Random triangulations of the d-sphere with minimum volume
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Georgakopoulos, Agelos Haslegrave, John Danielsson, Joel Larsson |
| author_facet | Georgakopoulos, Agelos Haslegrave, John Danielsson, Joel Larsson |
| contents | We study a higher-dimensional analogue of the {Random Travelling Salesman Problem}: let the complete $d$-dimensional simplicial complex $K_n^{d}$ on $n$ vertices be equipped with i.i.d.\ volumes on its facets, uniformly random in $[0,1]$. What is the minimum volume $M_{n,d}$ of a sub-complex homeomorphic to the $d$-dimensional sphere $\mathbb{S}^d$, containing all vertices? We determine the growth rate of $M_{n,2}$, and prove that it is well-concentrated. For $d>2$ we prove such results to the extent that current knowledge about the number of triangulations of $\mathbb{S}^d$ allows.
We remark that this can be thought of as a model of random geometry in the spirit of Angel \& Schramm's UIPT, and provide a generalised framework that interpolates between our model and the uniform random triangulation of $\mathbb{S}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random triangulations of the d-sphere with minimum volume Georgakopoulos, Agelos Haslegrave, John Danielsson, Joel Larsson Probability Combinatorics 60C05 (Primary), 05C65, 60D05, 90C27 (Secondary) We study a higher-dimensional analogue of the {Random Travelling Salesman Problem}: let the complete $d$-dimensional simplicial complex $K_n^{d}$ on $n$ vertices be equipped with i.i.d.\ volumes on its facets, uniformly random in $[0,1]$. What is the minimum volume $M_{n,d}$ of a sub-complex homeomorphic to the $d$-dimensional sphere $\mathbb{S}^d$, containing all vertices? We determine the growth rate of $M_{n,2}$, and prove that it is well-concentrated. For $d>2$ we prove such results to the extent that current knowledge about the number of triangulations of $\mathbb{S}^d$ allows. We remark that this can be thought of as a model of random geometry in the spirit of Angel \& Schramm's UIPT, and provide a generalised framework that interpolates between our model and the uniform random triangulation of $\mathbb{S}^d$. |
| title | Random triangulations of the d-sphere with minimum volume |
| topic | Probability Combinatorics 60C05 (Primary), 05C65, 60D05, 90C27 (Secondary) |
| url | https://arxiv.org/abs/2409.00235 |