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

• Main Authors: M. Widom, R. Mosseri, N. Destainville, F. Bailly
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 1989
• Subjects: Arctic; Linde
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.545.9799; http://euler.phys.cmu.edu/widom/pubs/PDF/jsp109_2002_p945.pdf

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 sfree/sfixed=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. KEY WORDS: Random tilings; integer partitions; configurational entropy; boundary effects; transition matrix Monte Carlo algorithms.