Normal and Poisson approximation for Gibbs point processes with pair potentials

TL;DR

This work develops Poisson and normal approximation results for Gibbs point processes with local and pairwise interactions in marked Euclidean spaces. It advances the disagreement coupling framework to infinite-volume domains via non-percolation and, for pair potentials, through a random connection model (RCM), enabling explicit thinnings and couplings. The authors establish both qualitative and quantitative CLTs for a broad class of stabilizing functionals and provide concrete examples (e.g., isolated points, k-NN edges, Gilbert graphs, and persistent Betti numbers) to illustrate the applicability. The results significantly extend prior Gibbs-process asymptotics by accommodating long-range interactions and marks, with precise rates and robust coupling constructions that can impact spatial statistics and stochastic geometry practice.

Abstract

We provide a Poisson approximation result for dependent thinnings of Gibbs point processes as well as qualitative and quantitative central limit theorems for geometric functionals of Gibbs point processes in increasing observation windows. The present paper extends prior work on finite-range Gibbs processes to processes with repulsive pairwise interaction of unbounded interaction range as well as processes on marked Euclidean space. The proofs rely on coupling different Gibbs processes using the disagreement coupling technique, which we generalize to infinite-volume domains under a suitable non-percolation condition. For the case of repulsive pairwise interactions, we introduce a version of disagreement coupling that constructs the Gibbs process by thinning a random connection model thus making previous approximation methods more flexible.

Figures (6)

• Figure 1: Illustration of the construction of $\Gamma_\nu(\bar{\varphi}^*,\psi)$. The green point is $\nu$, and the red points are $\psi$. The orange points are $V$, which consists of the points connected to $\nu$. $\Gamma^{\mathsf{Init}}$ is shown in blue as the subgraph with vertex set $V\cup \psi$ and $\Gamma^{\mathsf{Bulk}}$ is the graph shown in black.
• Figure 2: Left: Successful exploration $V_{k+1}\ne \varnothing$. Right: Failed exploration $V_{k+1} = \varnothing$.
• Figure 3: Left: $P_1=Q$ is the whole square while $P_{1,-}=Q\setminus B_s(x)$ is the grey region, and $P_3=P_{2,-}=\varnothing$. The PI $\kappa'$ lives on all of $P_1=Q$, while $\kappa"$ vanishes on the ball and in the grey region, it ignores boundary points in the grey ball. Theorem \ref{['thm:Talpha']} bounds the probability that the two processes disagree outside the larger ball. Right: $\kappa'$ vanishes on the smallest ball $P_3$ around ${\boldsymbol x}$, while $\kappa"$ vanishes outside the two grey annuli. Theorem \ref{['thm:Talpha']} bounds the probability that the two processes disagree on the medium sized ball $A$ around ${\boldsymbol x}$.
• Figure 4: The set $U$ is the union of the white and the light grey region consisting af all squares succeeding $Q_{z,1}$ lexicographically. The window $W_n$ is the middle $3\times 3$ square, so $U_n$ is the white region and $U\setminus U_n$ is the light grey region. The points in the dark grey region are the boundary points $\psi$ while the points in the light grey region are $\nu_n$. The points in the white region is the Poisson process that we are thinning.
• Figure 5: Coupled configurations $\mathcal{P}$ and $\check\mathcal{P}$ in the domain $Q$. Black points belong to $\mathcal{P}$. Inside $B_{2r}({\boldsymbol x})$, the two processes agree, while green points represent $\check\mathcal{P} \setminus B_{2r}({\boldsymbol x})$. Circles indicate members of the set $\mathcal{Y}$, that is, points connected to $Q\setminus B_{2r}({\boldsymbol x})$ by a path in either $\mathcal{P}$ or $\check\mathcal{P}$. The clusters in $\mathcal{P}$ intersecting $Q\setminus B_{2r}({\boldsymbol x})$ are shown with black edges, while clusters in $\check \mathcal{P}$ intersecting $Q\setminus B_{2r}({\boldsymbol x})$. The dashed black and green edge represents two points that form their own cluster in $\mathcal{P}$ but are part of a larger cluster in $\check \mathcal{P}$. The blue clusters inside $B_{2r}({\boldsymbol x})$ are common for the two processes. These are thinned in the same order, but different black or green clusters may be visited in between.

Theorems & Definitions (93)

• Definition 2.1
• Remark 2.2
• Theorem 2.3: Existence and uniqueness of Gibbs processes
• Definition 1
• Example 2.4: Euclidean point processes with finite interaction range
• Example 2.5: Interacting particle models
• Definition 2.6: RCM $\Gamma(Q, \psi)$
• Theorem 2.7: Existence and uniqueness of Gibbs processes with pair potential betschgibbsexist
• Definition 2
• Example 2.8