[Paper Review] Existence of Gibbsian point processes with geometry-dependent interactions

David Dereudre, Rémy Drouilhet+1 more|arXiv (Cornell University)|Mar 15, 2010

Point processes and geometric inequalities11 references4 citations

TL;DR

This paper establishes the existence of stationary Gibbsian point processes with geometry-dependent interactions—such as those based on Delaunay triangulations, Voronoi cells, or k-nearest neighbors—using a global approach combining entropy bounds and stationarity. It extends prior results by proving existence for non-bounded and hard-exclusion potentials without geometric restrictions, significantly generalizing classical many-body interaction frameworks in statistical mechanics.

ABSTRACT

We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of $k$-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.

Motivation & Objective

• To establish the existence of stationary Gibbsian point processes with interactions dependent on local geometric structures such as Delaunay edges, Voronoi cells, and k-nearest neighbor clusters.
• To overcome limitations of prior infinitesimal approaches that fail under non-hereditary or hard-exclusion interactions.
• To generalize classical many-body interaction models by allowing non-additive, geometry-dependent interactions that alter hyperedge configurations upon particle addition.
• To provide a global existence proof using stationarity and entropy density compactness, avoiding restrictive geometric or boundedness assumptions on the potential.

Proposed method

• Formalize interactions via a hypergraph structure $\mathcal{E}(\omega)$ on point configurations $\omega \subset \mathbb{R}^d$, where each hyperedge $\eta \in \mathcal{E}(\omega)$ contributes a potential $\varphi(\eta, \omega)$ depending on local neighborhood.
• Impose the finite horizon property, ensuring $\varphi(\eta, \omega)$ depends only on $\eta$ and points within a bounded distance, guaranteeing locality.
• Use a global approach based on stationarity and entropy density to control the thermodynamic limit, replacing infinitesimal Papangelou intensity methods.
• Apply entropy bounds to exploit compactness of entropy level sets, enabling convergence of finite-volume Gibbs measures to a stationary limit.
• Establish measurability of key quantities (e.g., partition functions, conditional kernels) via universal measurability and Fubini-type arguments on product spaces.
• Leverage stationarity to control interaction range and ensure uniformity in the thermodynamic limit, even when hyperedges are destroyed or created upon particle insertion.

Experimental results

Research questions

• RQ1 Can stationary Gibbsian point processes exist for geometry-dependent interactions that are non-additive and non-hereditary, such as those based on Delaunay or Voronoi structures?
• RQ2 Does the existence of such Gibbs measures hold under general bounded or unbounded potentials, including hard-exclusion cases?
• RQ3 Can the classical existence theory for finite-range many-body interactions be extended to non-additive, geometry-driven interactions without geometric restrictions?
• RQ4 Is it possible to prove existence without relying on infinitesimal characterization via Papangelou intensities, especially in non-hereditary settings?
• RQ5 How can stationarity and entropy compactness be used to control the interaction range and ensure convergence in the thermodynamic limit?

Key findings

• The paper proves the existence of stationary Gibbsian point processes for general geometry-dependent interactions, including Delaunay, Voronoi, and k-nearest neighbor interactions.
• It establishes existence for non-bounded and hard-exclusion potentials, overcoming limitations of previous methods that required boundedness or geometric constraints.
• The entropy bound technique enables compactness of entropy level sets, which is crucial for proving convergence of finite-volume Gibbs measures.
• Stationarity is used to control the interaction range uniformly, allowing the proof to avoid restrictive assumptions on the interaction geometry.
• Measurability of conditional kernels and partition functions is rigorously established using universal extensions and Fubini-type arguments on product spaces.
• The results generalize classical many-body interaction frameworks by relaxing the superstability assumption, extending the scope to non-additive, geometry-driven interactions.