Maximum Cluster Diameter in Non-critical Bond Percolation
Abstract
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 ( $$p\ne p_c$$ ). We prove that the maximum diameter $$R_n$$ satisfies $$R_n / \log n \rightarrow \varkappa (p)$$ almost surely, where $$\varkappa (p)$$ is determined by the exponential decay rate $$\xi (p)$$ of $$P_p(0 \leftrightarrow \partial B_n, |\mathcal {C}_0| \rho \log n\}$$ for $$\rho > \varkappa (p)$$ . Finally, we consider the asymptotics of the number of vertices in clusters with large diameters.
Fields of Study & Subjects
Stochastic processes and statistical mechanics
Random Matrices and Applications
Theoretical and Computational Physics
Publication details
- Journal / source
- Journal of Statistical Physics
- Publisher
- Springer Science and Business Media LLC
- Volume
- 193
- Issue
- 7
- Publication date
- 4 July 2026
- Language
- EN