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

2024/09/23 by J. Vergès, Verges, Julien
Earth and Planetary Sciences · Economics, Econometrics and Finance · #Complex Systems and Time Series Analysis #FOS: Mathematics #Marine and environmental studies #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2409.14773

openalex publication_date 2024/09/23 · openalex created_date 2024/09/25 · openalex updated_date 2026/08/01

Abstract

Consider a family of random masses 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≥ 0 such that the maximal mass A_n of an animal containing 0 with cardinal n satisfies A_n/n → A when n→ ∞, 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.

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