Maximum Cluster Diameter in Non-Critical Bond Percolation

TL;DR

This work analyzes the geometry of finite clusters in non-critical Bernoulli bond percolation on $\mathbb{Z}^d$, focusing on the maximal cluster diameter within a box. It proves that the maximum diameter $R_n$ scales like $\varkappa(p)\log n$, with $\varkappa(p)=d/\xi(p)$ where $\xi(p)$ is the exponential decay rate of the finite-cluster connection probability, and establishes a large deviation principle for $R_n$ together with precise asymptotics for the count of vertices in large-diameter clusters. The results extend prior studies from maximal volume to diameter, linking geometric properties to one-arm decay and offering quantitative insights into rare large-diameter events in the non-critical regime. These findings enhance understanding of finite-cluster geometry and provide tools for analyzing extreme-scale connectivity in 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 ($p\neq p_c$). 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 $ξ(p)$ of $P_p(0 \leftrightarrow \partial B_n, |\mathcal C_0|<\infty)$. Furthermore, we establish a large deviation principle for the event $\{R_n > ρ\log n\}$ for $ρ> \varkappa (p)$. Finally, we consider the asymptotics of the number of vertices in clusters with large diameters.

Figures (6)

• Figure 1.1 : (a) $p=0.25$
• Figure 1.2 : (b) $p=0.49$
• Figure 1.3 : (c) $p=0.51$
• Figure 1.4 : (d) $p=0.75$
• Figure 1.6 : A sketch of the event $\{0 \leftrightarrow \partial B_n , |\mathcal{C}_0| < \infty\}$: the origin belongs to a finite cluster which intersects $\partial B_n$.

Theorems & Definitions (12)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1: Borel–Cantelli
• Lemma 3.2: Boole's inequality
• proof : Proof of Theorem \ref{['thm:LLN:Rn']}
• Lemma 3.3
• proof : Proof of Lemma \ref{['lem:forpfofthm']}
• proof : Proof of Theorem \ref{['thm:Rnfb=Rnzb']}