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We continue our analysis of Ising models on the (directed) Erd\H{o}s-R\'enyirandom graph $G(N,p)$. We prove a quenched Central Limit Theorem for themagnetization and describe the fluctuations of the log-partition function. Inthe current note we consider the low temperature regime $\beta>1$ and the casewhen an external magnetic field is present. In both cases, we assume that$p=p(N)$ satisfies $p^3N \to \infty$.

Fluctuations of the Magnetization for Ising models on Erd\H{o}s-R\'enyi Random Graphs -- the Regimes of Low Temperature and External Magnetic Field