Discrete Beta Ensembles based on Gauss Type Quadratures

Let µ be a measure with support on the real line and n ≥ 1, β > 0. In the theory of random matrices, one considers a probability distribution on the eigenvalues t1, t2,. .. , tn of random matrices, of the form P (n) (tj − ti). This is the so-called β ensemble with temperature 1/β. We explicitly evaluate the m−point correlation functions when µ is a Gauss quad-rature type measure, and use this to investigate universality limits for sequences of such measures.