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

TL;DR

This paper studies the eigenvalue distribution and diameter of random Cayley graphs of SL_2 over finite fields, demonstrating spectral optimality and providing new density theorems for exceptional eigenvalues.

Contribution

It proves a density theorem for exceptional eigenvalues and provides numerical evidence for spectral optimality in large prime cases.

Findings

Density theorem for exceptional eigenvalues

Numerical evidence of optimal spectral gap

Diameter approaches theoretical bounds as p increases

Abstract

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.