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This research investigates the behavior of maximum cluster diameter in non-critical bond percolation.

December 22, 2025Open Access

Maximum Cluster Diameter in Non-Critical Bond Percolation

Key Points

This research investigates the behavior of maximum cluster diameter in non-critical bond percolation.
Studied independent Bernoulli bond percolation in dimensions d ≥ 2
Proved the relation R_n / log n → varkappa(p) almost surely
Established a large deviation principle for R_n > ρ log n when ρ > varkappa(p)
Analyzed asymptotics of clusters with large diameters
Identified a relationship between maximum diameter and logarithmic scales
Demonstrated exponential decay rates influence the maximum diameter
Provided insights into the frequency of large clusters in non-critical scenarios.

Abstract

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.

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Authors

Ken‐ichiro Kobayashi

Japanese Red Cross Society Wakayama Medical Center

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Maximum Cluster Diameter in Non-Critical Bond Percolation

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Abstract

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