A lower bound on the displacement of particles in 2D Gibbsian particle systems

TL;DR

This paper establishes a lower bound on particle displacement in 2D Gibbsian systems, showing particles near the center of a large box typically fluctuate at least on the order of a0a0log n, extending previous results to more general models.

Contribution

It provides a new lower bound on particle displacement in 2D Gibbsian systems, applicable to a wide class of models with arbitrary spins and interactions.

Findings

Particles near the center fluctuate at least a0a0log n in displacement.

Results extend to models like continuum Potts, Widom-Rowlinson, and Lennard-Jones.

Positional order is absent despite possible orientational order.

Abstract

While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size $2n \times 2n$ may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order $\sqrt{log n}$. Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom-Rowlinson or Lenard-Jones-type interaction.