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

##### MSC:
 82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics 60D05 Geometric probability and stochastic geometry 60F17 Functional limit theorems; invariance principles 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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