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Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions

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

Three-dimensional integer partitions provide a convenient representation of codimension-one three-dimensional random rhombus tilings. Calculating the entropy for such a model is a notoriously difficult problem. We apply transition matrix Monte Carlo simulations to evaluate their entropy with high precision. We consider both free- and fixed-boundary tilings. Our results suggest that the ratio of free- and fixed-boundary entropies is $σ_{free}/σ_{fixed}=3/2$, and can be interpreted as the ratio of the volumes of two simple, nested, polyhedra. This finding supports a conjecture by Linde, Moore and Nordahl concerning the ``arctic octahedron phenomenon'' in three-dimensional random tilings.

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M. Widom, R. Mosseri, N. Destainville, F. Bailly. 2003-01-14. Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions. https://doi.org/10.1023/a%3A1020464224385

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