In this talk, we present a notion of contour for long-range systems introduced by L. Affonso, R. Bissacot, E.O. Endo, and S. Handa. These new contours were inspired by the work of Frohlich-Spencer. With them, the authors presented a direct proof for the phase transition for ferromagnetic long-range Ising models on Z^d, d>1. The argument works in the sharp region a>d, where a is the power of the coupling constant. The phase transition persists even if we add a polynomial decaying external field.

After that, we present a recent result from J. Ding and Z. Zhuang on the phase transition for the random field Ising model and discuss their implications to the model with long-range interaction.

Bio: João Maia is a Ph.D. student at the Institute of Mathematics and Statistics, University of São Paulo (IME-USP), under the supervision of professor Rodrigo Bissacot. His research interests are in Classical Statistical Mechanics, with a focus on ferromagnetic random models.