The Critical Density for Activated Random Walks is always less than 1
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912009182773248 |
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| author | Asselah, Amine Forien, Nicolas Gaudillière, Alexandre |
| author_facet | Asselah, Amine Forien, Nicolas Gaudillière, Alexandre |
| contents | Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during an exponential time of parameter $λ$, after which it stays still in the frozen state, until another active particle shares its location, and turns it instantaneously back into activity. This model is known to have a phase transition, and we show that the critical density, controlling the phase transition, is less than one in any dimension and for any value of the sleep rate $λ$. We provide upper bounds for the critical density in both the small $λ$ and large $λ$ regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04779 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Critical Density for Activated Random Walks is always less than 1 Asselah, Amine Forien, Nicolas Gaudillière, Alexandre Probability 60K35, 82B26 Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during an exponential time of parameter $λ$, after which it stays still in the frozen state, until another active particle shares its location, and turns it instantaneously back into activity. This model is known to have a phase transition, and we show that the critical density, controlling the phase transition, is less than one in any dimension and for any value of the sleep rate $λ$. We provide upper bounds for the critical density in both the small $λ$ and large $λ$ regimes. |
| title | The Critical Density for Activated Random Walks is always less than 1 |
| topic | Probability 60K35, 82B26 |
| url | https://arxiv.org/abs/2210.04779 |