发表机构
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, University of Chinese Academy of Sciences; School of Mathematics, Shanghai University of Finance and Economics（中国科学院数学与系统科学研究院; 中国科学院大学数学科学学院; 上海财经大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机级联图边界顶点间距离与电阻的对数速度，证明其确定性极限，并揭示电阻速度近临界指数2/3，与距离的1/2指数形成对比。
AI 中文摘要
我们研究深度为$n$的随机级联图的两个边界顶点之间的图距离$D_n(p)$和有效电阻$R_n(p)$，该图通过以概率$p$串联、以概率$1-p$并联递归连接两个独立副本而获得。我们证明了确定性对数速度的存在性：对于每个$p\in[0,1]$，$n^{-1}\log D_n(p)$和$n^{-1}\log R_n(p)$几乎必然且依$L^1$收敛到确定性极限$v_D(p)$和$v_R(p)$。在距离情形下，对于每个$p\in[0,1]$，极限速度与相应的一阶矩对数速率一致；在电阻情形下，对于$p\in[1/2,1]$，二者一致。我们进一步确定了电阻速度的近临界行为：当$\delta\downarrow0$时，$v_R(\frac12+\delta)\sim 2\zeta(3)^{1/3}\lambda_*\delta^{2/3}$，其中$\lambda_*>0$由一个显式的非线性边值问题刻画。该指数$2/3$与Chen、Derrida、Duquesne和Shi（2026）获得的距离的指数$1/2$形成对比。本工作的主要思想由ChatGPT 5.6 Sol提出，作者对数学内容承担全部责任。三个主要定理及其支撑证明依赖关系已在Lean 4中形式化，相对于两个明确记录的外部数学输入。
英文摘要
We study the graph distance $D_n(p)$ and effective resistance $R_n(p)$ between the boundary vertices of a depth-$n$ random series--parallel graph, obtained by recursively joining two independent copies in series with probability $p$ and in parallel with probability $1-p$. We prove the existence of deterministic logarithmic speeds: for every $p\in[0,1]$, $n^{-1}\log D_n(p)$ and $n^{-1}\log R_n(p)$ converge to deterministic limits $v_D(p)$ and $v_R(p)$, respectively, almost surely and in $L^1$. The limiting speeds agree with the corresponding first-moment logarithmic rates for every $p\in[0,1]$ in the distance case and for $p\in[1/2,1]$ in the resistance case. We further determine the near-critical behavior of the resistance speed: $v_R(\frac12+δ)\sim 2ζ(3)^{1/3}λ_*δ^{2/3}$ as $δ\downarrow0$, where $λ_*>0$ is characterized by an explicit nonlinear boundary-value problem. This exponent $2/3$ contrasts with the exponent $1/2$ for distance obtained by Chen, Derrida, Duquesne, and Shi (2026). The main idea of this work was proposed by ChatGPT 5.6 Sol, and the authors take full responsibility for the mathematical content. The three main theorems and their supporting proof dependencies have been formalized in Lean 4, relative to two explicitly documented external mathematical inputs.
Comments41 pages, 2 figures
