Unitary Random-Matrix Ensemble with Governable Level Confinement

Abstract

A family of unitary $α$-Ensembles of random matrices with governable confinement potential $V(x) ~ |x|^α$ is studied employing exact results of the theory of non-classical orthogonal polynomials. The density of levels, two-point kernel, locally rescaled two-level cluster function and smoothed connected correlations between the density of eigenvalues are calculated for strong ($α> 1$) and border ($α= 1$) level confinement. It is shown that the density of states is a smooth function for $α> 1$, and has a well-pronounced peak at the band center for $α<= 1$. The case of border level confinement associated with transition point $α= 1$ is reduced to the exactly solvable Pollaczek random-matrix ensemble. Unlike the density of states, all the two-point correlators remain (after proper rescaling) to be universal down to and including $α= 1$.