On the convergence of the drainage network with branching
The Drainage Network is a system of coalescing random walks, exhibiting long-range dependence before coalescence, introduced by Gangopadhyay, Roy, and Sarkar. Coletti, Fontes, and Dias proved its convergence to the Brownian Web under diffusive scaling. In this work, we introduce a perturbation of th...
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Zusammenfassung: | The Drainage Network is a system of coalescing random walks, exhibiting
long-range dependence before coalescence, introduced by Gangopadhyay, Roy, and
Sarkar. Coletti, Fontes, and Dias proved its convergence to the Brownian Web
under diffusive scaling. In this work, we introduce a perturbation of the
system allowing branching of the random walks with low probabilities varying
with the scaling parameter. When the branching probability is inversely
proportional to the scaling parameter, we show that this drainage network with
branching consists of a tight family such that any weak limit point contains a
Brownian Net. We conjecture that the limit is indeed the Brownian Net. |
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DOI: | 10.48550/arxiv.2206.04235 |