Throughput in inhomogeneous planar drainage networks
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911018879287296 |
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| author | Ghosh, Partha Pratim Jahnel, Benedikt Steenbeck, Yannic |
| author_facet | Ghosh, Partha Pratim Jahnel, Benedikt Steenbeck, Yannic |
| contents | We consider navigation schemes on planar diluted lattices and semi lattices with one discrete and one continuous component. More precisely, nodes that survive inhomogeneous Bernoulli site percolation, or are placed as inhomogeneous Poisson points on shifted copies of $\mathbb{Z}$, forward their individually generated traffic to their respective closest neighbors to the left in the next layer. The resulting drainage network is a tree and we study the amount of traffic that goes through an increasing window at the origin. Our main results show that, properly rescaled, the total traffic, jointly with the total length of the contributing tree part, converges to the area under a time-inhomogeneous Brownian motion until it hits zero. The hitting time corresponds to the limiting maximal path length. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Throughput in inhomogeneous planar drainage networks Ghosh, Partha Pratim Jahnel, Benedikt Steenbeck, Yannic Probability Primary: 60D05, 60K30, 60F05, Secondary: 60K35, 90B20 We consider navigation schemes on planar diluted lattices and semi lattices with one discrete and one continuous component. More precisely, nodes that survive inhomogeneous Bernoulli site percolation, or are placed as inhomogeneous Poisson points on shifted copies of $\mathbb{Z}$, forward their individually generated traffic to their respective closest neighbors to the left in the next layer. The resulting drainage network is a tree and we study the amount of traffic that goes through an increasing window at the origin. Our main results show that, properly rescaled, the total traffic, jointly with the total length of the contributing tree part, converges to the area under a time-inhomogeneous Brownian motion until it hits zero. The hitting time corresponds to the limiting maximal path length. |
| title | Throughput in inhomogeneous planar drainage networks |
| topic | Probability Primary: 60D05, 60K30, 60F05, Secondary: 60K35, 90B20 |
| url | https://arxiv.org/abs/2506.18132 |