The $\ell_{\infty}$ Directed Spanning Forest
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911328588791808 |
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| author | Pal, Dipranjan Saha, Kumarjit |
| author_facet | Pal, Dipranjan Saha, Kumarjit |
| contents | We study the $\ell_{\infty}$\textit{ directed spanning forest}(DSF), which is a directed forest with vertex set given by a homogeneous Poisson point process such that each Poisson point connects to the nearest Poisson point (in $\ell_{\infty}$ distance) with a strictly larger $y$-coordinate. In this paper, we prove that the $\ell_{\infty}$ DSF is connected and we find optimal estimates on the tail distribution of coalescing time of two $\ell_{\infty}$ DSF paths. Similar estimates were earlier obtained in \cite{coupier20212d} for the $\ell_2$ (Euclidean) DSF and showed that when properly scaled, it converges in distribution to the Brownian web. The geometry of $\ell_\infty$ balls compel us to develop new argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02594 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $\ell_{\infty}$ Directed Spanning Forest Pal, Dipranjan Saha, Kumarjit Probability We study the $\ell_{\infty}$\textit{ directed spanning forest}(DSF), which is a directed forest with vertex set given by a homogeneous Poisson point process such that each Poisson point connects to the nearest Poisson point (in $\ell_{\infty}$ distance) with a strictly larger $y$-coordinate. In this paper, we prove that the $\ell_{\infty}$ DSF is connected and we find optimal estimates on the tail distribution of coalescing time of two $\ell_{\infty}$ DSF paths. Similar estimates were earlier obtained in \cite{coupier20212d} for the $\ell_2$ (Euclidean) DSF and showed that when properly scaled, it converges in distribution to the Brownian web. The geometry of $\ell_\infty$ balls compel us to develop new argument. |
| title | The $\ell_{\infty}$ Directed Spanning Forest |
| topic | Probability |
| url | https://arxiv.org/abs/2503.02594 |