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

• Main Authors: Widom, M., Mosseri, R., Destainville, N., Bailly, F.
• Format: Text
• Language: unknown
• Published: arXiv 2002
• Subjects: Statistical Mechanics cond-mat.stat-mech; FOS Physical sciences; Arctic; Linde
• Online Access: https://dx.doi.org/10.48550/arxiv.cond-mat/0201309; https://arxiv.org/abs/cond-mat/0201309

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.