5 research outputs found
Stochastic Flips on Dimer Tilings
International audienceThis paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called , which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a bound, where is the number of tiles of the tiling. We prove a upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case
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
Stochastic Flips on Dimer Tilings
This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called , which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a bound, where is the number of tiles of the tiling. We prove a upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case