Phase transitions in Ising models: the semi-infinite with decaying field and the random field long-range - Maia, João Vitor Teixeira (2024)
In this thesis, we present results on phase transition for two models: the semi-infinite Ising model with a decaying field, and the long-range Ising model with a random field. We study the semi-infinite Ising model with an external field h_i = λ |i_d|^{-δ}, λ is the wall influence, and δ>0. This external field decays as it gets further away from the wall. We are able to show that when δ>1 and \\β > \\β_c(d), there exists a critical value 0< λ_c:=λ_c(δ,\\β) such that, for λ<λ_c there is phase transition and for λ>λ_c we have uniqueness of the Gibbs state. In addition, when δ<1 we have only one Gibbs state for any positive \\β and λ. For the model with a random field, we extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting proposed by Affonso, Bissacot, Endo and Handa, with the coarse-graining procedure introduced by Fisher, Fröhlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions J_=|x-y|^{- \\α} when \\α > d in dimension d>= 3 if we have a suitable system of contours, yielding an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM should also work on this generality. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions.