  The spectral approach to infinite disordered crystals is applied to an
Anderson-type Hamiltonian to demonstrate the existence of extended states for
nonzero disorder in 2D lattices of different geometries. The numerical
simulations shown prove that extended states exist for disordered honeycomb,
triangular, and square crystals. This observation stands in contrast to the
predictions of scaling theory, and aligns with experiments in photonic lattices
and electron systems. The method used is the only theoretical approach aimed at
showing delocalization. A comparison of the results for the three geometries
indicates that the triangular and honeycomb lattices experience transition in
the transport behavior for same amount of disorder, which is to be expected
from planar duality. This provides justification for the use of
artificially-prepared triangular lattices as analogues for honeycomb materials,
such as graphene. The analysis also shows that the transition in the honeycomb
case happens more abruptly as compared to the other two geometries, which can
be attributed to the number of nearest neighbors. We outline the advantages of
the spectral approach as a viable alternative to scaling theory and discuss its
applicability to transport problems in both quantum and classical 2D systems.
