The density of eigenvalues seen from the soft edge of random matrices in the Gaussian beta-ensembles

We characterize the phenomenon of crowding near the largest eigenvalue $lambda_{max}$ of random $N times N$ matrices belonging to the Gaussian $beta$-ensemble of random matrix theory, including in particular the Gaussian orthogonal ($beta=1$), unitary ($beta=2$) and symplectic ($beta = 4$) ensembles. We focus on two distinct quantities: (i) the density of states (DOS) near $lambda_{max}$, $rho_{rm DOS}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $lambda_{max}$ (or the density of eigenvalues seen from $lambda_{max}$) and (ii) the probability density function of the gap between the first two largest eigenvalues, $p_{rm GAP}(r,N)$. Using heuristic arguments as well as well numerical simulations, we generalize our recent exact analytical study of the Hermitian case (corresponding to $beta = 2$). We also discuss some applications of these two quantities to statistical physics models.