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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2512.06640 |
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| _version_ | 1866915752104165376 |
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| author | Angel, Omer de la Riva, Daniel Hermon, Jonathan Shi, Yuliang |
| author_facet | Angel, Omer de la Riva, Daniel Hermon, Jonathan Shi, Yuliang |
| contents | We consider a slight modification of the frog model. For a given graph, each vertex has $\mathrm{Poisson}(λ)$ particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent, continuous-time simple random walk, becoming inactive after time $t$. Once an active frog jumps to a vertex, it activates all of its particles. The survival of active particles can be studied as a dependent percolation model with two parameters $λ$ and $t$. In the present work, we establish the existence of a phase transition with respect to each parameter for non-amenable graphs of bounded degrees and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06640 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and sharpness of the phase transition for the frog model on transitive graphs Angel, Omer de la Riva, Daniel Hermon, Jonathan Shi, Yuliang Probability We consider a slight modification of the frog model. For a given graph, each vertex has $\mathrm{Poisson}(λ)$ particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent, continuous-time simple random walk, becoming inactive after time $t$. Once an active frog jumps to a vertex, it activates all of its particles. The survival of active particles can be studied as a dependent percolation model with two parameters $λ$ and $t$. In the present work, we establish the existence of a phase transition with respect to each parameter for non-amenable graphs of bounded degrees and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs. |
| title | Existence and sharpness of the phase transition for the frog model on transitive graphs |
| topic | Probability |
| url | https://arxiv.org/abs/2512.06640 |