48 research outputs found
Distances on Rhombus Tilings
The rhombus tilings of a simply connected domain of the Euclidean plane are
known to form a flip-connected space (a flip is the elementary operation on
rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi).
Motivated by the study of a quasicrystal growth model, we are here interested
in better understanding how "tight" rhombus tiling spaces are flip-connected.
We introduce a lower bound (Hamming-distance) on the minimal number of flips to
link two tilings (flip-distance), and we investigate whether it is sharp. The
answer depends on the number n of different edge directions in the tiling:
positive for n=3 (dimer tilings) or n=4 (octogonal tilings), but possibly
negative for n=5 (decagonal tilings) or greater values of n. A standard proof
is provided for the n=3 and n=4 cases, while the complexity of the n=5 case led
to a computer-assisted proof (whose main result can however be easily checked
by hand).Comment: 18 pages, 9 figures, submitted to Theoretical Computer Science
(special issue of DGCI'09
Rapid Mixing for Lattice Colorings with Fewer Colors
We provide an optimally mixing Markov chain for 6-colorings of the square
lattice on rectangular regions with free, fixed, or toroidal boundary
conditions. This implies that the uniform distribution on the set of such
colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet
has a finite correlation length and a unique Gibbs measure at zero temperature.
Four and five are now the only remaining values of q for which it is not known
whether there exists a rapidly mixing Markov chain for q-colorings of the
square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA
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
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
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure