  We study the phenomenon of "crowding" near the largest eigenvalue
$\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian
Unitary Ensemble (GUE) of random matrix theory. 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 $\lambda_{\max}$ and (ii) the probability density function of the gap
between the first two largest eigenvalues, $p_{\rm GAP}(r,N)$. In the edge
scaling limit where $r = {\cal O}(N^{-1/6})$, which is described by a double
scaling limit of a system of unconventional orthogonal polynomials, we show
that $\rho_{\rm DOS}(r,N)$ and $p_{\rm GAP}(r,N)$ are characterized by scaling
functions which can be expressed in terms of the solution of a Lax pair
associated to the Painlev\'e XXXIV equation. This provides an alternative and
simpler expression for the gap distribution, which was recently studied by
Witte, Bornemann and Forrester in Nonlinearity 26, 1799 (2013). Our expressions
allow to obtain precise asymptotic behaviors of these scaling functions both
for small and large arguments.
