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

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