Percolation and limit theory for the Poisson lilypond model

TL;DR

This paper studies the lilypond model on Poisson point processes, proving central limit theorems for volume and component counts, tail bounds for cluster sizes, and percolation thresholds for enhanced models.

Contribution

It establishes new limit theorems, tail bounds, and percolation thresholds for the lilypond model on Poisson and binomial processes, advancing understanding of its probabilistic properties.

Findings

Central limit theorems for total volume and number of components

Subexponentially decaying tail bounds for cluster size at the origin

Positive critical enhancement parameter for percolation in higher dimensions

Abstract

The lilypond model on a point process in $d$-space is a growth-maximal system of non-overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for $d > 1 $ the critical value of this parameter, above which the enhanced model percolates, is strictly positive.