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}.