Discrete Models: Combinatorics, Computation, and Geometry, DMCCG 2001
Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.)
DMTCS Conference Volume AA (2001), pp. 2342
author:  Joakim Linde, Cristopher Moore and Mats G. Nordahl 

title:  An nDimensional Generalization of the Rhombus Tiling 
keywords:  Tilings, Discrete Dynamical Systems, Quasicrystals 
abstract: 
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
on regions of size
(n+2)
log L)
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 twodimensional
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.

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reference:  Joakim Linde and Cristopher Moore and Mats G. Nordahl (2001), An nDimensional Generalization of the Rhombus Tiling, in Discrete Models: Combinatorics, Computation, and Geometry, DMCCG 2001, Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AA, pp. 2342 
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