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

Understand

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.