  A new paradigm of Anderson localization caused by correlations in the
long-range hopping along with uncorrelated on-site disorder is considered which
requires a more precise formulation of the basic localization-delocalization
principles. A new class of random Hamiltonians with translation-invariant
hopping integrals is suggested and the localization properties of such models
are established both in the coordinate and in the momentum spaces alongside
with the corresponding level statistics. Duality of translation-invariant
models in the momentum and coordinate space is uncovered and exploited to find
a full localization-delocalization phase diagram for such models. The crucial
role of the spectral properties of hopping matrix is established and a new
matrix inversion trick is suggested to generate a one-parameter family of
equivalent localization/delocalization problems. Optimization over the free
parameter in such a transformation together with the
localization/delocalization principles allows to establish exact bounds for the
localized and ergodic states in long-range hopping models. When applied to the
random matrix models with deterministic power-law hopping this transformation
allows to confirm localization of states at all values of the exponent in
power-law hopping and to prove analytically the symmetry of the exponent in the
power-law localized wave functions.
