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

Consider a family of random masses m(v) indexed by vertices of the lattice Zd. 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≥0 such that the maximal mass An of an animal containing 0 with cardinal n satisfies An∕n→A when n→∞, almost surely. The same also goes for self-avoiding paths. We extend these results to the case where the family of masses is an ergodic marked point process, with a suitable definition for animals and paths in this context. Contrary to previous works, our proof does not rely on a concentration bound. Special cases include the initial discrete model with ergodic instead of i.i.d. masses and marked Poisson point processes. We also discuss some sufficient or necessary conditions for integrability.