  We investigate the behavior of the quasi-particle energy gap near quantum
phase transitions in a two-dimensional disordered boson Hubbard model at a
commensurate filling. Via Monte Carlo simulations of ensembles with fixed
numbers of particles, we observe the behavior of the gap as a function of the
tuning parameter for various strength of diagonal disorder. For weak disorder,
we find that gapped Mott insulating phase is sustained up to the transition
point and disappears only in a superfluid, strongly supporting a direct
Mott-insulator-to-superfluid transition. Bose glass behavior, insulating with
vanishing gap, appears only when the strength of disorder is bigger than a
critical value.
