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Matrix Models for Beta Ensembles

Ioana Dumitriu, Alan Edelman

TL;DR

The paper develops real tridiagonal random-matrix models for general β-ensembles, specifically β-Hermite (Gaussian) and β-Laguerre (Wishart), thereby extending beyond the traditional β=1,2,4 cases and enabling continuous Laguerre parameters. It derives explicit joint eigenvalue densities consistent with these ensembles and provides key structural results, including the Jacobian relations and independence between eigenvalues and the first eigenvector row. These constructions yield tractable, sparse matrix models that facilitate analysis of eigenvalue statistics, moment computations, and connections to Selberg-type integrals, while highlighting open problems for β-Jacobi and universality questions. The work offers concrete tools for studying level densities, spacings, and scaling limits in general β settings, with potential applications in physics and multivariate statistics.

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.

Matrix Models for Beta Ensembles

TL;DR

The paper develops real tridiagonal random-matrix models for general β-ensembles, specifically β-Hermite (Gaussian) and β-Laguerre (Wishart), thereby extending beyond the traditional β=1,2,4 cases and enabling continuous Laguerre parameters. It derives explicit joint eigenvalue densities consistent with these ensembles and provides key structural results, including the Jacobian relations and independence between eigenvalues and the first eigenvector row. These constructions yield tractable, sparse matrix models that facilitate analysis of eigenvalue statistics, moment computations, and connections to Selberg-type integrals, while highlighting open problems for β-Jacobi and universality questions. The work offers concrete tools for studying level densities, spacings, and scaling limits in general β settings, with potential applications in physics and multivariate statistics.

Abstract

This paper constructs tridiagonal random matrix models for general () -Hermite (Gaussian) and -Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for . Furthermore, in the cases of the -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.

Paper Structure

This paper contains 15 sections, 11 theorems, 49 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview
  3. Background
  4. Our Results
  5. The $\beta$-Hermite (Gaussian) Ensembles
  6. Motivation: Tridiagonalizing the GOE, GUE, and GSE
  7. Tridiagonal Matrix Lemmas
  8. The Eigendistribution of the $\beta$-Hermite ensemble
  9. The $\beta$-Laguerre (Wishart) Ensembles
  10. Motivation: Tridiagonalizing the Wishart ensembles
  11. The Eigendistribution of $\beta$-Laguerre Ensemble
  12. Applications and Open Problems
  13. Applications
  14. Open Problems
  15. Acknowledgments

Key Result

Theorem 2.1

If A is an $n \times n$ matrix from the GOE, then reduction of $A$ to tridiagonal form shows that the matrix $T$ from the $1$-Hermite ensemble has joint eigenvalue p.d.f. given by (once_again) with $\beta = 1$.

Figures (2)

  • Figure 1: Random Matrix Ensembles. As a guide to MATLAB notation, randn$(m,n)$ produces an $m \times n$ matrix with i.i.d. standard normal entries, conj$(X)$ produces the complex conjugate of the matrix $X$, and the $'$ operator produces the conjugate transpose of a matrix. Also $[X ~Y; ~Z ~W]$ produces a $2\times 2$ block matrix.
  • Figure 2: A dense symmetric matrix $A$ can be tridiagonalized (left side) or diagonalized (right side). In brackets, we provide the distributions starting with that of $A$ (GOE or Wishart real).

Theorems & Definitions (23)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • ...and 13 more