Solvable models for entropic quasicrystals

Beside the well known crystallographic solid, nature has in the last two decades revealed to us another solid phase: the quasicrystal. It behaves in almost all aspects as an ordinary crystal: it can be cleaved only in certain discrete directions, it grows in nicely symmetric structures, and its diffraction patterns consist of Bragg peaks. The defining difference with ordinary crystals is that its rotational symmetries are forbidden by the theory of crystallography. Quasicrystals seem to show both periodicity and rotational symmetry, but mathematically these symmetries are not compatible. In this seminar I present some tiling models that show these same properties of physical quasicrystals, albeit in two dimensions. A tiling is a complete and non-overlapping covering of space by copies of a few geometrical objects. Here they are studied as statistical models, so that a whole ensemble of tilings is considered. They are quite close to dense packings of hard disks of two different sizes. We will focus on cases where the thermodynamic quantities can be calculated exactly.