Asymptotic Densities in Statistical Ensembles

An expression for the single-eigenvalue (level) density is obtained for a class of ensembles previously considered by the author. The method involves a continuum approximation of a discrete spectrum, and leads to the asymptotic level density as the formal solution of an integral equation. Specific results are calculated for the Jacobi, Hermite, and Laguerre ensembles. These agree with prior calculations for $\beta =1\mathrm{}$ and $\beta =2\mathrm{}$, but have the feature of containing $\beta$ as an arbitrary continuously variable parameter. An analogy with a one-dimensional Coulomb gas is used to interpret results and to initiate a search for ensembles with physically realistic densities. A class of ensembles is found which has asymptotic densities whose first and second derivatives are non-negative.