The frog model on Galton-Watson trees
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2019
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| _version_ | 1866914648126652416 |
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| author | Michelen, Marcus Rosenberg, Josh |
| author_facet | Michelen, Marcus Rosenberg, Josh |
| contents | We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d.~$\mathrm{Poiss}(λ)$ many inactive particles are placed at each non-root vertex. Active particles perform discrete time simple random walk and activate the inactive particles they encounter. We show that for Galton-Watson trees with offspring distributions $Z$ satisfying $\mathbf{P}(Z \geq 2) = 1$ and $\mathbf{E}[Z^{4 + ε}] < \infty$ for some $ε> 0$, there is a critical value $λ_c\in(0,\infty)$ separating recurrent and transient regimes for almost surely every tree, thereby answering a question of Hoffman-Johnson-Junge. In addition, we also establish that this critical parameter depends on the entire offspring distribution, not just the maximum value of $Z$, answering another question of Hoffman-Johnson-Junge and showing that the frog model and contact process behave differently on Galton-Watson trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_02367 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The frog model on Galton-Watson trees Michelen, Marcus Rosenberg, Josh Probability 60K35 We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d.~$\mathrm{Poiss}(λ)$ many inactive particles are placed at each non-root vertex. Active particles perform discrete time simple random walk and activate the inactive particles they encounter. We show that for Galton-Watson trees with offspring distributions $Z$ satisfying $\mathbf{P}(Z \geq 2) = 1$ and $\mathbf{E}[Z^{4 + ε}] < \infty$ for some $ε> 0$, there is a critical value $λ_c\in(0,\infty)$ separating recurrent and transient regimes for almost surely every tree, thereby answering a question of Hoffman-Johnson-Junge. In addition, we also establish that this critical parameter depends on the entire offspring distribution, not just the maximum value of $Z$, answering another question of Hoffman-Johnson-Junge and showing that the frog model and contact process behave differently on Galton-Watson trees. |
| title | The frog model on Galton-Watson trees |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/1910.02367 |