Translation-invariance of two-dimensional Gibbsian point processes

TL;DR

This paper proves that two-dimensional Gibbsian point processes with various interactions, including singular and hard-core types, are invariant under translations, extending classical results in statistical mechanics.

Contribution

It establishes translation-invariance for a broad class of Gibbsian particle systems with complex interactions, including singular and hard-core cases.

Findings

Translation-invariance holds for Gibbsian systems with two-body interactions.

Hard-core and discontinuous interactions are included in the invariance proof.

The approach handles singular interactions like hard cores explicitly.

Abstract

The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general.