  We investigate the statistical behavior of the eigenvalues and diameter of
random Cayley graphs of ${\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]$ %and the Symmetric
group $S_n$ as the prime number $p$ goes to infinity. We prove a density
theorem for the number of exceptional eigenvalues of random Cayley graphs i.e.
the eigenvalues with absolute value bigger than the optimal spectral bound. Our
numerical results suggest that random Cayley graphs of ${\rm
SL}_2[\mathbb{Z}/p\mathbb{Z}]$ and the explicit LPS Ramanujan projective graphs
of $\mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ have optimal spectral gap and
diameter as the prime number $p$ goes to infinity.
