Quasicrystals cooling: from random tilings to aperiodic tilings – QuasiCool

We rely on computer simulations to conjecture results, as well as to provide theoretical proof for these results (looking at the real dynamics on some examples helps to intuitively catch its main properties and then to formalize them).

Promising simulations, sketched theoretical results

Theorical results

1 conference proceeding, 1 book chapter

The full title of the proposal QuasiCool is “Quasicrystals cooling: from random tilings to ape-
riodic tilings”. This is a four-year project grouping together seven researchers in combinatorics,
probability theory and dynamical systems, with a Ph.D. grant being also asked (to be started
the second year). The coordinator of this project, Thomas Fernique, is a CNRS researcher in
computer science since october 2008. The aim of this project is to understand from a theoretical
viewpoint, namely in terms of tilings, the structure and the growth of quasicrystals.
Quasicrystals are non-periodic but ordered materials discovered by D. Shechtman in 1982 (what
earned him the last chemistry Nobel prize). They are now generally obtained (as crystals) by
slowly cooling a melt. From a theoretical viewpoint, the problem is to understand which qua-
sicrystalline structures can be obtained, in order to classify them as does the Bravais-Fedorov
classification of crystalline structures.
Tilings are covering of a space without overlap by compacts called tiles. In the early 60’s, it
has been discovered that some non-periodic tilings of the plane can be completely characterized
only by specifying the way tiles can be neighboor, that is, by local constraints. Such tilings
are said to be aperiodic. The connection with quasicrystals was quickly done after they were
discovered, with local constraints modeling finite range interactions. However, the existence of
aperiodic tilings does not ensure that they can be really (i.e., easily) obtained. This led scien-
tists to study an alternative model of quasicrystals, namely random tilings. This model however
recently turned out to be limited, as some real quasicrystals look much more like aperiodic than
random tilings.
That is where our project comes into the picture. We model the cooling of quasicrystals by a
process where local transformations called flips are stochastically performed on an initial random
tiling, with a flip being all the more probable as the resulting tiling locally looks more aperiodic.
Understanding the convergence rate – hence the realism – of this process is a major objective
of our project. The two other major objectives of our project are strongly related: classify
aperiodic tilings (the “target” of the process) and understand random tilings (the initial point
of the process).