Maximum Cluster Diameter in Non-Critical Bond Percolation

2025/12/18 by Kobayashi, Kaito
#60K35 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.16174

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≠ p_c). We prove that the maximum diameter R_n satisfies R_n / log n → \varkappa(p) almost surely, where \varkappa(p) is determined by the exponential decay rate ξ(p) of P_p(0 ↔ ∂ B_n, |\mathcal C₀|<∞). Furthermore, we establish a large deviation principle for the event \R_n > ρlog n\ for ρ> \varkappa (p). Finally, we consider the asymptotics of the number of vertices in clusters with large diameters.

Citations

• On the upper tail large deviation rate function for chemical distance in supercritical percolation
• Selected problems in probability theory
• Sixty years of percolation
• Bootstrap percolation, and other automata
• Random graph asymptotics on high-dimensional tori
• Maximal clusters in non-critical percolation and related models

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