Canal-U

Absence of percolation for Poisson outdegree-one graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

A Poisson outdegree-one graph is a directed graph based on a marked Poisson point process such that each vertex has only one outgoing edge. We state the absence of percolation for such graphs satisfying two assumptions. The Shield assumption roughly says that the graph is locally determined with possible random horizons. The Loop assumption ensures that any forward branch merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. This result allows to solve a conjecture by D. Daley, S. Ebert and G. Last on the absence of percolation for the “line-segment model”. In this planar model, a segment is growing from any point of the Poisson process and stops its growth whenever it hits another segment. The random directions are picked independently and uniformly on the unit sphere. This is a joint work with D. Dereudre and S. Le Stum.