Quantum Chaos on Random Cayley Graphs of SL 2[Z/pZ]

ABSTRACT

We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of ${\mathrm{SL}}_2[\mathbb{Z}/p\mathbb{Z}]$ 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 ${\mathrm{SL}}_2[\mathbb{Z}/p\mathbb{Z}]$ 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.

Keywords:

• Quantum chaos
• representations of finite symmetric groups
• graphs and groups
• explicit machine computation and programs (not the theory of com-putation or programming)

2000 AMS Subject Classification:

• 81Q50
• 20C30
• 05C25
• 20-04

Acknowledgments

We would like to thank Prof. Sarnak for suggesting this project to us and also his insightful comments on the earlier versions of this work. The computations in this article were performed using the Mathematica system. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the second named author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

Additional information

Funding

National Science Foundation under Grant No. DMS-1440140 (1440140)