Spectrum of Random $d$-regular Graphs Up to the Edge

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $dgeq3$. We prove that, with probability $1-N^{-1+{varepsilon}}$ for any ${varepsilon} >0$, the following two properties hold as $N to infty$ provided that $dgeq3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in $N$, i.e. $lambda_2, |lambda_N|leq 2+N^{-c}$. (ii) All eigenvectors of random $d$-regular graphs are completely delocalized.