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Matrix models for beta ensembles. (English) Zbl 1060.82020
Summary: This paper constructs tridiagonal random matrix models for general $$(\beta > 0) \;\beta$$-Hermite (Gaussian) and $$\beta$$-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for $$\beta = 1,2,4$$. Furthermore, in the cases of the $$\beta$$-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.

##### MSC:
 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 82B10 Quantum equilibrium statistical mechanics (general)
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