Hermitean Matrix Ensembles and Orthogonal Polynomials By Yang Chen and Mourad E. H. Ismail In this article, we investigate orthogonal polynomials associated with com- plex Hermitean matrix ensembles using the combination of the methods of Ž . Coulomb fluid or potential theory , chain sequences, and Birkhoff  Trjitzinsky theory. We give a general formula for the largest eigenvalue of Ž the N  N Jacobi matrices which is equivalent to estimating the largest . zero of a sequence of orthogonal polynomials and the two-level correlation Ž . function for the  ensembles   0 introduced previously for  1. In the case of 0   1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems. 1. Introduction Random matrix ensembles, originally proposed by Wigner as a phenomeno- logical description of the statistical properties of the energy levels of heavy   nuclei 20 have recently seen applications in other areas of physics such as    quantum chaos 3 and 2D quantum gravity 19 . From the random matrix point-of-view a fundamental quantity of interest is the probability that an Address for correspondence: Professor Mourad E. H. Ismail, Department of Mathematics, Univer- sity of South Florida, Tampa, FL 33620. Ž . STUDIES IN APPLIED MATHEMATICS 100:33  52 1998 33  1998 by the Massachusetts Institute of Technology Published by Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA, and 108 Cowley Road, Oxford, OX4 1JF, UK.