Long-range first-passage percolation on the torus
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908598078013440 |
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| author | van der Hofstad, Remco Lodewijks, Bas |
| author_facet | van der Hofstad, Remco Lodewijks, Bas |
| contents | We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph $\mathcal K_n$ are embedded in the $d$-dimensional torus $\mathbb T_n^d$, and each edge $e$ is assigned an independent transmission time $T_e=\|e\|_{\mathbb T_n^d}^αE_e$, where $E_e$ is a rate-one exponential random variable associated with the edge $e$, $\|\cdot\|_{\mathbb T_n^d}$ denotes the torus-norm, and $α\geq0$ is a parameter. We are interested in the case $α\in[0,d)$, which corresponds to the instantaneous percolation regime for long-range first-passage percolation on $\mathbb Z^d$ studied by Chatterjee and Dey, and which extends first-passage percolation on the complete graph (the $α=0$ case) studied by Janson. We consider the typical distance, flooding time, and diameter of the model. Our results show a $1,2,3$-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on $\mathbb Z^d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_16088 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Long-range first-passage percolation on the torus van der Hofstad, Remco Lodewijks, Bas Probability We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph $\mathcal K_n$ are embedded in the $d$-dimensional torus $\mathbb T_n^d$, and each edge $e$ is assigned an independent transmission time $T_e=\|e\|_{\mathbb T_n^d}^αE_e$, where $E_e$ is a rate-one exponential random variable associated with the edge $e$, $\|\cdot\|_{\mathbb T_n^d}$ denotes the torus-norm, and $α\geq0$ is a parameter. We are interested in the case $α\in[0,d)$, which corresponds to the instantaneous percolation regime for long-range first-passage percolation on $\mathbb Z^d$ studied by Chatterjee and Dey, and which extends first-passage percolation on the complete graph (the $α=0$ case) studied by Janson. We consider the typical distance, flooding time, and diameter of the model. Our results show a $1,2,3$-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on $\mathbb Z^d$. |
| title | Long-range first-passage percolation on the torus |
| topic | Probability |
| url | https://arxiv.org/abs/2311.16088 |