Law of large numbers for greedy animals and paths in a Poissonian environment

TL;DR

This paper extends the law of large numbers for greedy animals and paths in a Poissonian environment, analyzing their maximal mass and monotonicity properties, and introduces a penalized model as an interpolation.

Contribution

It generalizes previous results by extending the law of large numbers to the closed unit ball and studies the properties of a new penalized model.

Findings

Law of large numbers extended to the closed unit ball.

Monotonicity of the limit function along a radius established.

A penalized model interpolates between two existing models.

Abstract

We study two continuous and isotropic analogues of the model of greedy lattice animals introduced by Cox, Gandolfi, Griffin and Kesten in 1993. In our framework, animals collect masses scattered on a Poisson point process on $\mathbb R^d$, and are allowed to have vertices outside the process or not, depending on the model. The author recently proved in a more general setting that for all $u$ in the Euclidean open unit ball, the maximal mass of animals with length $\ell$, containing $0$ and $\ell u$ satisfies a law of large numbers. We prove some additional properties in the Poissonian case, including an extension of the functional law of large numbers to the closed unit ball, and study strict monotonicity of the limit function along a radius. Moreover, we prove that a third, penalized model is a suitable interpolation between the former two.