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Annealed central limit theorems for the Ising model on random graphs

The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by $\sqrt{N}$ of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature $0 \leq β^{\mathrm \scriptscriptstyle an}_c < \infty$ and then prove our results in the uniqueness regime, i.e., the values of inverse temperature $β$ and external magnetic field $B$ for which either $β< β^{\mathrm \scriptscriptstyle an}_c$ and $B=0$, or $β>0$ and $B \neq 0$. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters $β$ and $B$, because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.