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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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