Fluctuations of the Magnetization for Ising models on Erdos-Renyi random graphs - the regimes of low temperature and external magnetic field

We continue our analysis of Ising models on the (directed) Erdos-Renyi 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 -> infinity.