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

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.