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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 hᵢ = |id|^-, is the wall influence, and >0. This external field decays as it gets further away from the wall. We are able to show that when >1 and > c (d), there exists a critical value 0c we have uniqueness of the Gibbs state. In addition, when 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.
João M. Maia (Thu,) studied this question.