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Zbl 1067.60099
Newman, Charles M.; Ravishankar, Krishnamurthi; Sun, Rongfeng
Convergence of coalescing nonsimple random walks to the Brownian web. (English)
[J] Electron. J. Probab. 10, Paper No. 2, 21-60, electronic only (2005). ISSN 1083-6489

The paper addresses discrete time processes of coalescing (annihilating) random walks. A specific subclass of non-simple random walks allows for multiple intersections before they eventually annihilate. The main goal of the paper is to prove the weak convergence of coalescing non-simple random walks to the so-called Brownian web. The idea of the latter originates from the unpublished {\it R. Arratia}'s thesis (1979), where a process of coalescing Brownian motions on a line, starting from every point on $R$ at time zero, has been constructed. The process of annihilating Brownian motions starting from every point in space and time is called the Brownian web. Major convergence proofs were formulated for the case of simple random walks. Because of multiple crossings presupposed for non-simple walks, the convergence analysis is more subtle. The new convergence criteria for the case of crossing paths are formulated and verified for non-simple random walks satisfying a finite fifth moment condition. Several corollaries pertain to the scaling limit of voter model interfaces, extending results of {\it J. T. Cox} and {\it R. Durrett} [Bernoulli 1, 343-370 (1995; Zbl 0849.60088)].
[Piotr Garbaczewski (Zielona Góra)]

MSC 2000:: *60K35 Interacting random processes
60J65 Brownian motion
60D05 Geometric probability
82C22 Interacting particle systems

Keywords: annihilating Brownian motions; crossing paths; scaling limits; non-simple random walks; Brownian networks

Citations: Zbl 0849.60088

Cited in: Zbl 1113.60092

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