Two directed non-planar random networks and their scaling limits. (English) Zbl 1533.60180
Summary: We study two directed non-planar random graphs, each of which has a dependence structure. We prove that each of these models, under diffusive scaling, converges to the Brownian web. To obtain this, we first obtain a Markovian renewal structure of the paths of the graph and then study the coalescence time of any two paths. Finally, we show that the condition required by C. Coletti and G. Valle [Ann. Inst. Henri Poincaré, Probab. Stat. 50, No. 3, 899–919 (2014; Zbl 1296.60080)] in their study of the diffusive scaling limit of a generalized Howard model of drainage can be relaxed.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60D05 | Geometric probability and stochastic geometry |
| 60F17 | Functional limit theorems; invariance principles |
| 60J65 | Brownian motion |
Citations:
Zbl 1296.60080References:
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