Visions in Arithmetic and Beyond: Celebrating Peter Sarnak's Work and Impact

In this lecture, we will review recent works regarding  spectral  statistics of the normalized adjacency matrices of random  $d$-regular graphs on $N$ vertices.

Denote their eigenvalues by $\lambda_1=d/\sqrt{d-1}\geq \la_2\geq\la_3\cdots\geq \la_N$ and let  $\gamma_i$ be the classical location of the $i$-th  eigenvalue under the  Kesten-McKay law.  Our main  result asserts that for any $d \ge 3$ the optimal eigenvalue rigidity holds in the sense that  

$|\lambda_i-\gamma_i|\leq \frac{N^{\oo_N(1)}}{N^{2/3} (\min\{i,N-i+1\})^{1/3}},\quad \forall i\in \{2,3,\cdots,N\}.$

 with probability $1-N^{-1+\oo_N(1)}$.  In particular, the characteristic $N^{-2/3}$ fluctuations for Tracy-Widom law is established for the second largest eigenvalue. 

Furthermore, for $d \ge N^\varepsilon $ for any $\varepsilon > 0$  fixed,   the extremal eigenvalues  obey  the Tracy-Widom law. This is a joint work with Jiaoyang Huang and Theo McKenzie.