Hermite and Laguerre $\beta$-ensembles: asymptotic corrections to the eigenvalue density

Abstract

We consider Hermite and Laguerre $\beta$-ensembles of large $N\times N$ random matrices. For all $\beta$ 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 $\beta$. At the edges, the density can be expressed as a multiple integral of the Konstevich type which constitutes a $\beta$-deformation of the Airy function. This allows us to obtain the main contribution to the soft edge density when the spectral parameter tends to $\pm \infty$.

26 pages, 6 figures; published version, minor changes, 2 refs added

References in corpus (2)

• Asymptotic corrections to the eigenvalue density of the GUE and LUE
• MOPS: Multivariate Orthogonal Polynomials (symbolically)

Cited by in corpus (25)

• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models
• Duality in random matrix ensembles for all Beta
• Moments of the Gaussian $\beta$ Ensembles and the large-$N$ expansion of the densities
• Products of Independent Gaussian Random Matrices
• Spectral density asymptotics for Gaussian and Laguerre $\beta$-ensembles in the exponentially small region
• Eigenvalue Separation in Some Random Matrix Models
• Nearest-neigbor spacing distributions of the beta-Hermite ensemble of random matrices
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
• Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices
• Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $\beta$-ensembles
• Spectral fluctuations of tridiagonal random matrices from the beta-Hermite ensemble
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Asymptotics for products of characteristic polynomials in classical $\beta$-Ensembles
• Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
• Asymptotic forms for hard and soft edge general $\beta$ conditional gap probabilities
• Scaling limits of correlations of characteristic polynomials for the Gaussian beta-ensemble with external source
• Fixed trace $\beta$-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• Limits for circular Jacobi beta-ensembles
• Edge effects in some perturbations of the GUE