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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.09766 |
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| _version_ | 1866915649809285120 |
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| author | de Carvalho, Gustavo O. Machado, Fábio P. |
| author_facet | de Carvalho, Gustavo O. Machado, Fábio P. |
| contents | We study the frog model on \( \mathbb{Z} \) with geometric lifetimes, introducing a random survival parameter. Active and inactive particles are placed at the vertices of \( \mathbb{Z} \). The lifetime of each active particle follows a geometric random variable with parameter \( 1-p \), where \( p \) is randomly sampled from a distribution \( π\). Each active particle performs a simple random walk on \( \mathbb{Z} \) until it dies, activating any inactive particles it encounters along its path. In contrast to the usual case where \( p \) is fixed, we show that there exist non-trivial distributions \( π\) for which the model survives with positive probability. More specifically, for $π\sim Beta(α,β)$, we establish the existence of a critical value \( β=0.5 \), that separates almost sure extinction from survival with positive probability. Furthermore, we show that the model is recurrent whenever it survives with positive probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09766 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Frog model on $\mathbb{Z}$ with random survival parameter de Carvalho, Gustavo O. Machado, Fábio P. Probability 60K35, 05C81 We study the frog model on \( \mathbb{Z} \) with geometric lifetimes, introducing a random survival parameter. Active and inactive particles are placed at the vertices of \( \mathbb{Z} \). The lifetime of each active particle follows a geometric random variable with parameter \( 1-p \), where \( p \) is randomly sampled from a distribution \( π\). Each active particle performs a simple random walk on \( \mathbb{Z} \) until it dies, activating any inactive particles it encounters along its path. In contrast to the usual case where \( p \) is fixed, we show that there exist non-trivial distributions \( π\) for which the model survives with positive probability. More specifically, for $π\sim Beta(α,β)$, we establish the existence of a critical value \( β=0.5 \), that separates almost sure extinction from survival with positive probability. Furthermore, we show that the model is recurrent whenever it survives with positive probability. |
| title | Frog model on $\mathbb{Z}$ with random survival parameter |
| topic | Probability 60K35, 05C81 |
| url | https://arxiv.org/abs/2503.09766 |