Long range order for random field Ising and Potts models

TL;DR

This paper provides a new, simplified proof for long-range order in random field Ising and Potts models in dimensions three and above at low temperatures, extending known results and establishing new bounds.

Contribution

It introduces a novel proof technique extending Peierls argument, proving long-range order for the random field Potts model and lower bounds for the correlation length in two dimensions.

Findings

Long-range order exists for the random field Potts model in dimensions three and above.

A lower bound on correlation length in 2D RFIM matches previous upper bounds.

The proof avoids renormalization group methods, simplifying the theoretical approach.

Abstract

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).