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

Abstract

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 $({\alpha }_c,h_c),$ the centered fluctuations satisfy $n^{2/3}(Q-\mathbb{E}Q)\Rightarrow \mathcal{N}(0,{\sigma }_0^2)$, with explicit variance ${\sigma }_0^2>0$. This disproves the quartic fluctuation conjecture of Bianchi, Collet and Magnanini (2024). Gaussian limits also hold $({\alpha }_c+\lambda /n,h_c+\gamma /n)$, except when the coefficients satisfy $-\frac{11}{16}+\frac{2}{9}(\gamma +\frac{4}{9}\lambda )=0$. Under this linearity condition, the limiting laws of $\sqrt{n}(Q-2/3)$ have densities proportional to $\exp \{-81z^4/64+9\theta z^2/8\}$, where $\theta =-57/16+8\lambda /27$. 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.

arXiv comment66 pages