The arctic octahedron phenomenon in three-dimensional codimension-one rhombohedral tilings

We calculate the configurational entropy of codimension-one three-dimensional random rhombus tilings. We use three-dimensional integer partitions to represent these tilings. We apply transition matrix Monte Carlo simulations to evaluate their entropy with high precision. We explore free- as well as fixed-boundary conditions and our numerical results suggest that the ratio of free- and fixed-boundary entropies is σfree/σfixed=3/2, and can be interpreted as the ratio of the volumes of two simple, nested, polyhedra. This ratio confirms a conjecture by Linde et al. concerning the ‘arctic octahedron phenomenon’ in three-dimensional codimension-one random tilings.