Critical temperature of non-interacting Bose gases on disordered lattices

For a non-interacting Bose gas on a lattice we compute the shift of the critical temperature for condensation when random-bond and on-site disorder are present. We evidence that the shift depends on the space dimensionality D and the filling fraction f. For D ->∞ (infinite range model), using results from the theory of random matrices, we show that the shift of the critical temperature is negative, depends on f, and vanishes only for large f. The connections with analogous results obtained for the spherical model are discussed. For D = 3 we find that, for large f, the critical temperature Tc is enhanced by disorder and that the relative shift does not appreciably depend on f; in contrast, for small f, Tc decreases, in agreement with the results obtained for a Bose gas in the continuum. We also provide numerical estimates for the shift of the critical temperature due to disorder induced in a non-interacting Bose gas by a bichromatic incommensurate potential.