  We use the method of eigenvalue level spacing developed by Dietlein and
Elgart (arXiv:1712.03925) to prove that the local eigenvalue statistics (LES)
for the Anderson model on $Z^d$, with uniform higher-rank $m \geq 2$,
single-site perturbations, is given by a Poisson point process with intensity
measure $n(E_0)~ds$, where $n(E_0)$ is the density of states at energy $E_0$ in
the region of localization near the spectral band edges. This improves the
result of Hislop and Krishna (arXiv:1809.01236), who proved that the LES is a
compound Poisson process with L\'evy measure supported on the set $\{1, 2,
\ldots, m \}$. Our proofs are an application of the ideas of Dieltein and
Elgart to these higher-rank lattice models with two spectral band edges, and
illustrate, in a simpler setting, the key steps of the proof of Dieltein and
Elgart.
