Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice

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.