  The tilings of lozenges in 2 dimensions and of rhomboedra in 3 dimensions are
studied when they are constrained by fixed boundary conditions. We establish a
link between those conditions and free or periodic boundary ones: the entropy
is written as a functional integral which is treated via a saddle-point method.
Then we can exhibit the dominant states of the statistical ensemble of tilings
and show that they can display a strong structural inhomogeneity caused by the
boundary. This inhomogeneity is responsible for a difference of entropy between
the studied fixed boundary tilings and free boundary ones. This method uses a
representation of tilings by membranes embedded in a higher-dimensional
hypercubic lattice. It is illustrated in the case of 60 degree lozenge tilings.
