  We study the transport and spectral properties of a non-Hermitian
one-dimensional disordered lattice, the diagonal matrix elements of which are
random complex variables taking both positive (loss) and negative (gain)
imaginary values: Their distribution is either the usual rectangular one or a
binary pair-correlated one possessing, in its Hermitian version, delocalized
states, and unusual transport properties. Contrary to the Hermitian case, all
states in our non-Hermitian system are localized. In addition, the eigenvalue
spectrum, for the binary pair-correlated case, exhibits an unexpected intricate
fractallike structure on the complex plane and with increasing non-Hermitian
disorder, the eigenvalues tend to coalesce in particular small areas of the
complex plane, a feature termed "eigenvalue condensation". Despite the strong
Anderson localization of all eigenstates, the system appears to exhibit
transport not by diffusion but by a new mechanism through sudden jumps between
states located even at distant sites. This seems to be a general feature of
open non-Hermitian random systems. The relation of our findings to recent
experimental results is also discussed.
