Distance and resistance on random series-parallel graphs: logarithmic speeds and near-critical asymptotics
Abstract
We study the graph distance Dn(p) and effective resistance Rn(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∈[0,1], n−1log Dn(p) and n−1log Rn(p) converge to deterministic limits vD(p) and vR(p), respectively, almost surely and in L1. The limiting speeds agree with the corresponding first-moment logarithmic rates for every p∈[0,1] in the distance case and for p∈[1/2,1] in the resistance case. We further determine the near-critical behavior of the resistance speed: vR(1/2+δ)∼ 2ζ(3)1/3λ*δ2/3 as δ↓0, 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.
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BibTeX
@misc{ding2026distance,
title = {{Distance and resistance on random series-parallel graphs: logarithmic speeds and near-critical asymptotics}},
author = {Ruiqi Ding and Zehua He and Yutao Liang and Yushu Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23802},
eprint = {2609.23802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23802},
}
RIS
TY - UNPB
TI - Distance and resistance on random series-parallel graphs: logarithmic speeds and near-critical asymptotics
AU - Ruiqi Ding
AU - Zehua He
AU - Yutao Liang
AU - Yushu Zheng
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23802
UR - https://arxiv.org/abs/2609.23802
AB - 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∈[0,1], n^−1log D_n(p) and n^−1log 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∈[0,1] in the distance case and for p∈[1/2,1] in the resistance case. We further determine the near-critical behavior of the resistance speed: v_R(1/2+δ)∼ 2ζ(3)^1/3λ_*δ^2/3 as δ↓0, 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.
ER -