Long range order for random field Ising and Potts models

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 re...

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Bibliographic Details
Published inarXiv.org
Main Authors Ding, Jian, Zhuang, Zijie
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.10.2021
Subjects
Entropy
Fields (mathematics)
Group theory
Ising model
Long range order
Low temperature
Lower bounds
Upper bounds
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Summary: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).
ISSN:2331-8422