Entropy and boundary conditions in random rhombus tilings

The tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied when they are constrained by Bred 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. 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 rhombus tilings.