Directed animals as branching-annihilating particles systems, through intertwining
Résumé
We consider the uniform infinite pyramid (UIP) defined in a previous article (Hénard, Maurel-Segala, Singh 2024) as the local limit of uniformly sampled directed animals on the square lattice, when their size grows to infinity. We interpret this random object as an interacting particle system and show that it satisfies a remarkable intertwining property: the particle system representing the UIP, which has long-range interactions, is the trace of a simple branching/annihilating system with only local interactions. We prove similar statements for the uniform infinite non-negative pyramid (IHP+) and the critical Boltzmann half-pyramid (BHP).
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