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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.16174 |
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Table of 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.