Electronic Journal of Probability

Scaling limit of the radial Poissonian web

Glauco Valle, Luiz Renato Fontes, and Leon Valencia

Full-text: Open access


We consider a variant of the radial spanning tree introduced by Baccelli and Bordenave. Like the original model, our model is a tree rooted at the origin, built on the realization of a planar Poisson point process. Unlike it, the paths of our model have independent jumps. We show that locally our diffusively rescaled tree, seen as the collection of the paths connecting its sites to the root, converges in distribution to the Brownian Bridge Web, which is roughly speaking a collection of coalescing Brownian bridges starting from all the points of a planar strip perpendicular to the time axis, and ending at the origin.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 31, 40 pp.

Accepted: 28 March 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments

spanning trees Brownian web Poisson point processes Poisson tree Coalescing processes Invariance Principle

This work is licensed under aCreative Commons Attribution 3.0 License.


Valle, Glauco; Fontes, Luiz Renato; Valencia, Leon. Scaling limit of the radial Poissonian web. Electron. J. Probab. 20 (2015), paper no. 31, 40 pp. doi:10.1214/EJP.v20-3395. https://projecteuclid.org/euclid.ejp/1465067137

Export citation


  • R. Arratia: Coalescing Brownian motions and the voter model on Z. Unpublished partial manuscript (circa 1981), available at rarratia@math.usc.edu.
  • Baccelli, Francois; Bordenave, Charles. The radial spanning tree of a Poisson point process. Ann. Appl. Probab. 17 (2007), no. 1, 305–359.
  • Belhaouari, S.; Mountford, T.; Sun, Rongfeng; Valle, G. Convergence results and sharp estimates for the voter model interfaces. Electron. J. Probab. 11 (2006), no. 30, 768–801 (electronic).
  • A. Bhatt; R. Roy: On a random directed spanning tree.
  • Coletti, C. F.; Fontes, L. R. G.; Dias, E. S. Scaling limit for a drainage network model. J. Appl. Probab. 46 (2009), no. 4, 1184–1197.
  • C. Coletti; L. Valencia: The Radial Brownian Web, preprint.
  • Coletti, Cristian; Valle, Glauco. Convergence to the Brownian Web for a generalization of the drainage network model. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 3, 899–919.
  • Ferrari, P. A.; Landim, C.; Thorisson, H. Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 2, 141–152.
  • Ferrari, P. A.; Fontes, L. R. G.; Wu, Xian-Yuan. Two-dimensional Poisson trees converge to the Brownian web. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 5, 851–858.
  • Fontes, L. R. G.; Isopi, M.; Newman, C. M.; Ravishankar, K. The Brownian web. Proc. Natl. Acad. Sci. USA 99 (2002), no. 25, 15888–15893 (electronic).
  • Fontes, L. R. G.; Isopi, M.; Newman, C. M.; Ravishankar, K. The Brownian web: characterization and convergence. Ann. Probab. 32 (2004), no. 4, 2857–2883.
  • Fontes, L. R. G.; Isopi, M.; Newman, C. M.; Ravishankar, K. Coarsening, nucleation, and the marked Brownian web. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 1, 37–60.
  • Fontes, Luiz Renato; Newman, Charles M. The full Brownian web as scaling limit of stochastic flows. Stoch. Dyn. 6 (2006), no. 2, 213–228.
  • Gangopadhyay, Sreela; Roy, Rahul; Sarkar, Anish. Random oriented trees: a model of drainage networks. Ann. Appl. Probab. 14 (2004), no. 3, 1242–1266.
  • Monroe, Itrel. Processes that can be embedded in Brownian motion. Ann. Probability 6 (1978), no. 1, 42–56.
  • Newman, C. M.; Ravishankar, K.; Sun, Rongfeng. Convergence of coalescing nonsimple random walks to the Brownian web. Electron. J. Probab. 10 (2005), no. 2, 21–60.
  • Penrose, Mathew D.; Wade, Andrew R. Random minimal directed spanning trees and Dickman-type distributions. Adv. in Appl. Probab. 36 (2004), no. 3, 691–714.
  • Penrose, Mathew D.; Wade, Andrew R. On the total length of the random minimal directed spanning tree. Adv. in Appl. Probab. 38 (2006), no. 2, 336–372.
  • R. Roy; K. Saha; A. Sarkar: Random directed forest and the Brownian web, arXiv:1301.3766 [math.PR]
  • Sun, Rongfeng. Convergence of coalescing nonsimple random walks to the Brownian Web. Thesis (Ph.D.) - New York University. ProQuest LLC, Ann Arbor, MI, 2005. 64 pp. ISBN: 978-0496-90323-8.
  • Sarkar, Anish; Sun, Rongfeng. Brownian web in the scaling limit of supercritical oriented percolation in dimension $1+1$. Electron. J. Probab. 18 (2013), no. 21, 23 pp.
  • Sun, Rongfeng; Swart, Jan M. The Brownian net. Ann. Probab. 36 (2008), no. 3, 1153–1208.
  • Jiang, Tie Feng. Large deviations for renewal processes. Stochastic Process. Appl. 50 (1994), no. 1, 57–71.
  • Toth, Balint; Werner, Wendelin. The true self-repelling motion. Probab. Theory Related Fields 111 (1998), no. 3, 375–452.
  • L. A. Valencia: A Teia Browniana Radial, PhD. thesis (in protuguese), IME-USP (2012)