  Consider Hermitian and symmetric random band matrices $H=(\sigma_{xy}A_{xy})$
on the $d$-dimensional lattice $\left(\mathbb{Z}/{L\mathbb{Z}}\right)^d$, where
$A_{xy}=\overline{A_{yx}}$ are independent uniformly distributed random
variables on $S^1$ or $\{+1, -1\}$, and the variance profile $\sigma^{2}_{xy}$
is characterized by the bandwidth $W$ and $\alpha$-stable density with
$\alpha\in (0,2]$. We investigate local eigenvalue statistics at the spectral
edge as $W\to \infty$ and observe the critical dimension $d_c=3\alpha$ and the
critical bandwidth $W_c=L^{(1-\frac{d}{3\alpha})_{+}}$, possibly with a $\log
L$ correction when $d=\alpha$ or $2\alpha$. In the Hermitian case, we establish
that (i) when $d<2\alpha$, GUE edge, interpolating, and Poisson statistics
emerge in the supercritical ($W\gg W_c$), critical ($W\sim W_c$), and
subcritical ($W\ll W_c$) regimes, respectively; (ii) when $d\ge 2\alpha$, as
long as $W\ge L^{\frac{1}{3}+\epsilon}$ for a small constant $\epsilon>0$, GUE
edge universality holds. In the symmetric case, we also establish similar but
subtle phenomena. In both $d=1$ and $\alpha=2$, the subcritical and
supercritical results have been proven by Sodin for the band model with a
cutoff variance profile \cite{sodin2010spectral}. Our proof builds upon Sodin's
program and new techniques of taming the singularity of Feynman diagrams and
graph integrals through a connection to the $\phi^3$ model.
