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

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 ${\sigma }_{free}/{\sigma }_{fixed}=3/2$, 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.

References in corpus (1)

• Total-energy-based prediction of a quasicrystal structure

Cited by in corpus (7)

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• Random tilings of high symmetry: II. Boundary conditions and numerical studies
• Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry