Random band matrices are a natural model for studying the quantum 
propagation in disordered systems as they interpolate between random 
Schrödinger operators and Wigner matrices. We study the random band matrix 
ensemble introduced by Disertori--Pinson--Spencer, characterised by the 
variance profile associated with the operator $(-W^{2}\Delta+1)^{-1}$, 
$\Delta$ being the Laplacian on the three-dimensional lattice and $W$ being 
the characteristic width of the band. For any energy $E$ close to the 
spectral edge, namely for $1.8 <|E| <2$, we rigorously prove that the 
density of states follows Wigner semicircle law with precision $W^{-2}$, 
provided $W$ is sufficiently large, depending on $E$. The proof relies on 
the supersymmetric approach of Disertori--Pinson--Spencer, and extends their 
result, which was previously established for energies $|E| \leq 1.8$. Joint 
work with M. Disertori.