The local limit of rooted directed animals on the square lattice

• 成果类型: Article; Early Access
• 署名作者: Henard, Olivier; Maurel-Segala, Edouard; Singh, Arvind
• 署名单位: Universite Paris Saclay; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01355-4
• 发表日期: 2024
• 关键词: site animals; enumeration; EQUIVALENCE; BEHAVIOR

摘要

We consider the local limit of uniformly distributed directed animals with size n on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed by letting gravity act on a right-continuous random walk, and one as a Markov process, obtained by slicing the animal horizontally. We look at geometric properties of this local limit and establish, in particular, that it consists of a single vertex at infinitely many (random) levels. The proof relies on martingales arguments and showcases the strength of this probabilistic approach.