Classical skew orthogonal polynomials and random matrices

Abstract

Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the cases that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre and Jacobi cases.

21 pages, no figures

References in corpus (3)

• Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff

Cited by in corpus (65)

• High Dimensional Statistical Inference and Random Matrices
• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence
• Integrable Structure of Ginibre's Ensemble of Real Random Matrices and a Pfaffian Integration Theorem
• Moments of the transmission eigenvalues, proper delay times and random matrix theory I
• Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices
• Moments of the Gaussian $\beta$ Ensembles and the large-$N$ expansion of the densities
• Moments of the transmission eigenvalues, proper delay times and random matrix theory II
• Fast approach to the Tracy-Widom law at the edge of GOE and GUE
• Vicious Random Walkers and a Discretization of Gaussian Random Matrix Ensembles
• Entanglement in random pure states: Spectral density and average von Neumann entropy
• Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry
• Full counting statistics for interacting trapped fermions
• Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions
• Moments of random matrices and hypergeometric orthogonal polynomials
• The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• {\bf $\tau$-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• Inter-relationships between orthogonal, unitary and symplectic matrix ensembles
• Correlation Kernels for Discrete Symplectic and Orthogonal Ensembles
• Universal scaling limits of the symplectic elliptic Ginibre ensemble
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• The rank 1 real Wishart spiked model
• Correlation Functions for β=1 Ensembles of Matrices of Odd Size
• An Application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type
• Pfaffian Expressions for Random Matrix Correlation Functions
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• GUE-chGUE Transition preserving Chirality at finite Matrix Size
• Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems
• Relationships between $\tau$-function and Fredholm determinant expressions for gap probabilities in random matrix theory
• Spiked Models in Wishart Ensemble
• Relations between moments for the Jacobi and Cauchy random matrix ensembles
• A geometrical triumvirate of real random matrices
• Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Wronskian structures of planar symplectic ensembles
• Linear Statistics of Matrix Ensembles in Classical Background
• Universality in Random Matrix Theory for orthogonal and symplectic ensembles
• Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory
• Skew-orthogonal polynomials, differential systems and random matrix theory
• Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble
• The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff
• The Partition Function of Multicomponent Log-Gases
• Universal Distribution of Would-be Topological Zero Modes in Coupled Chiral Systems
• Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions
• Evaluations of certain Catalan-Hankel Pfaffians via classical skew orthogonal polynomials
• Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter
• The rank 1 real Wishart spiked model I. Finite N analysis
• Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge II: Level Densities
• Ensemble Averages when βis a Square Integer
• Generalized Christoffel-Darboux formula for classical skew-orthogonal polynomials
• Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions
• Painlevé expressions for LOE, LSE and interpolating ensembles
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• A Solvable Mixed Charge Ensemble on the Line: Global Results
• The Partition Function of Log-Gases with Multiple Odd Charges
• A Riemann-Hilbert problem for skew-orthogonal polynomials
• SL(2,R) matrix model and supersymmetric Yang-Mills integrals
• Random matrix ensembles associated to compact symmetric spaces
• $q$-Pearson pair and moments in $q$-deformed ensembles
• Global and Local Scaling Limits for Linear Eigenvalue Statistics of Jacobi $\beta$-Ensembles
• Hard and soft edge spacing distributions for random matrix ensembles with orthogonal and symplectic symmetry
• On the relation between Darboux transformations and polynomial mappings