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The Brownian net. (English) Zbl 1143.82020
Consider a collection of branching-coalescing random walks starting from each point of space and time ${\Bbb Z}^2$. Under diffusive rescaling, the collection is shown to converge in law to a certain random object which is called the Brownian net ${\cal N}_b$ with branching parameter $b$. The Brownian net is a generalization of the Brownian web ${\cal W}$, which is developed e.g. in {\it L. R. G. Fontes, M. Isopi, C. M. Newman} and {\it K. Ravishankar} [Ann. Prob. 32, 2857--2883 (2004; Zbl 1105.60075)]. In fact, ${\cal N}_0$ is equal to ${\cal W}$ in distribution, while ${\cal N}_b$ with $b\ne 0$ differ from ${\cal W}$. Some properties of the Brownian net are also given.

MSC:
82C21Dynamic continuum models (systems of particles, etc.)
60D05Geometric probability and stochastic geometry
60F17Functional limit theorems; invariance principles
60K35Interacting random processes; statistical mechanics type models; percolation theory
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Full Text: DOI arXiv
References:
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