917 research outputs found
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.
Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions
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
, 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
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
An aperiodic hexagonal tile
We show that a single prototile can fill space uniformly but not admit a
periodic tiling. A two-dimensional, hexagonal prototile with markings that
enforce local matching rules is proven to be aperiodic by two independent
methods. The space--filling tiling that can be built from copies of the
prototile has the structure of a union of honeycombs with lattice constants of
, where sets the scale of the most dense lattice and takes all
positive integer values. There are two local isomorphism classes consistent
with the matching rules and there is a nontrivial relation between these
tilings and a previous construction by Penrose. Alternative forms of the
prototile enforce the local matching rules by shape alone, one using a
prototile that is not a connected region and the other using a
three--dimensional prototile.Comment: 32 pages, 24 figures; submitted to Journal of Combinatorial Theory
Series A. Version 2 is a major revision. Parts of Version 1 have been
expanded and parts have been moved to a separate article (arXiv:1003.4279
Enumeration of octagonal tilings
Random tilings are interesting as idealizations of atomistic models of
quasicrystals and for their connection to problems in combinatorics and
algorithms. Of particular interest is the tiling entropy density, which
measures the relation of the number of distinct tilings to the number of
constituent tiles. Tilings by squares and 45 degree rhombi receive special
attention as presumably the simplest model that has not yet been solved exactly
in the thermodynamic limit. However, an exact enumeration formula can be
evaluated for tilings in finite regions with fixed boundaries. We implement
this algorithm in an efficient manner, enabling the investigation of larger
regions of parameter space than previously were possible. Our new results
appear to yield monotone increasing and decreasing lower and upper bounds on
the fixed boundary entropy density that converge toward S = 0.36021(3)
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
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