The Frog Model on $\mathbb{Z}$ with General Random Survival Parameter
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917137986093056 |
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| author | Carvalho, Gustavo O. Machado, Fábio P. González, J. Hermenegildo R. |
| author_facet | Carvalho, Gustavo O. Machado, Fábio P. González, J. Hermenegildo R. |
| contents | We study the frog model on $\mathbb{Z}$ with particle-wise random geometric lifetimes: each particle has a survival parameter $π\in(0,1)$ sampled i.i.d., whose density near $1$ satisfies $f_π(u)\sim (1-u)^{β-1}L\big((1-u)^{-1}\big)$ with $β>0$, and $L$ slowly varying. This strictly extends the $\mathrm{Beta}(α,β)$ case. Let $η$ denote the common law of the i.i.d.\ initial number of particles $\{η_x\}_{x\in\mathbb{Z}}$. Using a percolation comparison and sharp one-particle displacement tails, we obtain a universal threshold at $β=\tfrac12$. If $β>\tfrac12$ and $E(η)<\infty$, extinction occurs almost surely. If $β<\tfrac12$ and $\mathbb{P}(η=0)<1$, survival has positive probability. At the boundary $β=\tfrac12$ we give sharp criteria: extinction if $E(η)<\infty$ and $8\,\limsup_{n\to\infty}L(n^2)<1/E(η)$; survival if $\mathbb{P}(η=0)<1$ and $\sqrt{2}\,\liminf_{n\to\infty}L(n^2)>1/E(η)$. These results recover the Carvalho-Machado threshold for Beta laws and show that only the exponent $β$ governs the phase transition, while $L$ impacts the critical regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Frog Model on $\mathbb{Z}$ with General Random Survival Parameter Carvalho, Gustavo O. Machado, Fábio P. González, J. Hermenegildo R. Probability 60K35, 60K37, 60G50, 60J80 We study the frog model on $\mathbb{Z}$ with particle-wise random geometric lifetimes: each particle has a survival parameter $π\in(0,1)$ sampled i.i.d., whose density near $1$ satisfies $f_π(u)\sim (1-u)^{β-1}L\big((1-u)^{-1}\big)$ with $β>0$, and $L$ slowly varying. This strictly extends the $\mathrm{Beta}(α,β)$ case. Let $η$ denote the common law of the i.i.d.\ initial number of particles $\{η_x\}_{x\in\mathbb{Z}}$. Using a percolation comparison and sharp one-particle displacement tails, we obtain a universal threshold at $β=\tfrac12$. If $β>\tfrac12$ and $E(η)<\infty$, extinction occurs almost surely. If $β<\tfrac12$ and $\mathbb{P}(η=0)<1$, survival has positive probability. At the boundary $β=\tfrac12$ we give sharp criteria: extinction if $E(η)<\infty$ and $8\,\limsup_{n\to\infty}L(n^2)<1/E(η)$; survival if $\mathbb{P}(η=0)<1$ and $\sqrt{2}\,\liminf_{n\to\infty}L(n^2)>1/E(η)$. These results recover the Carvalho-Machado threshold for Beta laws and show that only the exponent $β$ governs the phase transition, while $L$ impacts the critical regime. |
| title | The Frog Model on $\mathbb{Z}$ with General Random Survival Parameter |
| topic | Probability 60K35, 60K37, 60G50, 60J80 |
| url | https://arxiv.org/abs/2512.10171 |