The variational principle for a marked Gibbs point process with infinite-range multibody interactions

TL;DR

This paper establishes the Gibbs variational principle for a complex particle system with infinite-range, multibody interactions, proving existence of consistent infinite-volume Gibbs measures under non-overlap constraints.

Contribution

It extends the Gibbs variational principle to a model with unbounded particle sizes and infinite-range interactions, which cannot be expressed as fixed k-body potentials.

Findings

Proved the Gibbs variational principle for the model.

Established existence of infinite-volume Gibbs point processes.

Controlled boundary influence using geometric and hard-core constraints.

Abstract

We prove the Gibbs variational principle for the Asakura--Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a $k$-body interaction for fixed $k$. As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.