Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2512.16174 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912838018138112 |
|---|---|
| author | Kobayashi, Kaito |
| author_facet | Kobayashi, Kaito |
| contents | In this paper, we study independent (Bernoulli) bond percolation in dimensions $d \ge 2$, focusing on the maximum diameter of finite clusters in the non-critical regime ($p\neq p_c$). We prove that the maximum diameter $R_n$ satisfies $R_n / \log n \to \varkappa(p)$ almost surely, where $\varkappa(p)$ is determined by the exponential decay rate $ξ(p)$ of $P_p(0 \leftrightarrow \partial B_n, |\mathcal C_0|<\infty)$. Furthermore, we establish a large deviation principle for the event $\{R_n > ρ\log n\}$ for $ρ> \varkappa (p)$. Finally, we consider the asymptotics of the number of vertices in clusters with large diameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximum Cluster Diameter in Non-Critical Bond Percolation Kobayashi, Kaito Probability Mathematical Physics 60K35 In this paper, we study independent (Bernoulli) bond percolation in dimensions $d \ge 2$, focusing on the maximum diameter of finite clusters in the non-critical regime ($p\neq p_c$). We prove that the maximum diameter $R_n$ satisfies $R_n / \log n \to \varkappa(p)$ almost surely, where $\varkappa(p)$ is determined by the exponential decay rate $ξ(p)$ of $P_p(0 \leftrightarrow \partial B_n, |\mathcal C_0|<\infty)$. Furthermore, we establish a large deviation principle for the event $\{R_n > ρ\log n\}$ for $ρ> \varkappa (p)$. Finally, we consider the asymptotics of the number of vertices in clusters with large diameters. |
| title | Maximum Cluster Diameter in Non-Critical Bond Percolation |
| topic | Probability Mathematical Physics 60K35 |
| url | https://arxiv.org/abs/2512.16174 |