Improved Survival Results for the One-Dimensional Renewal Contact Process
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918529667694592 |
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| author | de Carvalho, Gustavo O. de Lima, Lucas R. |
| author_facet | de Carvalho, Gustavo O. de Lima, Lucas R. |
| contents | The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $μ$. We establish new sufficient conditions ensuring finiteness of the critical infection parameter $λ_c(μ)$ for the one-dimensional model. In particular, we prove that $λ_c(μ)<+\infty$ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30121 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improved Survival Results for the One-Dimensional Renewal Contact Process de Carvalho, Gustavo O. de Lima, Lucas R. Probability 60K35, 82C22, 60G55 The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $μ$. We establish new sufficient conditions ensuring finiteness of the critical infection parameter $λ_c(μ)$ for the one-dimensional model. In particular, we prove that $λ_c(μ)<+\infty$ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument. |
| title | Improved Survival Results for the One-Dimensional Renewal Contact Process |
| topic | Probability 60K35, 82C22, 60G55 |
| url | https://arxiv.org/abs/2605.30121 |