# Existence of Gibbs point processes with stable infinite range interaction

## TL;DR

This paper proves the existence of Gibbs point processes with infinite range interactions using an entropy-based approach, relaxing previous stability assumptions and accommodating multi-body interactions.

## Contribution

It introduces a new proof method for Gibbs processes with infinite range interactions, including a novel intensity regularity condition and extends Ruelle's results.

## Key findings

- Established existence of Gibbs point processes with infinite range interactions.
- Relaxed superstability assumptions compared to Ruelle's classical results.
- Applicable to multi-body interactions beyond pairwise cases.

## Abstract

We provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is super-linear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite range multi-body counterparts.