Z. Zhuang: Long range order for random field Ising and Potts models

Abstract: In this talk I will 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)).

In the second part, I will review some techniques we use to analyze the supreme of Gaussian and sub-Gaussian processes that naturally arise from our extension of Peierls argument. This part is independent of the first one.

This talk is based on https://arxiv.org/abs/2110.04531, which is a joint work with Jian Ding (Upenn).

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