  We study the weakly disordered Bose-Hubbard model on a cubic lattice through
a one-loop renormalization group analysis of the corresponding effective field
theory which is explicitly derived by combining a strong-coupling expansion
with a replica average over the disorder. The method is applied not only to
generic uncorrelated on-site disorder but also to simultaneous hopping disorder
correlated with the differences of adjacent disorder potentials. Such
correlations are inherent in fine-grained optical speckle potentials used as a
source of disorder in optical lattice experiments. As a result of strong
coupling, the strength of the replica mixing disorder vertex, responsible for
the emergence of a Bose glass, crucially depends on the chemical potential and
the Hubbard repulsion and vanishes to leading order in the disorder at
commensurate boson fillings. As a consequence, at such fillings a direct
transition between the Mott-insulator and the superfluid in the presence of
disorder cannot be excluded on the basis of a one-loop calculation. At
incommensurate fillings, at a certain length scale, the Mott insulator will
eventually become unstable towards the formation of a Boss glass. Phase
diagrams as a function of the microscopic parameters are presented and the
finite-size crossover between the Mott-insulating state and the Bose glass is
analyzed.
