  [Remark: Much more natural and important models are studied in
arXiv:1405.2083. Furthermore, the physical picture given there is much more
complete.] The quantum N-portrait -- black holes pictured as Bose-Einstein
condensates -- is studied for a special class of models in arbitrary
dimensions. In the presence of extra dimensions, the depletion rate is shown to
be significantly enhanced in these models as compared to standard
semi-classical results. Consequences for large as well as micro condensates are
discussed. Speculations are made on a possible connection of higher-dimensional
graviton condensates, strings and branes.
