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

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 $A_n/n\to A$ 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.