BETA LAGUERRE ENSEMBLES IN GLOBAL REGIME

Beta Laguerre ensembles, generalizations of Wishart and Laguerre ensembles, can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart (beta = 1) case and the Laguerre (beta = 2) case, for fixed beta, it is known that the empirical distribution of the eigenvalues of the ensembles converges weakly to Marchenko-Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter beta is allowed to vary as a function of the matrix size N. We show that the above Marchenko-Pastur law holds as long as beta N -> infinity. When beta N -> 2c is an element of (0, infinity), the limiting measure is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.