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

For \(d\geq 3\), 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.