Interaction processes for unions of facets, a limit behavior

TL;DR

This paper introduces a new facet process model in stochastic geometry, analyzes its properties, and studies its behavior as intensity increases, including a central limit theorem for Poisson cases.

Contribution

It develops a novel framework for facet processes in arbitrary dimensions, utilizing L2 expansion and correlation functions, and explores asymptotic behaviors and limit theorems.

Findings

Facet processes exhibit local stability and repulsiveness.

A central limit theorem is established for Poisson-based models as intensity grows.

Correlation functions show diverse asymptotics depending on model parameters.

Abstract

In the series of models with interacting particles in stochastic geometry, a new contribution presents the facet process which is defined in arbitrary Euclidean dimension. In 2D, 3D specially it is a process of interacting segments, flat surfaces, respectively. Its investigation is based on the theory of functionals of finite spatial point processes given by a density with respect to a Poisson process. The methodology based on L2 expansion of the covariance of functionals of Poisson process is developed for U-statistics of facet intersections which are building blocks of the model. The importance of the concept of correlation functions of arbitrary order is emphasized. Some basic properties of facet processes, such as local stability and repulsivness are shown and a standard simulation algorithm mentioned. Further the situation when the intensity of the process tends to infinity is studied. In the case of Poisson processes a central limit theorem follows from recent results of Wiener-Ito theory. In the case of non-Poisson processes we restrict to models with finitely many orientations. Detailed analysis of correlation functions exhibits various asymptotics for different combination of U-statistics and submodels of the facet process.