Asymptotic behavior of the critical density of activated random walk
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908681958850560 |
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| author | Kaufman, Harley Meisel, Josh |
| author_facet | Kaufman, Harley Meisel, Josh |
| contents | We study the asymptotic behavior of the critical density of the activated random walk model as the sleep rate $λ$ tends to $0$ and $\infty$. For large $λ$, we prove new lower bounds in dimensions 1 and 2, showing that in one dimension the critical density approaches $1$ superpolynomially fast. For small $λ$, we prove a new lower bound in two dimensions for how fast the critical density vanishes. We also obtain the first-order approximation for transient walks in both regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00720 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of the critical density of activated random walk Kaufman, Harley Meisel, Josh Probability 82C22, 60K35, 82C26 We study the asymptotic behavior of the critical density of the activated random walk model as the sleep rate $λ$ tends to $0$ and $\infty$. For large $λ$, we prove new lower bounds in dimensions 1 and 2, showing that in one dimension the critical density approaches $1$ superpolynomially fast. For small $λ$, we prove a new lower bound in two dimensions for how fast the critical density vanishes. We also obtain the first-order approximation for transient walks in both regimes. |
| title | Asymptotic behavior of the critical density of activated random walk |
| topic | Probability 82C22, 60K35, 82C26 |
| url | https://arxiv.org/abs/2512.00720 |