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

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

2002-01-17

arxiv.org/abs/cond-mat/0201309v2

J. Stat. Phys. 109, 945 (2002)

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

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.

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