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 $R_n/\log n\to \varkappa (p)$ almost surely, where $\varkappa (p)$ is determined by the exponential decay rate $\hat{I}¾(p)$ of $P_p(0\leftrightarrow \partial B_n,|{\mathcal{C}}_0|<\infty )$. Furthermore, we establish a large deviation principle for the event $\{R_n>\ddot{I}\log n\}$ for $\ddot{I}>\varkappa (p)$. Finally, we consider the asymptotics of the number of vertices in clusters with large diameters.

16 pages, 3 figures