Long range order for random field Ising model

Abstract

We study existence of long range order in the random field Ising model. We define the classical Ising model and use the so-called Griffiths--Peierls argument to prove existence of long range order for low temperatures, in dimension two and above. Then, we introduce the random field Ising model. We proceed as in J. Ding and Z. Zhuang recent work in \cite{dingzhuang} to extend Peierls argument and show that long range order also exists in this model, at low temperatures with the presence of a weak disorder, in dimension three and above.