On the convergence of the drainage network with branching
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910538284400640 |
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| author | Santos, Rafael Valle, Glauco Zuaznábar, Leonel |
| author_facet | Santos, Rafael Valle, Glauco Zuaznábar, Leonel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04235 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the convergence of the drainage network with branching Santos, Rafael Valle, Glauco Zuaznábar, Leonel Probability 60K35 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. |
| title | On the convergence of the drainage network with branching |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2206.04235 |