930 research outputs found
Combinatorial Space Tiling
The present article studies combinatorial tilings of Euclidean or spherical
spaces by polytopes, serving two main purposes: first, to survey some of the
main developments in combinatorial space tiling; and second, to highlight some
new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science
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
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
Scaling Dynamics of Domain Walls in the Cubic Anisotropy Model
We have investigated the dynamics of domain walls in the cubic anisotropy
model. In this model a global O(N) symmetry is broken to a set of discrete
vacua either on the faces, or vertices of a (hyper)cube. We compute the scaling
exponents for in two dimensions on grids of points and
compare them to the fiducial model of symmetry breaking. Since the model
allows for wall junctions lattice structures are locally stable and
modifications to the standard scaling law are possible. However, we find that
since there is no scale which sets the distance between walls, the walls appear
to evolve toward a self-similar regime with .Comment: 16 pages, 12 figure
Phason elasticity of a three-dimensional quasicrystal: transfer-matrix method
We introduce a new transfer matrix method for calculating the thermodynamic
properties of random-tiling models of quasicrystals in any number of
dimensions, and describe how it may be used to calculate the phason elastic
properties of these models, which are related to experimental measurables such
as phason Debye-Waller factors, and diffuse scattering wings near Bragg peaks.
We apply our method to the canonical-cell model of the icosahedral phase,
making use of results from a previously-presented calculation in which the
possible structures for this model under specific periodic boundary conditions
were cataloged using a computational technique. We give results for the
configurational entropy density and the two fundamental elastic constants for a
range of system sizes. The method is general enough allow a similar calculation
to be performed for any other random tiling model.Comment: 38 pages, 3 PostScript figures, self-expanding uuencoded compressed
tar file, LaTeX using RevTeX macros and epsfig.st
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