Ramanujan property of random regular graphs and delocalization of random band matrices

In this lecture, we  review two recent works on random matrices. 

1.  We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We establish the edge universality for random $d$-regular graphs, namely,  the distributions of $\lambda_2$ and $-\lambda_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. 
As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices
are Ramanujan, meaning $\max\{\lambda_2,|\lambda_N|\}\leq 2$. This resolves a conjecture by  Sarnak and Miller-Novikoff-Sabelli.


2. Consider an $ N \times N$ Hermitian one-dimensional random band matrix with  band width $W > N^{1 / 2 + \varepsilon} $ for any $ \varepsilon > 0$. 
In the bulk of the spectrum and in the large $ N $ limit, we prove that  all $ L^2 $- normalized eigenvectors are delocalized, meaning their $ L^\infty$ norms are simultaneously bounded by $ N^{-\frac{1}{2} + \varepsilon} $ with overwhelming probability, for any $ \varepsilon > 0 $.  This resolves the delocalization of one-dimensional random band matrices in the full conjectured regime of the band width.