Séminaire de Probabilités commun ICJ/UMPA

The local limit of rooted directed animals on the square lattice

by Olivier Hénard

435 (ENS)




Since the pioneer work of Dhar, we know that directed animals on the square lattice can be enumerated by volume (beware to ask enumeration with respect to other parameters though...).
In this work, we introduce a probabilistic touch on the topic by considering the local limit of finite uniformly distributed directed animals on the square lattice viewed from the root. We are able to give two constructions of the resulting uniform infinite animal : one as a heap of dominoes ("à la Viennot") whose successive coordinates follow a  simple modification of a random walk, and one as a Markov process, obtained by slicing the animal horizontally. 
The relations between various conditioned versions of the uniform infinite animal give us some martingales which in return allow to prove some first geometric properties of the non conditioned object, notably the sausaging property.
This is joint work with Édouard Maurel Segala and Arvind Singh.