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Structural properties of Gibbsian point processes in abstract spaces

Steffen Betsch

TL;DR

This work provides a highly general theory of Gibbs point processes on arbitrary measurable spaces, defining Gibbs objects through GNZ equations without any topological assumptions. It unifies partition functions, Janossy densities, correlation functions, and a generalized Hamiltonian via factorial measures, and derives the DLR framework in full generality. A new convergence result enables extraction of locally convergent subsequences from sequences of point processes, yielding broad existence results for Gibbs processes, including cluster-dependent interactions and Gibbs particle processes. The theory recovers classical models (e.g., Strauss, hard-core, and Lennard–Jones-type systems) and confirms conjectures on Gibbs particle processes, thereby offering a versatile toolkit for stochastic geometry and spatial statistics in abstract spaces.

Abstract

In the language of random counting measures many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.

Structural properties of Gibbsian point processes in abstract spaces

TL;DR

This work provides a highly general theory of Gibbs point processes on arbitrary measurable spaces, defining Gibbs objects through GNZ equations without any topological assumptions. It unifies partition functions, Janossy densities, correlation functions, and a generalized Hamiltonian via factorial measures, and derives the DLR framework in full generality. A new convergence result enables extraction of locally convergent subsequences from sequences of point processes, yielding broad existence results for Gibbs processes, including cluster-dependent interactions and Gibbs particle processes. The theory recovers classical models (e.g., Strauss, hard-core, and Lennard–Jones-type systems) and confirms conjectures on Gibbs particle processes, thereby offering a versatile toolkit for stochastic geometry and spatial statistics in abstract spaces.

Abstract

In the language of random counting measures many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.
Paper Structure (19 sections, 52 theorems, 233 equations)

This paper contains 19 sections, 52 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Basic properties
  3. Janossy densities and correlation functions of Gibbs processes
  4. Finite Gibbs processes
  5. Hamiltonians in abstract measurable spaces
  6. The DLR equations
  7. Some facts about the local convergence topology and new convergence results
  8. An existence result for Gibbs point processes in general spaces
  9. The abstract result
  10. Initial examples and some remarks
  11. Cluster-dependent interactions in the subcritical regime
  12. Notation and preliminaries for the proof of Theorem \ref{['THM existence and uniqueness for subcritical cluster-dependent interactions']}
  13. A proof of Theorem \ref{['THM existence and uniqueness for subcritical cluster-dependent interactions']}
  14. Gibbs particle processes
  15. Factorial measures
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\eta$ be a Gibbs process with PI $\kappa$ and $m \in \mathbb{N}$. Then, for any measurable function $f : \mathbb{X}^m \times \mathbf{N} \to [0, \infty]$,

Theorems & Definitions (69)

  • Lemma 2.1: Multivariate GNZ equations
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 59 more