Limit theory for planar Gilbert tessellations

TL;DR

This paper develops limit theorems such as law of large numbers, variance asymptotics, and a central limit theorem for planar Gilbert tessellations, which model crack growth in a plane based on Poisson point processes.

Contribution

It introduces a rigorous probabilistic framework for analyzing geometric functionals of Gilbert tessellations using stabilization theory.

Findings

Established law of large numbers for tessellation functionals

Derived variance asymptotics for geometric measures

Proved a central limit theorem for the tessellation process

Abstract

A Gilbert tessellation arises by letting linear segments (cracks) in the plane unfold in time with constant speed, starting from a homogeneous Poisson point process of germs in randomly chosen directions. Whenever a growing edge hits an already existing one, it stops growing in this direction. The resulting process tessellates the plane. The purpose of the present paper is to establish law of large numbers, variance asymptotics and a central limit theorem for geometric functionals of such tessellations. The main tool applied is the stabilization theory for geometric functionals.