Two-dimensional random tilings of large codimension: new progress

Details Two-dimensional random tilings of large codimension: new progress Two-dimensional random tilings of large codimension: new progress

1999-12-15

arxiv.org/abs/cond-mat/9912275v1

Physics

Condensed Matter

Statistical Mechanics

Proceedings of the 7th International Conference on Quasicrystals (ICQ7, Stuttgart), 4 pages, 4 figures

Scientific paper

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 [ICQ6] that, in this limit, the thermodynamic properties of the tiling become 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.

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