Marked Gibbs point processes with unbounded interaction: an existence result

TL;DR

This paper establishes the existence of marked Gibbs point processes in Euclidean space with unbounded interactions and marks in a general normed space, using entropy and large deviation techniques.

Contribution

It extends the theory of Gibbs point processes to include unbounded interactions and marks with super-exponential moments, broadening applicability.

Findings

Constructed Gibbs processes with unbounded, possibly random interaction range.

Proved tightness of finite-volume Gibbs measures using entropy and large deviations.

Applied results to infinite-dimensional interacting diffusions.

Abstract

We construct marked Gibbs point processes in $\mathbb{R}^d$ under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks -- attached to the locations in $\mathbb{R}^d$ -- belong to a general normed space $\mathcal{S}$. They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented.