Two-dimensional random tilings of large codimension: new progress

N. Destainville; M. Widom; R. Mosseri; F. Bailly

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Two-dimensional random tilings of large codimension: new progress

N. Destainville·M. Widom·R. Mosseri·F. Bailly

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Abstract

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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N. Destainville, M. Widom, R. Mosseri, F. Bailly. 1999-12-15. Two-dimensional random tilings of large codimension: new progress. https://arxiv.org/abs/cond-mat/9912275

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