  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.
