  We consider near-critical two-dimensional statistical systems at phase
coexistence on the half plane with boundary conditions leading to the formation
of a droplet separating coexisting phases. General low-energy properties of
two-dimensional field theories are used in order to find exact analytic results
for one- and two-point correlation functions of both the energy density and
order parameter fields. The subleading finite-size corrections are also
computed and interpreted within an exact probabilistic picture in which
interfacial fluctuations are characterized by the probability density of a
Brownian excursion. The explicit analysis of the closed-form expression for
order parameter correlations reveals the long-ranged character of interfacial
correlations and their confinement within the interfacial region. The analysis
of correlations is then carried out in momentum space through the notion of
interface structure factor, which we extend to the case of systems bounded by a
flat wall. The presence of the wall and its associated entropic repulsion leads
to a specific term in the interface structure factor which we identify.
