Open Access
2020 Leaves on the line and in the plane
Mathew D. Penrose
Electron. J. Probab. 25: 1-40 (2020). DOI: 10.1214/20-EJP447


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.


Download Citation

Mathew D. Penrose. "Leaves on the line and in the plane." Electron. J. Probab. 25 1 - 40, 2020.


Received: 25 September 2018; Accepted: 6 April 2020; Published: 2020
First available in Project Euclid: 5 May 2020

zbMATH: 1447.60034
MathSciNet: MR4095049
Digital Object Identifier: 10.1214/20-EJP447

Primary: 60D05 , 60F05 , 60G55 , 60G57 , 82C22

Keywords: central limit theorem , dead leaves model , Ornstein-Uhlenbeck process , random measure , random tessellation

Vol.25 • 2020
Back to Top