<p>In this paper, we study independent (Bernoulli) bond percolation in dimensions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, focusing on the maximum diameter of finite clusters in the non-critical regime (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p\ne p_c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <msub> <mi>p</mi> <mi>c</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>). We prove that the maximum diameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R_n / \log n \rightarrow \varkappa (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo stretchy="false">/</mo> <mo>log</mo> <mi>n</mi> <mo stretchy="false">→</mo> <mi>ϰ</mi> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> almost surely, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varkappa (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϰ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is determined by the exponential decay rate <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\xi (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$P_p(0 \leftrightarrow \partial B_n, |\mathcal {C}_0|<\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">↔</mo> <mi>∂</mi> </mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mrow> <mo>,</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="script">C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">|</mo> <mo><</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we establish a large deviation principle for the event <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\{R_n > \rho \log n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo>></mo> <mi>ρ</mi> <mo>log</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\rho > \varkappa (p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>></mo> <mi>ϰ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Finally, we consider the asymptotics of the number of vertices in clusters with large diameters.</p>

Maximum Cluster Diameter in Non-critical Bond Percolation

期刊论文
出版模式：

开放获取
发表日期：

2026年7月4日

查看全文

Kaito Kobayashi

摘要

In this paper, we study independent (Bernoulli) bond percolation in dimensions $d \ge 2$ $d \geq 2$ , focusing on the maximum diameter of finite clusters in the non-critical regime ( $p\ne p_c$ $p \neq p_{c}$ ). We prove that the maximum diameter $R_n$ $R_{n}$ satisfies $R_n / \log n \rightarrow \varkappa (p)$ $R_{n} / log n \to ϰ (p)$ almost surely, where $\varkappa (p)$ $ϰ (p)$ is determined by the exponential decay rate $\xi (p)$ $ξ (p)$ of $P_p(0 \leftrightarrow \partial B_n, |\mathcal {C}_0|<\infty )$ $P_{p} (0 \leftrightarrow \partial B_{n}, | C_{0} | < \infty)$ . Furthermore, we establish a large deviation principle for the event $\{R_n > \rho \log n\}$ ${R_{n} > ρ log n}$ for $\rho > \varkappa (p)$ $ρ > ϰ (p)$ . Finally, we consider the asymptotics of the number of vertices in clusters with large diameters.