We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.
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Strictly speaking, should this fan be exactly regular, there would be multiple intersections, for example at its very center To avoid this difficulty, each family must be slightly shifted by a random distance much smaller than the interline separation. Such shifts have been performed in the Fig. DeBruijn4.2, even though it might not be clear because of resolution
A rectangular kite has two right angles between its unequal sides
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Widom, M., Destainville, N., Mosseri, R. et al. Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies. J Stat Phys 120, 837–873 (2005). https://doi.org/10.1007/s10955-005-6998-x
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DOI: https://doi.org/10.1007/s10955-005-6998-x