Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

For $d\ge 3$, we study the Ising model on ${\mathbb{Z}}^d$ with random field given by $\{\epsilon h_v:v\in {\mathbb{Z}}^d\}$ where $h_v$’s are independent normal variables with mean 0 and variance 1. We show that for any $T<T_c$ (here $T_c$ is the critical temperature without disorder), long range order exists as long as $\epsilon$ is sufficiently small depending on $T$. Our work extends previous results of Imbrie (1985) and Bricmont–Kupiainen (1988) from the very low temperature regime to the entire low temperature regime.