Lower Bound on the Mean Square Displacement of Particles in the Hard Disk Model

The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions 2n × 2n with arbitrary boundary configuration, and we show that the mean square displacement of particles near the center of the box is bounded from below by c log n. The result generalizes to a large class of models with fairly arbitrary interaction.