Direct Paths in the Temporal Hypercube
Fuente:
arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916964488708096 |
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| author | Eide, Austin Gösgens, Martijn Prałat, Paweł |
| author_facet | Eide, Austin Gösgens, Martijn Prałat, Paweł |
| contents | We consider the $n$-dimensional random temporal hypercube, i.e., the $n$-dimensional hypercube graph with its edges endowed with i.i.d. continuous random weights. We say that a vertex $w$ is accessible from another vertex $v$ if and only if there is a path from $v$ to $w$ with increasing edge weights. We study accessible "direct" paths from a fixed vertex to its antipodal point and show that as $n \to \infty$, the number of such paths converges in distribution to a mixed Poisson law with mixture given by the product of two independent exponentials with rate $1$. Our proof makes use of the Chen-Stein method, coupling arguments, as well as combinatorial arguments which show that typical pairs of accessible paths have small overlap. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Direct Paths in the Temporal Hypercube Eide, Austin Gösgens, Martijn Prałat, Paweł Probability Combinatorics We consider the $n$-dimensional random temporal hypercube, i.e., the $n$-dimensional hypercube graph with its edges endowed with i.i.d. continuous random weights. We say that a vertex $w$ is accessible from another vertex $v$ if and only if there is a path from $v$ to $w$ with increasing edge weights. We study accessible "direct" paths from a fixed vertex to its antipodal point and show that as $n \to \infty$, the number of such paths converges in distribution to a mixed Poisson law with mixture given by the product of two independent exponentials with rate $1$. Our proof makes use of the Chen-Stein method, coupling arguments, as well as combinatorial arguments which show that typical pairs of accessible paths have small overlap. |
| title | Direct Paths in the Temporal Hypercube |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2509.18931 |