64 research outputs found
Flip dynamics in octagonal rhombus tiling sets
We investigate the properties of classical single flip dynamics in sets of
two-dimensional random rhombus tilings. Single flips are local moves involving
3 tiles which sample the tiling sets {\em via} Monte Carlo Markov chains. We
determine the ergodic times of these dynamical systems (at infinite
temperature): they grow with the system size like ;
these dynamics are rapidly mixing. We use an inherent symmetry of tiling sets
and a powerful tool from probability theory, the coupling technique. We also
point out the interesting occurrence of Gumbel distributions.Comment: 5 Revtex pages, 4 figures; definitive versio
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
Molecular random tilings as glasses
We have recently shown [Blunt et al., Science 322, 1077 (2008)] that
p-terphenyl-3,5,3',5'-tetracarboxylic acid adsorbed on graphite self-assembles
into a two-dimensional rhombus random tiling. This tiling is close to ideal,
displaying long range correlations punctuated by sparse localised tiling
defects. In this paper we explore the analogy between dynamic arrest in this
type of random tilings and that of structural glasses. We show that the
structural relaxation of these systems is via the propagation--reaction of
tiling defects, giving rise to dynamic heterogeneity. We study the scaling
properties of the dynamics, and discuss connections with kinetically
constrained models of glasses.Comment: 5 pages, 5 figure
Broken symmetry and the variation of critical properties in the phase behaviour of supramolecular rhombus tilings
The degree of randomness, or partial order, present in two-dimensional
supramolecular arrays of isophthalate tetracarboxylic acids is shown to vary
due to subtle chemical changes such as the choice of solvent or small
differences in molecular dimensions. This variation may be quantified using an
order parameter and reveals a novel phase behaviour including random tiling
with varying critical properties as well as ordered phases dominated by either
parallel or non-parallel alignment of neighbouring molecules, consistent with
long-standing theoretical studies. The balance between order and randomness is
driven by small differences in the intermolecular interaction energies, which
we show, using numerical simulations, can be related to the measured order
parameter. Significant variations occur even when the energy difference is much
less than the thermal energy highlighting the delicate balance between entropic
and energetic effects in complex self-assembly processes
Mixing times of lozenge tiling and card shuffling Markov chains
We show how to combine Fourier analysis with coupling arguments to bound the
mixing times of a variety of Markov chains. The mixing time is the number of
steps a Markov chain takes to approach its equilibrium distribution. One
application is to a class of Markov chains introduced by Luby, Randall, and
Sinclair to generate random tilings of regions by lozenges. For an L X L region
we bound the mixing time by O(L^4 log L), which improves on the previous bound
of O(L^7), and we show the new bound to be essentially tight. In another
application we resolve a few questions raised by Diaconis and Saloff-Coste, by
lower bounding the mixing time of various card-shuffling Markov chains. Our
lower bounds are within a constant factor of their upper bounds. When we use
our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an
O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov
chain for linear extensions.Comment: 39 pages, 8 figure
Local statistics of lattice dimers
We show how to compute the probability of any given local configuration in a
random tiling of the plane with dominos. That is, we explicitly compute the
measures of cylinder sets for the measure of maximal entropy on the space
of tilings of the plane with dominos.
We construct a measure on the set of lozenge tilings of the plane, show
that its entropy is the topological entropy, and compute explicitly the
-measures of cylinder sets.
As applications of these results, we prove that the translation action is
strongly mixing for and , and compute the rate of convergence to
mixing (the correlation between distant events). For the measure we
compute the variance of the height function.Comment: 27 pages, 6 figure
Landau levels in quasicrystals
Two-dimensional tight-binding models for quasicrystals made of plaquettes
with commensurate areas are considered. Their energy spectrum is computed as a
function of an applied perpendicular magnetic field. Landau levels are found to
emerge near band edges in the zero-field limit. Their existence is related to
an effective zero-field dispersion relation valid in the continuum limit. For
quasicrystals studied here, an underlying periodic crystal exists and provides
a natural interpretation to this dispersion relation. In addition to the slope
(effective mass) of Landau levels, we also study their width as a function of
the magnetic flux per plaquette and identify two fundamental broadening
mechanisms: (i) tunneling between closed cyclotron orbits and (ii) individual
energy displacement of states within a Landau level. Interestingly, the typical
broadening of the Landau levels is found to behave algebraically with the
magnetic field with a nonuniversal exponent.Comment: 14 pages, 9 figure
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