Spontaneous breaking of rotational symmetry in a probabilistic hard disk model in Statistical Mechanics

Throughout this thesis, we are going to work in a hard disk model which is analogue to the setting in [MR09] and [HMR13]. In this model of two-dimensional crystals, every admissible configuration is a hard disk configuration, which, in addition, is a perturbed version of some triangular lattice with side length one. A triangular lattice with side length one is called standard. The objective of this thesis is to show that admissible configurations in a given box with side length N ∈ N are arbitrarily close to some standard triangular lattice whenever the particle density is chosen sufficiently high, and this choice can be made independent of the box size N . Since this result is independent of the choice of N , the existence of an infinite-volume Gibbs measure can be shown which, in spite of the rotational symmetry of the set of admissible configurations, breaks the rotational symmetry in a strong sense.