  We present a family of finite-volume criteria which cover the regime of
exponential decay for the fractional moments of Green functions of operators
with random potentials. Such decay is a technically convenient characterization
of localization for it is known to imply spectral localization, absence of
level repulsion, dynamical localization and a related condition which plays a
significant role in the quantization of the Hall conductance in two-dimensional
Fermi gases. The constructive criteria also preclude fast power-law decay of
the Green functions at mobility edges.
