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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.15208 |
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| _version_ | 1866914655486607360 |
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| author | Campos, Alberto M. Teixeira, Augusto |
| author_facet | Campos, Alberto M. Teixeira, Augusto |
| contents | In this article, we study a covering process of the discrete one-dimensional torus that uses connected arcs of random sizes in the covering. More precisely, fix a distribution μon \mathbb{N}, and for every n\geq 1 we will cover the torus \mathbb{Z}/n\mathbb{Z} as follows: at each time step, we place an arc with a length distributed as μand a uniform starting point. Eventually, the space will be covered entirely by these arcs. Changing the arc length distribution μcan potentially change the limiting behavior of the covering time. Here, we expose four distinct phases for the fluctuations of the cover time in the limit. These phases can be informally described as the Gumbel phase, the compactly support phase, the pre-exponential phase, and the exponential phase. Furthermore, we expose a continuous-time cover process that works as a limit distribution within the compactly support phase. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Covering Distributions Campos, Alberto M. Teixeira, Augusto Probability 60G18, 60G55, 60F05 In this article, we study a covering process of the discrete one-dimensional torus that uses connected arcs of random sizes in the covering. More precisely, fix a distribution μon \mathbb{N}, and for every n\geq 1 we will cover the torus \mathbb{Z}/n\mathbb{Z} as follows: at each time step, we place an arc with a length distributed as μand a uniform starting point. Eventually, the space will be covered entirely by these arcs. Changing the arc length distribution μcan potentially change the limiting behavior of the covering time. Here, we expose four distinct phases for the fluctuations of the cover time in the limit. These phases can be informally described as the Gumbel phase, the compactly support phase, the pre-exponential phase, and the exponential phase. Furthermore, we expose a continuous-time cover process that works as a limit distribution within the compactly support phase. |
| title | Covering Distributions |
| topic | Probability 60G18, 60G55, 60F05 |
| url | https://arxiv.org/abs/2401.15208 |