## Electronic Journal of Probability

### Brownian web in the scaling limit of supercritical oriented percolation in dimension 1 + 1

#### Abstract

We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice $Z^2_{\mathrm{even}}:=\{(x,i) in Z^2: x+i \mathrm{is even}\}$ converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 21, 23 pp.

Dates
Accepted: 5 February 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064246

Digital Object Identifier
doi:10.1214/EJP.v18-2019

Mathematical Reviews number (MathSciNet)
MR3035749

Zentralblatt MATH identifier
1290.60107

Keywords
Brownian web oriented percolation

Rights

#### Citation

Sarkar, Anish; Sun, Rongfeng. Brownian web in the scaling limit of supercritical oriented percolation in dimension 1 + 1. Electron. J. Probab. 18 (2013), paper no. 21, 23 pp. doi:10.1214/EJP.v18-2019. https://projecteuclid.org/euclid.ejp/1465064246

#### References

• Arratia, Richard Alejandro. COALESCING BROWNIAN MOTIONS ON THE LINE. Thesis (Ph.D.)â€“The University of Wisconsin - Madison. ProQuest LLC, Ann Arbor, MI, 1979. 134 pp.
• Arratia, R.: Coalescing Brownian motions and the voter model on Z. Unpublished partial manuscript, 1981.
• Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
• Baccelli, Francois; Bordenave, Charles. The radial spanning tree of a Poisson point process. Ann. Appl. Probab. 17 (2007), no. 1, 305–359.
• Bezuidenhout, Carol; Grimmett, Geoffrey. The critical contact process dies out. Ann. Probab. 18 (1990), no. 4, 1462–1482.
• 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).
• Birkner, M., Cerny, J., Depperschmidt, A. and Gantert, N.: Directed random walk on an oriented percolation cluster, ARXIV1204.2951
• 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.
• Coletti, C. and Valle, G.: Convergence to the Brownian Web for a generalization of the drainage network model, ARXIV1109.3517
• Coupier, D. and Tran, V.C.: The 2D-directed spanning forest is almost surely a tree. phRandom Structures Algorithms, 42, (2013), 59–72.
• Durrett, Richard. Oriented percolation in two dimensions. Ann. Probab. 12 (1984), no. 4, 999–1040.
• Ferrari, P. A.; Fontes, L. R. G.; Wu, Xian-Yuan. Two-dimensional Poisson trees converge to the Brownian web. Ann. Inst. H. PoincarÃ© Probab. Statist. 41 (2005), no. 5, 851–858.
• 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. PoincarÃ© Probab. Statist. 42 (2006), no. 1, 37–60.
• Ferrari, P. A.; Landim, C.; Thorisson, H. Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. PoincarÃ© Probab. Statist. 40 (2004), no. 2, 141–152.
• Garban, C., Pete, G. and Schramm, O.: Pivotal, cluster and interface measures for critical planar percolation, ARXIV1008.1378
• Gangopadhyay, Sreela; Roy, Rahul; Sarkar, Anish. Random oriented trees: a model of drainage networks. Ann. Appl. Probab. 14 (2004), no. 3, 1242–1266.
• Kuczek, Thomas. The central limit theorem for the right edge of supercritical oriented percolation. Ann. Probab. 17 (1989), no. 4, 1322–1332.
• Neuhauser, Claudia. Ergodic theorems for the multitype contact process. Probab. Theory Related Fields 91 (1992), no. 3-4, 467–506.
• 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.
• Norris, J. and Turner, A.: Weak convergence of the localized disturbance flow to the coalescing Brownian flow, ARXIV1106.3252
• Norris, James; Turner, Amanda. Hastingsâ€“Levitov Aggregation in the Small-Particle Limit. Comm. Math. Phys. 316 (2012), no. 3, 809–841.
• Sun, Rongfeng; Swart, Jan M. The Brownian net. Ann. Probab. 36 (2008), no. 3, 1153–1208.
• Sarkar, A. and Sun, R. Brownian web and oriented percolation: density bounds. RIMS Kokyuroku, No. 1805, Applications of Renormalization Group Methods in Mathematical Sciences, (2012), 90–101.
• Tóth, Bálint; Werner, Wendelin. The true self-repelling motion. Probab. Theory Related Fields 111 (1998), no. 3, 375–452.
• Wu, Xian-Yuan; Zhang, Yu. A geometrical structure for an infinite oriented cluster and its uniqueness. Ann. Probab. 36 (2008), no. 3, 862–875.