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

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