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PreprintCC BY 4.0arXiv

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

  • Krishnamurthi Ravishankar
  • Kumarjit Saha
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{ravishankar2026convergence,
  title = {{Convergence of a drainage network model with branching to the Brownian net}},
  author = {Krishnamurthi Ravishankar and Kumarjit Saha},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40555},
  eprint = {2610.40555},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40555},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Convergence of a drainage network model with branching to the Brownian net
AU  - Krishnamurthi Ravishankar
AU  - Kumarjit Saha
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40555
UR  - https://arxiv.org/abs/2610.40555
AB  - 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.
ER  -