Matrix models for beta ensembles

  title={Matrix models for beta ensembles},
  author={Ioana Dumitriu and Alan Edelman},
  journal={Journal of Mathematical Physics},
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. 

Figures and Tables from this paper

A matrix model for the -Jacobi ensemble
This model is a partial answer to an open problem presented by Dumitriu and Edelman, where they also presented models for the β-Laguerre and β-Hermite ensembles.
Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models
We study the global spectrum fluctuations for β-Hermite and β-Laguerre ensembles via the tridiagonal matrix models introduced previously by the present authors [J. Math. Phys. 43, 5830 (2002)], and
Eigenvalue statistics for beta-ensembles
Random matrix theory is a maturing discipline with decades of research in multiple fields now beginning to converge. Experience has shown that many exact formulas are available for certain matrices
Precise asymptotics for beta ensembles
We consider the extremal (largest and smallest) eigenvalues of random matrices in the β-Hermite and β-Laguerre ensembles. Using the general β Tracy-Widom law together with Ledoux and Rider’s small
Eigenvalue distributions of beta-Wishart matrices
We derive explicit expressions for the distributions of the extreme eigenvalues of the beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results
Fast sampling from β-ensembles
This work provides a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles and derives an approximate sampler for the simulation of β-ensembles, which supports a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size N mixes in O(logN).
Moderate deviations for spectral measures of random matrix ensembles
In this paper we consider the (weighted) spectral measure µn of a nn random matrix, distributed according to a classical Gaussian, Laguerre or Jacobi ensemble, and show a moderate deviation principle
C 2013 Society for Industrial and Applied Mathematics Eigenvalue Distributions of Beta-wishart Matrices *
We derive explicit expressions for the distributions of the extreme eigenvalues of the Beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results


The Distribution of the Largest Eigenvalue in the Gaussian Ensembles: β = 1, 2, 4
The focus of this survey is on the distribution function F Nβ (t) for the largest eigenvalue in the finite N Gaussian Orthogonal Ensemble (GOE, β = 1), the Gaussian Unitary Ensemble (GUE, β = 2), and
On orthogonal and symplectic matrix ensembles
The focus of this paper is on the probability,Eβ(O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (β=1) and Gaussian Symplectic
Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices
The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing
Universality of the distribution functions of random matrix theory
if their associatedBoltzmann weights satisfy the factorization or star-triangle equations of Mc-Guire [1], Yang [2] and Baxter [3]. For such models the free energy per siteand the one-point
Integrable lattices: random matrices and random permutations
These lectures present a survey of recent developments in the area of random matrices (finite and infinite) and random permutations. These probabilistic problems suggest matrix integrals (or Fredholm
Distribution of the determinant of a random real-symmetric matrix from the gaussian orthogonal ensemble
  • Delannay, Le Caer G
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 2000
The Mellin transform of the probability density of the determinant of NxN random real-symmetric matrices from the Gaussian orthogonal ensemble is calculated and is shown to be asymptotically Gaussian.
Probability density of the determinant of a random Hermitian matrix
The probability density function for the determinant of a nn random Hermitian matrix taken from the Gaussian unitary ensemble is calculated. It is found to be a Meijer G- function or a linear
Aspects Of Multivariate Statistical Theory
Tables. Commonly Used Notation. 1. The Multivariate Normal and Related Distributions. 2. Jacobians, Exterior Products, Kronecker Products, and Related Topics. 3. Samples from a Multivariate Normal
Riemannian symmetric superspaces and their origin in random‐matrix theory
Gaussian random‐matrix ensembles defined over the tangent spaces of the large families of Cartan’s symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics, as they
Pair correlation of zeros of the zeta function.
s T— »oo and U-+Q in such a way that UL = A; here L = ̂ —logT is the average 2n density of zeros up to T and A is an arbitrary positive constant. If the same number of points were distributed at