Limit theorems for geometric functionals of Gibbs point processes

Schreiber, T.; Yukich, J. E.

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1158-1182.

On observe un processus ponctuel $𝛯$ dans ${ℝ}^d$ dans une fenêtre $Q_{\lambda }$ de volume $\lambda$. L’observation en un point $x$ que l’on note $\xi (x,𝛯)$ est une fonction des points situés à une distance aléatoire de $x$. Quand $𝛯$ est un processus de Poisson ponctuel ou Binomial, la limite pour $\lambda$ grand de la somme totale ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$ (convenablement recentrée et normalisée) est bien comprise. Dans ce papier, nous étudions cette somme totale quand $𝛯$ est Gibbsien et prouvons la loi des grands nombres, la variance asymptotique et un théorème de la limite centrale. Les preuves reposent sur la simulation parfaite de processus ponctuels Gibbsiens pour établir leurs propriétés de mélange. Ces résultats généraux sont appliqués dans différents contextes comme des modèles de croissance et de percolation, des graphes de Voronoi et des problèmes de quantification pour des entrées Gibbsiennes.

Observations are made on a point process $𝛯$ in ${ℝ}^d$ in a window $Q_{\lambda }$ of volume $\lambda$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,𝛯)$, is a function of the points within a random distance of $x$. When the input $𝛯$ is a Poisson or binomial point process, the large $\lambda$ limit theory for the total score ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$, when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input $𝛯$. The proofs use perfect simulation of Gibbs point processes to establish their mixing properties. The general limit results are applied to random sequential packing and spatial birth growth models, Voronoi and other Euclidean graphs, percolation models, and quantization problems involving Gibbsian input.

Classification : 60F05, 60G55, 60D05
Mots clés : Perfect simulation, Gibbs point processes, exponential mixing, gaussian limits, hard core model, random packing, geometric graphs, Gibbs-Voronoi tessellations, quantization

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