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.

中文翻译：

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二维吉布斯粒子系统中粒子位移的下界

虽然 2D 吉布斯粒子系统可能会表现出取向顺序，导致类似格子的结构，但如果粒子之间的相互作用满足一些弱假设，这些粒子系统就不会表现出位置顺序。在这里，我们调查大小为 $2n \times 2n$ 的盒子内的粒子可能会从其理想的晶格位置波动的程度。我们表明，靠近盒子中心的粒子通常表现出至少为 $\sqrt{log n}$ 级的位移。因此，我们将硬盘模型的最新结果扩展到具有相当任意的粒子自旋和相互作用的粒子系统。我们的结果适用于诸如相当普遍的连续波茨型模型之类的模型，例如具有 Widom-Rowlinson 或 Lenard-Jones 型交互作用的模型。