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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.07816 |
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| _version_ | 1866912818702319616 |
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| author | Hoffman, Christopher Richey, Jacob Son, Hyojeong |
| author_facet | Hoffman, Christopher Richey, Jacob Son, Hyojeong |
| contents | We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07816 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Density of Activated Random Walk on $\mathbb{Z}$ Hoffman, Christopher Richey, Jacob Son, Hyojeong Probability We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW. |
| title | Local Density of Activated Random Walk on $\mathbb{Z}$ |
| topic | Probability |
| url | https://arxiv.org/abs/2601.07816 |