Lower bound on the mean square displacement of particles in the hard disk model

TL;DR

This paper establishes a lower bound on particle displacement in the 2D hard disk model, showing that fluctuations grow at least logarithmically with system size, indicating limited positional order at high densities.

Contribution

It provides a rigorous lower bound on particle displacement in the hard disk model, extending to models with similar interactions, thus advancing understanding of particle fluctuations.

Findings

Mean square displacement is bounded below by c log n.

Displacements grow at least logarithmically with system size.

Results apply to a broad class of interaction models.

Abstract

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 $2n \times 2n$ 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.