preserving the invariance under orthogonal, unitary or symplectic transformations. The spectral properties are derived analytically for finite matrix size NxM for all three beta, in terms of the orthogonal polynomials of the standard Wishart-Laguerre ensembles. For large-N in a certain double scaling limit we obtain a generalised Marcenko-Pastur distribution on the macroscopic scale, …
Classical multiple orthogonal polynomials for arbitrary number of weights and their explicit representation
本文研究了任意权数的经典多重正交多项式，为 Jacobi-Piñeiro、Laguerre 和 Hermite 多项式族提供了最近邻递推系数的显式表达式，并推导了第二类 Laguerre 和多重 Hermite 多项式的 I 型显式表达式。
We introduce the so-called Clifford-Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related to the Dunkl operators (see [Rösler M., Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006]) as well as with the basis of the weighted $L^{2}$ space introduced by Dunkl.
Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero-Moser-Sutherland type and are related with $β$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities $k$ tend to $\infty$, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form $\sqrt k\cdot x$ for $k\to\infty$ with $x$ in …
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。
Through an algebraic method using the Dunkl--Cherednik operators, the multivariable Hermite and Laguerre polynomials associated with the $A_{N-1}$- and $B_N$-Calogero models with bosonic, fermionic and distinguishable particles are investigated. The Rodrigues formulas of column type that algebraically generate the monic non-symmetric multivariable Hermite and Laguerre polynomials corresponding to the distinguishable case are presented. Symmetric and anti-symmetric polynomials that respectively give the eigenstates for bosonic and fermionic particles are also presented by the symmetrization and…
We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $β$-ensembles of $N\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as $N\to\infty$. In the bulk of the spectrum of each $β$-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_1F_{1}$ whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre $β$-ensembles is shown to be a multivariate Airy function, which i…
Calogero-Sutherland-type quantum systems, generalized hypergeometric functions and superintegrability for integral chains
重新研究CS型模型与广义超几何函数，构造对应广义CS算子及基于球面退化双仿射Hecke代数的算子族，建立广义对应关系、给出函数表示约束，还分析了β-形变积分及两类积分链的超可积性。
We present a derivation of classical Hermite, Laguerre, and Jacobi orthogonal polynomials directly through the Gram-Schmidt orthogonization process. The derivation uses certain generalized Vandermonde determinants with entries defined by Gamma and Beta functions. We also provide a geometric formulation of Gram-Schmidt orthogonalization using the Hodge star operator.
Computing marginal eigenvalue distributions for the Gaussian and Laguerre orthogonal ensembles
本文通过引入条件间隙概率生成函数并结合有限傅里叶级数与Pfaffian数值计算，实现了高斯和拉盖尔正交系综边缘特征值分布的高精度计算，并揭示了其均值与正交多项式零点的交错关系。
Two families of stochastic interacting particle systems, the interacting Brownian motions and Bessel processes, are defined as extensions of Dyson's Brownian motion models and the eigenvalue processes of the Wishart and Laguerre processes where the parameter $β$ from random matrix theory is taken as a real positive number. These are systems where several particles evolve as individual Brownian motions and Bessel processes while repelling mutually through a logarithmic potential. These systems are also special cases of Dunkl processes, a broad family of multivariate stochastic processes defined…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。