Convergence of a drainage network model with branching to the Brownian net

We consider a drainage network model represented by a system of coalescing random walks exhibiting long range dependence before coalescence. This model was introduced by Gangopadhyay et. al. [GRS04] and Coletti et. al. [CFD09] proved its convergence to the Brownian web under diffusive scaling. Santos et. al. [SVZ24] considered a branching version of this model where branching occurs with low probabilities varying with the scaling parameter and showed that any sub-sequential limit contains a copy of the Brownian net. It was conjectured that the limit is indeed the Brownian net. In this work we prove this conjecture. Further, compared to [SVZ24], we explore the question of scaling limit in a wider spectrum of scaling regime. This work contributes to the broader understanding of the basin of attraction of the Brownian net.