pREPRINTS

1. Ding, J., Huang, F. and Maia, J., Phase transitions in low-dimensional long-range random field Ising models. arXiv preprint arXiv:2412.19281.  2024  

arXiv Abstract

We consider the long-range random field Ising model in dimension \(d = 1, 2\), whereas the long-range interaction is of the form \(J_{xy} = |x-y|^{-\alpha}\) with \(1< \alpha < 3/2\) for \(d=1\) and with \(2 < \alpha \leq 3\) for \(d = 2\). Our main results establish phase transitions in these regimes. In one dimension, we employ a Peierls argument with some novel modification, suitable for dealing with the randomness coming from the external field; in two dimensions, our proof follows that of Affonso, Bissacot, and Maia (2023) with some adaptations, but new ideas are required in the critical case of \(\alpha=3\).

1. Affonso, L., Bissacot, R. and Maia, J., ​Phase Transitions in Multidimensional Long-Range Random Field Ising Models. arXiv preprint arXiv:2307.14150.  2024  

arXiv Abstract

We extend a recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in a class of ferromagnetic random field Ising mo-dels. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting, proposed by two of us, with the coarse-graining procedure introduced by Fisher, Fröhlich, and Spencer. Our result shows that the Ding-Zhuang strategy is also useful for interactions \(J_{xy}=|x-y|^{- \alpha}\) when \(\alpha > d\) in dimension \(d\geq 3\) if we have a suitable system of contours, yielding an alternative approach that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen suggested that the RGM should also work on this generality. We can consider i​.i.d. random fields with Gaussian or Bernoulli distributions.

2. Bissacot, R. and Maia, J., Phase Transitions in the semi-infinite Ising model with a decaying field. arXiv preprint arXiv:2311.18058. 2023 

arXiv Abstract

We study the semi-infinite Ising model with an external field \(h_i = \lambda |i_d|^{-\delta}\), \(\lambda\) is the wall influence, and \(\delta>0\). This external field decays as it gets further away from the wall. We are able to show that when \(\delta>1\) and \(\beta > \beta_c(d)\), there exists a critical value \(0< \lambda_c:=\lambda_c(\delta,\beta)\) such that, for \(\lambda<\lambda_c\) there is phase transition and for \(\lambda>\lambda_c\) we have uniqueness of the Gibbs state. In addition, when \(\delta<1\) we have only one Gibbs state for any positive \(\beta\) and \(\lambda\).

pUBLICATIONS

1. Maia, J., Phase Transitions in Ising models: the Semi-infinite with decaying field and the Random Field Long-range (Ph.D. Thesis). 2024

Full text Abstract

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 \(h_i = \lambda |i_d|^{-\delta}\), \(\lambda\) is the wall influence, and \(\delta>0\). This external field decays as it gets further away from the wall. We are able to show that when \(\delta>1\) and \(\beta > \beta_c(d)\), there exists a critical value \(0< \lambda_c:=\lambda_c(\delta,\beta)\) such that, for \(\lambda<\lambda_c\) there is phase transition and for \(\lambda>\lambda_c\) we have uniqueness of the Gibbs state. In addition, when \(\delta<1\) we have only one Gibbs state for any positive \(\beta\) and \(\lambda\).

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 \(J_{xy}=|x-y|^{- \alpha}\) when \(\alpha > d\) in dimension \(d\geq 3\) 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.