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Convergence of coalescing nonsimple random walks to the Brownian web. (English) Zbl 1067.60099
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 The idea of the latter originates from the unpublished {\it R. Arratia}’s thesis (1979), where a process of coalescing Brownian motions on a line, starting from every point on $R$ at time zero, has been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)].

MSC:
60K35Interacting random processes; statistical mechanics type models; percolation theory
60J65Brownian motion
60D05Geometric probability and stochastic geometry
82C22Interacting particle systems
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