Leaves on the line and in the plane

TL;DR

This paper analyzes the Dead Leaves Model in different dimensions, providing new formulas and limit theorems for the structure of leaves and their boundaries, with applications to related random measures.

Contribution

It introduces new analytical results for the DLM in 1D and 2D, including formulas for intensities, correlations, and CLTs, and generalizes to Dead Leaves Random Measures.

Findings

Derived formulas for point process intensities and correlations in 1D.

Established asymptotic covariances and functional CLTs.

Extended results to general Dead Leaves Random Measures.

Abstract

The Dead Leaves Model (DLM) provides a random tessellation of $d$-space, representing the visible portions of fallen leaves on the ground when $d=2$. For $d=1$, we establish formulae for the intensity, two-point correlations, and asymptotic covariances for the point process of cell boundaries, along with a functional CLT. For $d=2$ we establish analogous results for the random surface measure of cell boundaries, and also determine the intensity of cells in a more general setting than in earlier work of Cowan and Tsang. We introduce a general notion of Dead Leaves Random Measures and give formulae for means, asymptotic variances and functional CLTs for these measures; this has applications to various other quantities associated with the DLM.