13 research outputs found
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
Estimating the asymptotics of solid partitions
We study the asymptotic behavior of solid partitions using transition matrix
Monte Carlo simulations. If denotes the number of solid partitions of
an integer , we show that . This shows clear deviation from the value ,
attained by MacMahon numbers , that was conjectured to hold for solid
partitions as well. In addition, we find estimates for other sub-leading terms
in . In a pattern deviating from the asymptotics of line and plane
partitions, we need to add an oscillatory term in addition to the obvious
sub-leading terms. The period of the oscillatory term is proportional to
, the natural scale in the problem. This new oscillatory term might
shed some insight into why partitions in dimensions greater than two do not
admit a simple generating function.Comment: 21 pages, 8 figure
Fast domino tileability
Domino tileability is a classical problem in Discrete Geometry, famously
solved by Thurston for simply connected regions in nearly linear time in the
area. In this paper, we improve upon Thurston's height function approach to a
nearly linear time in the perimeter.Comment: Appeared in Discrete Comput. Geom. 56 (2016), 377-39
Flip invariance for domino tilings of three-dimensional regions with two floors
We investigate tilings of cubiculated regions with two simply connected
floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected
component for such tilings, and provide an algebraic invariant that "almost"
characterizes the flip connected components of such regions, in a sense that we
discuss in the paper. We also introduce a new local move, the trit, which,
together with the flip, connects the space of domino tilings when the two
floors are identical.Comment: 33 pages, 34 figures, 2 tables. We updated the reference lis
Local dimer dynamics in higher dimensions
We consider local dynamics of the dimer model (perfect matchings) on
hypercubic boxes . These consist of successively switching the dimers
along alternating cycles of prescribed (small) lengths. We study the
connectivity properties of the dimer configuration space equipped with these
transitions. Answering a question of Freire, Klivans, Milet and Saldanha, we
show that in three dimensions any configuration admits an alternating cycle of
length at most 6. We further establish that any configuration on
features order alternating cycles of length at most . We also
prove that the dynamics of dimer configurations on the unit hypercube of
dimension is ergodic when switching alternating cycles of length at most
. Finally, in the planar but non-bipartite case, we show that
parallelogram-shaped boxes in the triangular lattice are ergodic for switching
alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon
and R\'emila, which also uses 8-cycles. None of our proofs make reference to
height functions.Comment: 14 pages, 4 figure
Cubist Algebras
We construct algebras from rhombohedral tilings of Euclidean space obtained
as projections of certain cubical complexes. We show that these `Cubist
algebras' satisfy strong homological properties, such as Koszulity and
quasi-heredity, reflecting the combinatorics of the tilings. We construct
derived equivalences between Cubist algebras associated to local mutations in
tilings. We recover as a special case the Rhombal algebras of Michael Peach and
make a precise connection to weight 2 blocks of symmetric groups
Domino tilings of three-dimensional regions: flips, trits and twists
In this paper, we consider domino tilings of regions of the form , where is a simply connected planar region and . It turns out that, in nontrivial examples, the set of such
tilings is not connected by flips, i.e., the local move performed by removing
two adjacent dominoes and placing them back in another position. We define an
algebraic invariant, the twist, which partially characterizes the connected
components by flips of the space of tilings of such a region. Another local
move, the trit, consists of removing three adjacent dominoes, no two of them
parallel, and placing them back in the only other possible position: performing
a trit alters the twist by . We give a simple combinatorial formula for
the twist, as well as an interpretation via knot theory. We prove several
results about the twist, such as the fact that it is an integer and that it has
additive properties for suitable decompositions of a region.Comment: 38 pages, 17 figures. Most of this material is also covered in the
first author's Ph.D. thesis (arXiv:1503.04617
Faculty Publications and Creative Works 2001
One of the ways in which we recognize our faculty at the University of New Mexico is through Faculty Publications & Creative Works. An annual publication, it highlights our faculty\u27s scholarly and creative activities and achievements and serves as a compendium of UNM faculty efforts during the 2001 calendar year. Faculty Publications & Creative Works strives to illustrate the depth and breadth of research activities performed throughout our University\u27s laboratories, studios and classrooms. We believe that the communication of individual research is a significant method of sharing concepts and thoughts and ultimately inspiring the birth of new ideas. In support of this, UNM faculty during 2001 produced over 2,299* works, including 1,685 scholarly papers and articles, 69 books, 269 book chapters, 184 reviews, 86 creative works and 6 patented works. We are proud of the accomplishments of our faculty which are in part reflected in this book, which illustrates the diversity of intellectual pursuits in support of research and education at the University of New Mexico