Projet de Th`ese
Doctorant Ilya Galanov
Directeur de th`ese Thomas Fernique
Laboratoire LIPN (Paris 13)
Intitul´e du projet Self-assembly of tilings
We call tile a subset of the plane which is homeomorphic to a close ball and tiling a covering of the plane by interior-disjoint tiles. A set of tiles is then said to beaperiodic if
• there is only finitely many tiles up to isometries;
• there exists a tiling of the plane by these tiles;
• any tiling of the plane by these tiles is non-periodic.
Such tile sets exist (the first one has been found in the 60’s) and are used to model qua- sicrystals, which are non-periodic materials whose stability seems nevertheless be enforced only by short range energetic interactions. Indeed, aperiodic tiles enforce non-periodicity only via local constraints (on the way neighboor tiles can fit together).
Although an aperiodic tile set can, by definition, form a non-periodic tiling, it seems hard to naturally build such a tiling, that is, just by adding tiles one by one - one speaks about self-assembly. Indeed, any aperiodic tile set can form deceptions, that are covering of a finite region of the plane by interior-disjoint tile which cannot be extended to tiling of the whole plane ([1]). Some interesting solutions, based on priority rules which add constraint on the order the tiles are added, have been proposed (see [3, 6]), but this often leads to very complicated tile sets, hence rather unrealistic in order to model quasicrystals.
A first axe of this Ph.D. should be to understand more deeply deceptions. Indeed, the proof that deceptions always exist is based on covering of very specific regions (not at all convex, for example). Moreover, one only knows that the probability to obtain a deception by adding tile one is positive. But how large it is? Are deceptions a rare event or could we hope to add many tiles before getting stucked onto a deception? Formally, how the propor- tion of the interior-disjoint coverings of a ball varies with the diameter of this ball? Does it converge towards zero? At which rate?
But aperiodic tile set are not the unique way to model quasicrystals. An interesting alternative is indeed provided byrandom tilings. The general idea is that a quasicrystalline structure, instead of being deterministically enforced by aperiodic tile set, could appear sta- tistically. To illustrate this, consider all the way you can put unit cubes in a n×n×n box.
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Up to a projection, this can be seen as a tiling (namely a dimer tiling of a hexagonal region).
Then, pick uniformly at random one of these tilings. It turns out that, with a probability which tends to 1 with n, the chosen tiling shows a circular region outside of wich tiles will be ordered in a prescribed directions (they are said to be frozen). This is referred to as the arctic circle phenomenon [4] and show how a macroscopic property (the circle) can just statistically arise. Another example is given in [5]: integer-valued function defined on the d-dimensional grid of size n and constrained to take close values on neighboor vertices and value zero on the boundary are stastistically ”flat” (for big enough d), that is, take values close to zero almost everywhere with a probability which tends towards 1 with n.
It is an open question whether tilings with (at least rough) quasicrystalline structure can statistically appear as in the previous examples. This however deals only with the property of random tilings. But a first question should be: can random tilings be obtain (more eas- ily than aperiodic ones) by a self-assembly process (that is, by adding tiles one at once)?
Promising simulations have been done [7, 2], but no rigorous result has yet been proven.
This could be a second axe of this Ph.D: can random tilings be self-assembled (beyond sim- ulations) in a realistic way?
This project has clear physical motivations, but since the goal is to obtain rigorous mathematical results, there is a challenging trade-off between realism and tractability. It should be already very interesting to consider particular tile sets, as the celebrated Robinson, Penrose or Kari- ˇCulik ones.
References
[1] S. Dworkin, J.-I Shieh, Deceptions in quasicrystal growth, Comm. Math. Phys. 168 (1995), pp. 337–352
[2] V. Elser, D. Joseph, A Model of quasicrystal growth, Phys. Rev. Lett. 79 (1997), pp.
1066–1069.
[3] Ch. Goodman-Strauss,Self-assembly of hierarchical non-periodic tilings, in preparation [4] W. Jockusch, J. Propp, P. Shor,Random domino tilings and the arctic circle theorem,
preprint (1995).
[5] R. Peled, High-dimensional Lipschitz functions are typically flat, to appear in Annals of Probability.
[6] J. E. S. Socolar,Growth rules for quasicrystals, in Quasicrystals: The State of the Art, D. P. DiVincenzo and P. J. Steinhardt eds, World Scientific, Singapore, 1991.
[7] A. Stannard, M. O. Blunt, P. H. Beton, J. P. Garrahan, Entropically stabilized growth of a two-dimensional random tiling, Phys. Rev. Lett. 82 (2010), pp. 041109.
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