Two-dimensional Poisson Trees converge to the Brownian web

The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times in $\mathbb{R}$ and at all points of $\mathbb{R}$. It was introduced by Arratia; a variant was then studied by Toth and Werner; another variant was analyzed recently by Fontes, Isopi, Newman and Ravishankar. The two-dimensional \emph{Poisson tree} is a family of continuous time one-dimensional random walks with uniform jumps in a bounded interval. The walks start at the space-time points of a homogeneous Poisson process in ${\mathbb{R}}^2$ and are in fact constructed as a function of the point process. This tree was introduced by Ferrari, Landim and Thorisson. By verifying criteria derived by Fontes, Isopi, Newman and Ravishankar, we show that, when properly rescaled, and under the topology introduced by those authors, Poisson trees converge weakly to the Brownian web.