Matrix Models for Beta Ensembles

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. Department of Mathematics, room 2-331, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 ([email protected]) Department of Mathematics, room 2-388, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 ([email protected]; http://math.mit.edu/ ̃edelman)