The local limit of rooted directed animals on the square lattice

2024/01/23 by Olivier Hénard, Hénard, Olivier, Édouard Maurel-Segala +3
Mathematics · #60K35 #82B41 #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2401.12935

openalex publication_date 2024/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the local limit of finite uniformly distributed directed animals 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 prove, in particular, that it consists of a single vertex at infinitely many (random) levels. Several martingales are found in connection with the confinement of the infinite directed animal on the non-negative coordinates.

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