  The aim of this paper is to provide a novel proof for the Local Semicircle
Law for the Wigner ensemble. The core of the proof is the intensive use of the
algebraic structure that arises, i.e. resolvent expansions and resolvent
identities. On the analytic side, concentration of measure results and high
probability bounds are used. The conclusion is obtained using a bootstrapping
argument that provides information about the change of the bounds from large to
small scales. This approach leads to a new and shorter proof of the Weak Local
Law for Wigner Matrices, that exploits heavily the algebraic structure
appearing in the setting.
