Edge rigidity and universality of random regular graphs of intermediate degree

Understand

For random $d$-regular graphs on $N$ vertices with $1 \ll d \ll N^{2/3}$, we develop a $d^{-1/2}$ expansion of the local eigenvalue distribution about the Kesten-McKay law up to order $d^{-3}$.

• This result is valid up to the edge of the spectrum.
• It implies that the eigenvalues of such random regular graphs are more rigid than those of Erd\H{o}s-R\'enyi graphs of the same average degree.
• As a first application, for $1 \ll d \ll N^{2/3}$, we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by $(2 + o(1)) \sqrt{d - 1}$.