This paper proves that waves get trapped near the bottom of the energy range for the lattice Anderson model with a Bernoulli random potential in every dimension d ≥ 2. The Anderson model describes a quantum particle hopping on the integer lattice Z^d with a random on-site potential. In the Bernoulli case each site independently takes the value 0 or 1 with probability 1/2. The authors show that, for energies close to the bottom of the deterministic spectrum [0,4d], the operator almost surely has only point spectrum and its eigenfunctions decay exponentially. They also prove a form of dynamical localization expressed as uniform moment bounds in expectation.

The proof follows the multiscale-analysis framework developed by Fröhlich–Spencer and later by Bourgain and Kenig. Multiscale analysis is a technique that controls the operator step by step on larger and larger boxes, ruling out resonances that would allow a wave to spread. The main new ingredient in this paper is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schrödinger equation. A unique continuation principle gives a lower bound on how much of an eigenfunction’s mass must appear in a region, and that information is important to show that changing the potential at some sites will shift eigenvalues and break resonances.

On the lattice a direct analogue of the continuum unique continuation fails, so the authors replace it by a probabilistic statement. Their PDUC is obtained by a bootstrap argument together with a key probabilistic lemma. That lemma is proved by an “adaptive revealing” procedure: the random potential is revealed step by step in a way that exploits independence to get the needed lower bounds on where eigenfunctions must be nonnegligible. With the PDUC in hand, the multiscale iteration controls bad events at each scale and produces the exponential decay and dynamical bounds.