The structure of finite clusters in high intensity Poisson Boolean stick process

Abstract

Sticks at one of different orientation are placed in an i.i.d. fashion at points of a Poisson point process of intensity $\lambda$. Sticks of the same direction have the same length, while sticks in different directions may have different lengths. We study the geometry of finite cluster as $\lambda \to \infty$. The asymptotic shape of the custer being determined by the probabilities of the sticks in various direction and their lengths and orientations. We also obtain the limiting geometric structure of this component.