285 research outputs found
Around the Domino Problem â Combinatorial Structures and Algebraic Tools
Given a finite set of square tiles, the domino problem is the question of whether is it possible to tile the plane using these tiles. This problem is known to be undecidable in the planar case, and is strongly linked to the question of the periodicity of the tiling. In this thesis we look at this problem in two different ways: first, we look at the particular case of low complexity tilings and second we generalize it to more general structures than the plane, groups.
A tiling of the plane is said of low complexity if there are at most mn rectangles of size m Ă n appearing in it. Nivat conjectured in 1997 that any such tiling must be periodic, with the consequence that the domino problem would be decidable for low complexity tilings. Using algebraic tools introduced by Kari and Szabados, we prove a generalized version of Nivatâs conjecture for a particular class of tilings (a subclass of what is called of algebraic subshifts). We also manage to prove that Nivatâs conjecture holds for uniformly recurrent tilings, with the consequence that the domino problem is indeed decidable for low-complexity tilings.
The domino problem can be formulated in the more general context of Cayley graphs of groups. In this thesis, we develop new techniques allowing to relate the Cayley graph of some groups with graphs of substitutions on words. A first technique allows us to show that there exists both strongly periodic and weakly-but-not-strongly aperiodic tilings of the Baumslag-Solitar groups BS(1, n). A second technique is used to show that the domino problem is undecidable for surface groups. Which provides yet another class of groups verifying the conjecture saying that the domino problem of a group is decidable if and only if the group is virtually free
Perfect sampling algorithm for Schur processes
We describe random generation algorithms for a large class of random
combinatorial objects called Schur processes, which are sequences of random
(integer) partitions subject to certain interlacing conditions. This class
contains several fundamental combinatorial objects as special cases, such as
plane partitions, tilings of Aztec diamonds, pyramid partitions and more
generally steep domino tilings of the plane. Our algorithm, which is of
polynomial complexity, is both exact (i.e. the output follows exactly the
target probability law, which is either Boltzmann or uniform in our case), and
entropy optimal (i.e. it reads a minimal number of random bits as an input).
The algorithm encompasses previous growth procedures for special Schur
processes related to the primal and dual RSK algorithm, as well as the famous
domino shuffling algorithm for domino tilings of the Aztec diamond. It can be
easily adapted to deal with symmetric Schur processes and general Schur
processes involving infinitely many parameters. It is more concrete and easier
to implement than Borodin's algorithm, and it is entropy optimal.
At a technical level, it relies on unified bijective proofs of the different
types of Cauchy and Littlewood identities for Schur functions, and on an
adaptation of Fomin's growth diagram description of the RSK algorithm to that
setting. Simulations performed with this algorithm suggest interesting limit
shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints
present in v2
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
Domino tilings and related models: space of configurations of domains with holes
We first prove that the set of domino tilings of a fixed finite figure is a
distributive lattice, even in the case when the figure has holes. We then give
a geometrical interpretation of the order given by this lattice, using (not
necessarily local) transformations called {\em flips}.
This study allows us to formulate an exhaustive generation algorithm and a
uniform random sampling algorithm.
We finally extend these results to other types of tilings (calisson tilings,
tilings with bicolored Wang tiles).Comment: 17 pages, 11 figure
Pinwheel patterns and powder diffraction
Pinwheel patterns and their higher dimensional generalisations display
continuous circular or spherical symmetries in spite of being perfectly
ordered. The same symmetries show up in the corresponding diffraction images.
Interestingly, they also arise from amorphous systems, and also from regular
crystals when investigated by powder diffraction. We present first steps and
results towards a general frame to investigate such systems, with emphasis on
statistical properties that are helpful to understand and compare the
diffraction images. We concentrate on properties that are accessible via an
alternative substitution rule for the pinwheel tiling, based on two different
prototiles. Due to striking similarities, we compare our results with the toy
model for the powder diffraction of the square lattice.Comment: 7 pages, 4 figure
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
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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