We study the localization properties of a test dipole feeling the disordered potential induced by dipolar impurities trapped at random positions in an optical lattice. This random potential is marked by correlations which are a convolution of short-range and long-range ones. We show that when short-range correlations are dominant, extended states can appear in the spectrum. Introducing long-range correlations, the extended states, if any, are wiped out and localization is restored over the whole spectrum. Moreover, long-range correlations can either increase or decrease the localization length at the center of the band, which indicates a richer behavior than previously predicted.
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