Two - Dimensional Random Tilings of Largecodimensionm

Random tilings of the plane by rhombi are projections into the plane of corrugated two-dimensional surfaces in a higher dimensional hypercubic crystal. We consider tilings of 2D-fold symmetry projected from D-dimensional space. Thermodynamic properties depend on the relative numbers of tiles with diierent angles and ori-entations, and also on the boundary conditions imposed on the tiling. Relative tile numbers deene the average slope of the corrugated two-dimensional surface and hence the average phason strain. We study tilings with large codimension and xed boundaries inside a regular 2D-sided polygon with p rhombi on each side. For D ! 1 we show that the thermodynamic properties become independent of p. Furthermore, we argue that the boundary conditions become thermodynam-ically irrelevant in the large D limit. For p=1 we use exact enumeration for D up to 10, and \mean eld theory" arguments, to propose an upper bound for the random tiling entropy of log(2)=0.693 per tile. The entropy for nite D increases monotonically and reaches a limit slightly below this bound.