Distance and resistance on random series-parallel graphs: logarithmic speeds and near-critical asymptotics

Abstract

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.