Two-dimensional Poisson Trees converge to the Brownian web
arXiv:math/0304247 · doi:10.1016/j.anihpb.2004.06.003
Abstract
The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times in and at all points of . 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 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.
22 pages, 1 figure. This version corrects an error in the previous proof. The results are the same
References in corpus (4)
Cited by in corpus (13)
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- Brownian Web in the Scaling Limit of Supercritical Oriented Percolation in Dimension 1+1
- A drainage network with dependence and the Brownian web
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- Continuum random tree as the scaling limit for a drainage network model: a Brownian web approach