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For $d\geq 3$, we study the Ising model on $\mathbb Z^d$ with random fieldgiven by $\{\epsilon h_v: v\in \mathbb Z^d\}$ where $h_v$'s are independentnormal 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 orderexists as long as $\epsilon$ is sufficiently small depending on $T$. Our workextends previous results of Imbrie (1985) and Bricmont--Kupiainen (1988) fromthe very low temperature regime to the entire low temperature regime.

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