Matrix models for beta ensembles

Abstract

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.

Nearest-neighbour spacing distributions of the β-Hermite ensemble of random matrices

“…It follows immediately that , a sum of independent chi-squared random variables, has a chi-squared distribution with , where X is a standard Gaussian [29]. Then, the properly rescaled eigenvalues, whose large β distribution tends to a multivariate Gaussian distribution, fluctuate around the N roots of the Nth Hermite polynomial.…”

“…where Dumitriu and Edelman [29] derived the β-Hermite ensemble of real-symmetric tridiagonal random matrices whose eigenvalue density is given by eq.4 for any β ≥0 [29][30][31][32]. Extensions of other classical ensembles were similarly performed, for the β-Laguerre ensemble among others [29].…”

“…A newcomer in RMT, the β-Hermite ensemble (β-HE) of tridiagonal random matrices [29] (section 2), whose temperature 1/β can take any value, offers an enlarged perspective. Its joint distribution of eigenvalues is identical with those of the previous Gaussian ensembles for β =1, 2, 4 respectively [29][30][31][32][33]. The β-HE owes its name to the Hermite orthogonal polynomials whose weight…”

“…Extensions of other classical ensembles were similarly performed, for the β-Laguerre ensemble among others [29]. A random matrix from the β-HE is defined as: N N × …”

A Random Matrix Decimation Procedure Relating β = 2/(r + 1) to β = 2(r + 1)

“…has its eigenvalue PDF given by ME β,N (e −x 2 /2 ) [4]. Taking various scaled N → ∞ limits (see Section 5.2) of these ensembles of random matrices leads to a description of the limiting eigenvalue distributions in terms of stochastic differential operators [7,19,16].…”