An n-Dimensional Generalization of the Rhombus Tiling

• Main Authors: Linde, Joakim, Moore, Cristopher, Nordahl, Mats G.
• Other Authors: Chalmers University of Technology Göteborg, UNM Computer Science department New Mexico, The University of New Mexico Albuquerque, Santa Fe Institute, Cori, Robert and Mazoyer, Jacques and Morvan, Michel and Mosseri, Rémy
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2001
• Subjects: Tilings; Discrete Dynamical Systems; Quasicrystals; [INFO]Computer Science [cs]; [INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG]; [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]; [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]; Arctic
• Online Access: https://hal.inria.fr/hal-01182973; https://hal.inria.fr/hal-01182973/document; https://hal.inria.fr/hal-01182973/file/dmAA0102.pdf

International audience Several classic tilings, including rhombuses and dominoes, possess height functions which allow us to 1) prove ergodicity and polynomial mixing times for Markov chains based on local moves, 2) use coupling from the past to sample perfectly random tilings, 3) map the statistics of random tilings at large scales to physical models of random surfaces, and and 4) are related to the "arctic circle"' phenomenon.However, few examples are known for which this approach works in three or more dimensions.Here we show that the rhombus tiling can be generalized to n-dimensional tiles for any $n ≥ 3$. For each $n$, we show that a certain local move is ergodic, and conjecture that it has a mixing time of $O(L^{n+2} log L)$ on regions of size $L$. For $n=3$, the tiles are rhombohedra, and the local move consists of switching between two tilings of a rhombic dodecahedron.We use coupling from the past to sample random tilings of a large rhombic dodecahedron, and show that arctic regions exist in which the tiling is frozen into a fixed state.However, unlike the two-dimensional case in which the arctic region is an inscribed circle, here it seems to be octahedral.In addition, height fluctuations between the boundary of the region and the center appear to be constant rather than growing logarithmically.We conjecture that this is because the physics of the model is in a "smooth" phase where it is rigid at large scales, rather than a "rough" phase in which it is elastic.