##
**Directed, cylindric and radial Brownian webs.**
*(English)*
Zbl 1456.60206

Summary: The Brownian web (BW) is a collection of coalescing Brownian paths \((W_{(x,t)},(x,t) \in \mathbb{R} ^2)\) indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in C.F. Coletti and L.A. Valencia [“Scaling limit for a family of random paths with radial behavior”, Preprint, arXiv:1310.6929] is shown to converge to the CBW.

### MSC:

60J65 | Brownian motion |

60D05 | Geometric probability and stochastic geometry |

60J05 | Discrete-time Markov processes on general state spaces |

05C80 | Random graphs (graph-theoretic aspects) |

60G52 | Stable stochastic processes |

60G57 | Random measures |

### Keywords:

Brownian web; navigation algorithm; random spanning forests; weak convergence of stochastic processes
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\textit{D. Coupier} et al., Electron. J. Probab. 24, Paper No. 20, 48 p. (2019; Zbl 1456.60206)

### References:

[1] | R. Arratia. Coalescing Brownian motions on the line. PhD thesis, University of Wisconsin, Madison, 1979. |

[2] | F. Baccelli and C. Bordenave. The radial spanning tree of a Poisson point process. Annals of Applied Probability, 17(1):305-359, 2007. · Zbl 1136.60007 |

[3] | F. Baccelli, D. Coupier, and V. Tran. Semi-infinite paths of the 2d-radial spanning tree. Advances in Applied Probability, 45(4):895-1201, 2013. · Zbl 1287.60016 |

[4] | N. Berestycki, C. Garban, and A. Sen. Coalescing Brownian flows: a new approach. Annals of Probability, 43(6):3177-3215, 2015. · Zbl 1345.60111 |

[5] | P. Billingsley. Convergence of probability measures. Wiley Series in probability and Mathematical Statistics: Tracts on probability and statistics. Wiley, 1968. · Zbl 0172.21201 |

[6] | P. Billingsley. Probability and Measure. Wiley Series in probability and Mathematical Statistics. Wiley, 3 edition, 1995. · Zbl 0822.60002 |

[7] | N. Bonichon and J.-F. Marckert. Asymptotics of geometrical navigation on a random set of points in the plane. Adv. in Appl. Probab., 43(4):899-942, 12 2011. · Zbl 1238.60014 |

[8] | C. Coletti, E. Dias, and L. Fontes. Scaling limit for a drainage network model. Journal of Applied Probability, 46(4):1184-1197, 2009. · Zbl 1186.60104 |

[9] | C. Coletti and L. Valencia. Scaling limit for a family of random paths with radial behavior. arXiv:1310.6929, 2014. |

[10] | C. Coletti and G. Valle. Convergence to the Brownian web for a generalization of the drainage network model. Annales de l’Institut Henri Poincaré, 50(3):899-919, 2014. · Zbl 1296.60080 |

[11] | C. F. Coletti, L. R. G. Fontes, and E. S. Dias. Scaling limit for a drainage network model. J. Appl. Probab., 46(4):1184-1197, 12 2009. · Zbl 1186.60104 |

[12] | D. Coupier, K. Saha, A. Sarkar, and V. Tran. The 2d-directed spanning forest converges to the Brownian web. Technical report, arXiv:1805.09399, 2018. |

[13] | D. Coupier and V. Tran. The 2d-directed spanning forest is almost surely a tree. Random Structures and Algorithms, 42(1):59-72, 2013. · Zbl 1257.05159 |

[14] | P. Ferrari, L. Fontes, and X.-Y. Wu. Two-dimensional Poisson trees converge to the Brownian web. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 41(5):851-858, 2005. · Zbl 1073.60094 |

[15] | P. Ferrari, C. Landim, and H. Thorisson. Poisson trees, succession lines and coalescing random walks. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 40(2):141-152, 2004. · Zbl 1042.60064 |

[16] | L. Fontes, L. Valencia, and G. Valle. Scaling limit of the radial poissonian web. Technical report, arXiv:1403.5286, 2014. · Zbl 1321.60209 |

[17] | L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar. The brownian web: Characterization and convergence. Ann. Probab., 32(4):2857-2883, 10 2004. · Zbl 1105.60075 |

[18] | L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar. Coarsening, nucleation, and the marked Brownian web. Ann. Inst. H. Poincaré Probab. Statist., 42(1):37-60, 2006. · Zbl 1087.60072 |

[19] | C. Fortuin, P. Kasteleyn, and J. Ginibre. Correlation inequalities on some partially ordered sets. Commun. Math. Phys., 22:89-103, 1970. · Zbl 0346.06011 |

[20] | M. Fulmek. Nonintersecting lattice paths on the cylinder. Séminaire Lotharingien de Combinatoire, 52:B52b, 2004. 16 pages. · Zbl 1064.05013 |

[21] | S. Gangopadhyay, R. Roy, and A. Sarkar. Random oriented trees: a model of drainage networks. Ann. App. Probab., 14(3):1242-1266, 2004. · Zbl 1047.60098 |

[22] | C. D. Howard and C. M. Newman. Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab., 29(2):577-623, 2001. · Zbl 1062.60099 |

[23] | C. Newman, K. Ravishankar, and R. Sun. Convergence of coalescing nonsimple random walks to the Brownian web. Electronic Journal of Probability, 10(2):21-60, 2005. · Zbl 1067.60099 |

[24] | J. Norris and A. Turner. Planar aggregation and the coalescing Brownian flow. arXiv:0810.0211, 2008. |

[25] | J. Norris and A. Turner. Hastings-levitov aggregation in the small-particle limit. Communication in Mathematical Physics, 316(3):809-841, 2012. · Zbl 1259.82026 |

[26] | J. Norris and A. Turner. Weak convergence of the localized disturbance flow to the coalescing Brownian flow. Annals of Probability, 43(3):935-970, 2015. · Zbl 1327.60086 |

[27] | D. Revuz and M. J. Yor. Continuous martingales and Brownian motion. Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg, 1994. · Zbl 0804.60001 |

[28] | R. Roy, K. Saha, and A. Sarkar. Random directed forest and the Brownian web. Annales de l’Institut Henri Poincaré, 52(3):1106-1143, 2016. · Zbl 1375.60038 |

[29] | E. Schertzer, R. Sun, and J. Swart. The Brownian web, the Brownian net, and their universality. In Advances in Disordered Systems, Random Processes and Some Applications, pages 270-368. Cambridge University Press, 2017. Survey based on a course given in the Institut Henri Poincaré trimestre program on Disordered Systems, Random Spatial Processes and Some Applications, Jan. 5-Apr. 3 2015. |

[30] | F. Soucaliuc and W. Werner. A Note on Reflecting Brownian Motions. Electron. Commun. Probab., 7:117-122, 2002. · Zbl 1009.60068 |

[31] | F. Soucaliuc, B. Tóth, and W. Werner. Reflection and coalescence between one-dimensional Brownian paths. Ann. Inst. Henri Poincaré. Probab. Statist., 36:509-536, 2000. · Zbl 0968.60072 |

[32] | R. Sun. Convergence of Coalescing Nonsimple Random Walks to the Brownian Web. PhD thesis, New York University, arXiv:physics/0501141v1, 2005. · Zbl 1067.60099 |

[33] | B. Tóth and W. Werner. The true self-repelling motion. Probability Theory and Related Fields, 111:375-452, 1998. · Zbl 0912.60056 |

[34] | G. Valle and L. Zuaznábar. A version of the random directed forest and its convergence to the Brownian web. arXiv:1704.05555, 2017. |

[35] | F. J. Viklund, A. Sola, and A. Turner. Scaling limits of anisotropic Hasting-Levitov clusters. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 48(1):235-257, 2012. · Zbl 1251.82025 |

[36] | F. J. Viklund, A. Sola, and A. Turner. Small particle limits in a regularized Laplacian random growth model. Communication in Mathematical Physics, 334(1):331-366, 2015. · Zbl 1310.82019 |

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