Critical fluctuations in the edge--triangle exponential random graph models

@misc{liu2026critical,
  title = {{Critical fluctuations in the edge--triangle exponential random graph models}},
  author = {Song-Hao Liu and Qi-Man Shao and Hao Shi and Jing-Yu Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41781},
  eprint = {2610.41781},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41781},
}

TY  - UNPB
TI  - Critical fluctuations in the edge--triangle exponential random graph models
AU  - Song-Hao Liu
AU  - Qi-Man Shao
AU  - Hao Shi
AU  - Jing-Yu Xu
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41781
UR  - https://arxiv.org/abs/2610.41781
AB  - We prove quantitative limit theorems for the edge density Q in the dense edge--triangle exponential random graph model at criticality and under parameter perturbations of order n^−1. At the fixed critical parameters (α_c,h_c), the centered fluctuations satisfy n^2/3(Q−𝔼 Q)⇒ 𝒩(0,σ_0^2), with explicit variance σ_0^2>0. This disproves the quartic fluctuation conjecture of Bianchi, Collet and Magnanini (2024). Gaussian limits also hold (α_c+ λ/n,h_c + γ/n), except when the coefficients satisfy −11/16+2/9(γ+4/9λ)=0. Under this linearity condition, the limiting laws of √n(Q−2/3) have densities proportional to exp{−81z^4/64+9θz^2/8}, where θ=−57/16+8λ/27. We obtain a sharp n^−1/3 rate in $1$-Wasserstein distance for the Gaussian limits and an O(n^−1/2) bound for the quartic limits. The proofs combine Stein's method for exchangeable pairs with conditional row estimates and a local Dobrushin contraction.
ER  -