Quantum Chaos on random Cayley graphs of ${\rm SL}_2[\mathbb{Z}/p\mathbb{Z}]$

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.