  We sudy the impact of a weak random potential with a Gaussian correlation
function on the thermodynamics of a two-dimensional (2D) dipolar bosonic gas.
Analytical expressions for the quantum depletion, anomalous density, the ground
state energy, the equation of state and the sound velocity are derived in the
roton regime within the framework of the Bogoliubov theory. Surprisingly, we
find that the condensate depletion and the anomalous density are comparable.
The structure factor and the superfluid fraction are also obtained analytically
and numerically. We show that these quantities acquire dramatically modified
profiles when the roton is close to zero yielding the transition to an unusual
quantum state.
