Anderson-Bernoulli localization near the edge on the 3D lattice

We study the Laplacian plus a Bernoulli potential, which can be used to model an electron hopping inside alloy type materials. Anderson localization of this model was studied by Bourgain-Kenig for continuous space, and by Ding-Smart for the 2D lattice. Following their framework, we prove almost sure Anderson localization for energies close to the lower edge of the spectrum, for the 3D lattice. Our main contribution is proving a 3D discrete unique continuation result, for Laplacian plus any deterministic bounded potential, while such result does not hold for the 2D lattice. Our proof is based on exploring the geometry of the 3D lattice, and arguments in the proof a Liouville theorem of Buhovsky–Logunov–Malinnikova–Sodin.

This is joint work with Linjun Li.