Long range order for random field Ising and Potts models

Understand

We present a new and simple proof for the classic results of Imbrie (1985) and Bricmont-Kupiainen (1988) that for the random field Ising model in dimension three and above there is long range order at low temperatures with presence of weak disorder.

• With the same method, we obtain a couple of new results: (1) we prove that long range order exists for the random field Potts model at low temperatures with presence of weak disorder in dimension three and above; (2) we obtain a lower bound on the correlation length for the random field Ising model at low temperatures in dimension two (which matches the upper bound in Ding-Wirth (2020)).
• Our proof is based on an extension of the Peierls argument with inputs from Chalker (1983), Fisher-Fr\"ohlich-Spencer (1984), Ding-Wirth (2020) and Talagrand's majorizing measure theory (1980s) (and in particular, our proof does not involve the renormalization group theory).