Two-dimensional random tilings of large codimension: new progress

Two-dimensional random tilings of rhombi can be seen as projections of two- dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a D-dimensional space. We study the limiting case, when the quantity D, and therefore the number of different species of tiles, become large. We had previously demonstrated [M . Widom, N. Destainville, R. Mosseri, F. Bailly, in: Proceedings of the Six th International Conference on Quasicrystals, World Scientific, Singapore, 1997.] that, in this limit, the thermodynamic properties of the tiling beco me independent of the boundary conditions. The exact value of the limiting entropy and finite D corrections remain open questions. Here, we develop a mean-field theory, which uses an iterative description of the tilings based on an analogy with avoiding oriented walks on a random tiling. We compare the quantities so-obtained with numerical calculations. We also discuss the role of spatial correlations. (C) 2000 Elsevier Science B.V. All lights re served.