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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2403.18592 |
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| _version_ | 1866915313344315392 |
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| author | Durrett, Rick |
| author_facet | Durrett, Rick |
| contents | In this paper we will consider the contact process in a very simple type of random environment that physicists call the random dilution model. We start with the contact process on a graph, here either $\mathbb{Z}^d$, a $d$-dimensional torus or an \ER graph, and then flip independent $(1-p)$ coins to delete edges, or delete vertices. Let $p^*$ be the threshold for percolation in the diluted graph. We will primarily be concerned with two phenomena. (i) The critical value for the contact process on the dliuted graph $λ_c(p)$ does not converge to $\infty$ as $p \downarrow p^*$. (ii) In contrast to the contact process on a homogeneous graph, the density of 1's starting from all sites occupied converges to 0 at a polynomial rate when $p<p^*$ (the ``Griffiths phase'') and like $c/(\log t)^a$ when $p=p^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18592 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unusual properties of contact processes on percolated graphs Durrett, Rick Probability 60K35 In this paper we will consider the contact process in a very simple type of random environment that physicists call the random dilution model. We start with the contact process on a graph, here either $\mathbb{Z}^d$, a $d$-dimensional torus or an \ER graph, and then flip independent $(1-p)$ coins to delete edges, or delete vertices. Let $p^*$ be the threshold for percolation in the diluted graph. We will primarily be concerned with two phenomena. (i) The critical value for the contact process on the dliuted graph $λ_c(p)$ does not converge to $\infty$ as $p \downarrow p^*$. (ii) In contrast to the contact process on a homogeneous graph, the density of 1's starting from all sites occupied converges to 0 at a polynomial rate when $p<p^*$ (the ``Griffiths phase'') and like $c/(\log t)^a$ when $p=p^*$. |
| title | Unusual properties of contact processes on percolated graphs |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2403.18592 |