  We study the two-dimensional weakly repulsive Bose gas at zero temperature in
the presence of correlated disorder. Using large-scale simulations, we show
that the low-energy Bogoliubov cumulative density of states remains quadratic
up to a critical disorder strength, beyond which a power law with
disorder-dependent exponent $\beta<2$ sets in. We associate this threshold
behavior with the transition from superfluid to Bose glass, and compare the
resulting mean-field phase diagram with scaling laws and the Thomas-Fermi
percolation threshold of the mean-field density profile.
