The Square-Triangle-Rhombus random tiling model
by Maria Tsarenko
Abstract: Crystals (such as diamond or amethyst) are invariant under discrete
translations and rotations. The study of their atomic structure can
therefore be described by models on periodic lattices. Rotational
symmetries compatible with translational symmetry in two dimensions are
known to be exclusively 2-, 3-, 4- or 6-fold.
In 1984, Schechtman discovered a `controversial' crystal possessing a
5-fold symmetry -- a naturally occurring Penrose-like tiling for which he
received the 2011 Nobel Prize in Chemistry. Since then, around 100
quasi-crystals (metal alloys) have been created in labs, and a few (like
khatyrkite) have been found naturally occurring.
Modelling the atomic structure of quasicrystals motivated the
study of Random Tilings. These are tilings of the plane that possess an
entropically favoured `forbidden' rotational symmetry in a statistical
sense, that is, at a macroscopic level.
In this talk I will present a new type of random tiling, the
\(square\)-\(triangle\)-\(rhombus\) tiling model. I shall briefly explain how
it is `solvable' by the algebraic nested Bethe Ansatz, derive an exact
expression for the line of maximum entropy, discuss the underlying
symmetries and the phase diagram.