Density of Levels in a Generalized Matrix Ensemble

Abstract

We present analytical results for the average density of levels $\rho \left(\lambda \right)$ in an ensemble of random matrices, which is a generalization of the standard Gaussian unitary ensemble. The generalized ensemble contains a parameter $\mu \left(N\right)$ which is allowed to scale, in an arbitrary way, with the matrix size $N$. The results crucially depend on the behavior of $\mu \left(N\right)$. For a sublinear dependence of $\mu$ on $N$ a modified Wigner semicircle is obtained, in the large $N$ limit. For a superlinear dependence the ensemble approaches the Poissonian limit of uncorrelated levels, with a Gaussian shape for $\rho \left(\lambda \right)$. For a strictly linear dependence, i.e., when $\underset{N\to \infty }{limop}\left(\mu /N\right)=\mathrm{const}$, an intermediate situation occurs.