On one-dimensional Cluster cluster model
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913927014645760 |
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| author | Berger, Noam Procaccia, Eviatar B. Sharon, Daniel |
| author_facet | Berger, Noam Procaccia, Eviatar B. Sharon, Daniel |
| contents | The Cluster-cluster model was introduced by Meakin et al in 1984. Each $x\in \mathbb{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently. Each cluster $C$ performs a continuous-time SRW with rate $|C|^{-α}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension $d=1$, we show that for $α>-2$, at time $t$, the cluster size is of order $t^\frac{1}{α+ 2}$, and for $α< -2$ we get an infinite cluster in finite time a.s. Additionally, for $α= 0$ we show convergence in distribution of the scaling limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On one-dimensional Cluster cluster model Berger, Noam Procaccia, Eviatar B. Sharon, Daniel Probability Mathematical Physics The Cluster-cluster model was introduced by Meakin et al in 1984. Each $x\in \mathbb{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently. Each cluster $C$ performs a continuous-time SRW with rate $|C|^{-α}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension $d=1$, we show that for $α>-2$, at time $t$, the cluster size is of order $t^\frac{1}{α+ 2}$, and for $α< -2$ we get an infinite cluster in finite time a.s. Additionally, for $α= 0$ we show convergence in distribution of the scaling limit. |
| title | On one-dimensional Cluster cluster model |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2507.03552 |