  We examine the one-dimensional Gross-Pitaevskii lattice at zero temperature
in the presence of uncorrelated disorder. We obtain analytical expressions for
the thermodynamic properties of the ground state field and compare them with
numerical simulations both in the weak and strong interaction regimes. We
analyze weak excitations above the ground state and compute the localization
properties of Bogoliubov-de Gennes modes. In the long-wavelength limit, these
modes delocalize in accordance with the extended nature of the ground state.
For strong interactions, we observe and derive a divergence of their
localization length at finite energy due to an effective correlated disorder
induced by the weak ground state field fluctuations. We derive effective strong
interaction field equations for the excitations and generalize to higher
dimensions.
