The contact process on Scale-Free Percolation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913841158291456 |
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| author | Barnier, Andree Hoscheit, Patrick Salvi, Michele Vergu, Elisabeta |
| author_facet | Barnier, Andree Hoscheit, Patrick Salvi, Michele Vergu, Elisabeta |
| contents | We consider the contact process on scale-free percolation, a spatial random graph model where the degree distribution of the vertices follows a power law with exponent $β$. We study the extinction time $τ_{G_n}$ of the contact process on the graph restricted to a d-dimensional box of volume n, starting from full occupancy. In the regime $β\in (2, 3)$, where the degrees have finite mean but infinite variance and the graph exhibits the ultra-small world behaviour, we adapt the techniques of [Linker et al., 2021] to show that $τ_{G_n}$ is exponential in n. Our main contribution, though, deals with the case $β\geq 3$, where the degrees have finite variance and the graph is small-world. We prove that also in this case $τ_{G_n}$ grows exponentially, at least up to a logarithmic correction reflecting the sparser graph structure. The proof requires the generalization of a result from [Mountford et al., 2016] and combines a multi-scale analysis of the graph, the study of the chemical distance between vertices and percolation arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The contact process on Scale-Free Percolation Barnier, Andree Hoscheit, Patrick Salvi, Michele Vergu, Elisabeta Probability We consider the contact process on scale-free percolation, a spatial random graph model where the degree distribution of the vertices follows a power law with exponent $β$. We study the extinction time $τ_{G_n}$ of the contact process on the graph restricted to a d-dimensional box of volume n, starting from full occupancy. In the regime $β\in (2, 3)$, where the degrees have finite mean but infinite variance and the graph exhibits the ultra-small world behaviour, we adapt the techniques of [Linker et al., 2021] to show that $τ_{G_n}$ is exponential in n. Our main contribution, though, deals with the case $β\geq 3$, where the degrees have finite variance and the graph is small-world. We prove that also in this case $τ_{G_n}$ grows exponentially, at least up to a logarithmic correction reflecting the sparser graph structure. The proof requires the generalization of a result from [Mountford et al., 2016] and combines a multi-scale analysis of the graph, the study of the chemical distance between vertices and percolation arguments. |
| title | The contact process on Scale-Free Percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2505.10582 |