Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble

Abstract

We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[\sigma, +\infty[$. In particular, we show that the probability that all the eigenvalues of an $(n\times n)$ random matrix are positive (negative) decreases for large $n$ as $\sim exp[-\beta\theta(\alpha)n^2]$ where the Dyson index $\beta$ characterizes the ensemble, $\alpha$ is some extra parameter and the exponent $\theta(\alpha)$ is a function of $\alpha$ which will be given explicitly. For $\alpha=0$, $\theta(0)= (\log 3)/4 = 0.274653...$ is universal. We compute the probability that the eigenvalues lie in the interval $[\sigma,+\infty[$ with $(\sigma>0,\; {\rm if}\;\alpha>0)$ and $(\sigma\in\mathbb R,\; {\rm if }\;\alpha=0)$. This generalizing the celebrated Wigner semicircle law to these restricted ensembles. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied in \cite{D}, \cite{S}.