Explosivity in 1-d Activated Random Walk
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909983186092032 |
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| author | Forien, Nicolas Hoffman, Christopher Johnson, Tobias Meisel, Josh Richey, Jacob Rolla, Leonardo T. |
| author_facet | Forien, Nicolas Hoffman, Christopher Johnson, Tobias Meisel, Josh Richey, Jacob Rolla, Leonardo T. |
| contents | We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial distributions to the setting of ergodic ones, thus completing the proof of a conjecture of Rolla's in dimension one. As a corollary we obtain that, for supercritical ergodic initial distributions with any positive density of particles initially active, the system will stay active almost surely. Our result is another piece of evidence attesting to the universality of the phase transition of Activated Random Walk on $\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03411 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explosivity in 1-d Activated Random Walk Forien, Nicolas Hoffman, Christopher Johnson, Tobias Meisel, Josh Richey, Jacob Rolla, Leonardo T. Probability Statistical Mechanics 60K35 (Primary) 82C22 (Secondary) We show that Activated Random Walk on $\mathbb{Z}$ is explosive above criticality. That is, activating a single particle in a supercritical state of sleeping particles triggers an infinite avalanche of activity with positive probability. This extends the same result recently proven by Brown, Hoffman, and Son for i.i.d. initial distributions to the setting of ergodic ones, thus completing the proof of a conjecture of Rolla's in dimension one. As a corollary we obtain that, for supercritical ergodic initial distributions with any positive density of particles initially active, the system will stay active almost surely. Our result is another piece of evidence attesting to the universality of the phase transition of Activated Random Walk on $\mathbb{Z}$. |
| title | Explosivity in 1-d Activated Random Walk |
| topic | Probability Statistical Mechanics 60K35 (Primary) 82C22 (Secondary) |
| url | https://arxiv.org/abs/2601.03411 |