  We use the Bogoliubov theory of Bose-Einstein condensation to study the
properties of dipolar particles (atoms or molecules) confined in a uniform
two-dimensional geometry at zero temperature. We find equilibrium solutions to
the dipolar Gross-Pitaevskii equation and the Bogoliubov-de Gennes equations.
Using these solutions we study the effects of quantum fluctuations in the
system, particularly focussing on the instability point, where the roton
feature in the excitation spectrum touches zero. Specifically, we look at the
behaviour of the noncondensate density, the phase fluctuations, and the density
fluctuations in the system. Near the instability, the density-density
correlation function shows a particularly intriguing oscillatory behaviour.
Higher order correlation functions display a distinct hexagonal lattice pattern
formation, demonstrating how an observation of broken symmetry can emerge from
a translationally symmetric quantum state.
