In this paper, we study independent (Bernoulli) bond percolation in dimensions d 2, focusing on the maximum diameter of finite clusters in the non-critical regime (p pc). We prove that the maximum diameter Rₙ satisfies Rₙ / n (p) almost surely, where (p) is determined by the exponential decay rate ξ (p) of Pₚ (0 Bₙ, | C₀| ρ n\} for ρ> (p). Finally, we consider the asymptotics of the number of vertices in clusters with large diameters.
Ken‐ichiro Kobayashi (Thu,) studied this question.