Fluctuations of the Magnetization for Ising Models on Dense ErdH{o}s-R\'enyi Random Graphs

We analyze Ising/Curie-Weiss models on the (directed) Erd\H{o}s-R\'enyi random graph on $N$ vertices in which every edge is present with probability $p$. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993]. We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime $\beta<1$ when $p=p(N)$ satisfies $p^3N^2\to +\infty$.