We associate to a graphon $γ$ the sequence of $W$-random graphs $(G_n(γ))_{n \geq 1}$. We say that the graphon is singular if, for any finite graph $F$, the homomorphism density $t(F,G_n(γ))$ has a variance of order $O(n^{-2})$. This behavior is singular because generically, the density of a fixed finite graph $F$ in a $W$-random graph has a variance of order $O(n^{-1})$. We conjecture that the only singular graphons are the constant graphons $γ_p$ with $p \in [0,1]$, corresponding to the Erdős-Rényi random graphs $G(n,p)$. In this paper, we investigate the general properties of the singular graphons, and we show that they share many properties with the Erdős-Rényi random graphs. In particular, if $γ$ is a singular graphon, then the scaled densities $n(t(F,G_n(γ))-\mathbb{E}[t(F,G_n(γ))])$ converge in joint distribution. This generalises the central limit theorem satisfied by the Erdős-Rényi random graphs $G(n,p)$; however, the limiting distribution might be non-Gaussian if the conjecture does not hold. We also establish an equation satisfied by the characteristic polynomial of the Laplacian of the graph $G_n(γ)$ associated to a singular graphon; this opens the way to a spectral approach of the conjecture.
Singular central limit theorems for the spherical ensemble and beyond
École normale supérieure, CEREMADE, Université Paris-Dauphine, PSL, CNRS（巴黎高等师范学院，CEREMADE，巴黎第九大学，PSL，法国国家科学研究中心）；LPSM, Sorbonne Université（LPSM，索邦大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation
