  In a diffusive conductor the eigenstates are spread over the entire sample.
However, with certain probability, an anomalously localized state (ALS) can
occur, i.e. the wave function assumes anomalously large values in some region
of space. Existing analytical theories of ALS are based on models described by
a continuous (Gaussian) random potential. In the present paper we study ALS in
a lattice (Anderson) model. We demonstrate that close to the center of the
band, E=0, a new type of ALS exist and calculate analytically their likelihood.
These ALS are lattice-specific and have no analog in the continuum. Our
findings are relevant to numerical simulations, which are necessarily performed
on a lattice. We demonstrate that inconsistencies with "continuous" results
reported in the previous numerical work on ALS can be explained within our
analytical theory. Finally, we point out that, in order to compare the numerics
with the "continuous" ALS theories, simulations must be carried out not too far
from the band edges, within the band, where the continuous description applies.
Simulations performed for $E$ close to the band center reveal lattice-specific
ALS that do not exist in continuous models.
