  We consider a Hamiltonian given by the Laplacian plus a Bernoulli potential
on the two dimensional lattice. We prove that, for energies sufficiently close
to the edge of the spectrum, the resolvent on a large square is likely to decay
exponentially. This implies almost sure Anderson localization for energies
sufficiently close to the edge of the spectrum. Our proof follows the program
of Bourgain--Kenig, using a new unique continuation result inspired by a
Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin.
