Subcritical Boolean percolation on graphs of bounded degree
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908671787663360 |
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| author | Faipeur, Corentin |
| author_facet | Faipeur, Corentin |
| contents | In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and vertices are said to be active independently with probability $p \in [0, 1]$. We consider $W$ to be the reunion of random balls with an active center. In certain circumstances, the model does not exhibit a phase transition, in the sense that $W$ almost surely contains an infinite component for all $p > 0$, or even $W$ covers the entire graph. In this paper, we give a sufficient condition on the radius distribution for the existence of a subcritical phase, namely a regime such that almost surely all the connected components of $W$ are finite. Additionally, we provide a sufficient condition for the exponential decay of the size of a typical component. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15722 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subcritical Boolean percolation on graphs of bounded degree Faipeur, Corentin Probability In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and vertices are said to be active independently with probability $p \in [0, 1]$. We consider $W$ to be the reunion of random balls with an active center. In certain circumstances, the model does not exhibit a phase transition, in the sense that $W$ almost surely contains an infinite component for all $p > 0$, or even $W$ covers the entire graph. In this paper, we give a sufficient condition on the radius distribution for the existence of a subcritical phase, namely a regime such that almost surely all the connected components of $W$ are finite. Additionally, we provide a sufficient condition for the exponential decay of the size of a typical component. |
| title | Subcritical Boolean percolation on graphs of bounded degree |
| topic | Probability |
| url | https://arxiv.org/abs/2410.15722 |