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

• Main Authors: Widom, M., Mosseri, R., Destainville, Nicolas, Bailly, F.
• Other Authors: Groupe de Physique des Solides (GPS), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Groupe de Physique Théorique (LPQ), Laboratoire de Physique Quantique (LPQ), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de physique des solides et de cristallogénèse (LPSC), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Centre National de la Recherche Scientifique (CNRS)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2002
• Subjects: [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]; Arctic; Linde
• Online Access: https://hal.science/hal-00012894

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.