Law of large numbers for greedy animals and paths in an ergodic environment

TL;DR

This paper extends the law of large numbers for maximal mass of animals and paths from i.i.d. to ergodic environments, including marked point processes, establishing almost sure convergence under certain conditions.

Contribution

It generalizes existing results to ergodic and marked point process environments, broadening the applicability of the law of large numbers for animals and paths.

Findings

Almost sure convergence of maximal animal mass per size in ergodic environments

Extension to marked Poisson point processes

Conditions for integrability of the mass distribution

Abstract

Consider a family of random masses $\mathbf{m}(v)$ indexed by vertices of the lattice $\mathbb Z^d$. In the case where the masses are i.i.d.\ and satisfy a certain moment condition, it is known that there exists a deterministic $A\ge 0$ such that the maximal mass $A_n$ of an animal containing $0$ with cardinal $n$ satisfies $A_n/n \rightarrow A$ when $n\to \infty$, almost surely. The same also goes for self-avoiding paths. We extend this result to the case where the family of masses is an ergodic marked point process, with a suitable definition for animals in this context. Special cases include the initial model with ergodic instead of i.i.d.\ masses and marked Poisson point processes. We also discuss some sufficient or necessary conditions for integrability.