1,733 research outputs found
Slopes of Tilings
We study here slopes of periodicity of tilings. A tiling is of slope if it is
periodic along direction but has no other direction of periodicity. We
characterize in this paper the set of slopes we can achieve with tilings, and
prove they coincide with recursively enumerable sets of rationals.Comment: Journ\'ees Automates Cellulaires 2010, Turku : Finland (2010
The Ammann-Beenker tilings revisited
This paper introduces two tiles whose tilings form a one-parameter family of
tilings which can all be seen as digitization of two-dimensional planes in the
four-dimensional Euclidean space. This family contains the Ammann-Beenker
tilings as the solution of a simple optimization problem.Comment: 7 pages, 4 figure
Numerical entropy and phason elastic constants of plane random tilings with any 2D-fold symmetry
We perform Transition matrix Monte Carlo simulations to evaluate the entropy
of rhombus tilings with fixed polygonal boundaries and 2D-fold rotational
symmetry. We estimate the large-size limit of this entropy for D=4 to 10. We
confirm analytic predictions of N. Destainville et al., J. Stat. Phys. 120, 799
(2005) and M. Widom et al., J. Stat. Phys. 120, 837 (2005), in particular that
the large size and large D limits commute, and that entropy becomes insensible
to size, phason strain and boundary conditions at large D. We are able to infer
finite D and finite size scalings of entropy. We also show that phason elastic
constants can be estimated for any D by measuring the relevant perpendicular
space fluctuations.Comment: Accepted for publication in Eur. Phys. J.
Local Rules for Computable Planar Tilings
Aperiodic tilings are non-periodic tilings characterized by local
constraints. They play a key role in the proof of the undecidability of the
domino problem (1964) and naturally model quasicrystals (discovered in 1982). A
central question is to characterize, among a class of non-periodic tilings, the
aperiodic ones. In this paper, we answer this question for the well-studied
class of non-periodic tilings obtained by digitizing irrational vector spaces.
Namely, we prove that such tilings are aperiodic if and only if the digitized
vector spaces are computable.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Random skew plane partitions with a piecewise periodic back wall
Random skew plane partitions of large size distributed according to an
appropriately scaled Schur process develop limit shapes. In the present work we
consider the limit of large random skew plane partitions where the inner
boundary approaches a piecewise linear curve with non-lattice slopes,
describing the limit shape and the local fluctuations in various regions. This
analysis is fairly similar to that in [OR2], but we do find some new behavior.
For instance, the boundary of the limit shape is now a single smooth (not
algebraic) curve, whereas the boundary in [OR2] is singular. We also observe
the bead process introduced in [B] appearing in the asymptotics at the top of
the limit shape.Comment: 24 pages. This version to appear in Annales Henri Poincar
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