  We show analytically that the spectral density of the $q$-body
Sachdeev-Ye-Kitaev (SYK) model agrees with that of Q-Hermite polynomials with Q
a non-trivial function of $q \ge 2$ and the number of Majorana fermions $N \gg
1$. Numerical results, obtained by exact diagonalization, are in excellent
agreement with the analytical spectral density even for relatively small $N
\sim 8$. For $N \gg 1$ and not close to the edge of the spectrum, we find the
macroscopic spectral density simplifies to $\rho(E) \sim
\exp[2\arcsin^2(E/E_0)/\log \eta]$, where $\eta$ is the suppression factor of
the contribution of intersecting Wick contractions relative to nested
contractions. This spectral density reproduces the known result for the free
energy in the large $q$ and $N$ limit. In the infrared region, where the SYK
model is believed to have a gravity-dual, the spectral density is given by
$\rho(E) \sim \sinh[2\pi \sqrt 2 \sqrt{(1-E/E_0)/(-\log \eta)}]$. It therefore
has a square-root edge, as in random matrix ensembles, followed by an
exponential growth, a distinctive feature of black holes and also of low energy
nuclear excitations. Results for level-statistics in this region confirm the
agreement with random matrix theory. Physically this is a signature that, for
sufficiently long times, the SYK model and its gravity dual evolve to a fully
ergodic state whose dynamics only depends on the global symmetry of the system.
Our results strongly suggest that random matrix correlations are a universal
feature of quantum black holes and that the SYK model, combined with
holography, may be relevant to model certain aspects of the nuclear dynamics.
