Romain Durand

In this comprehensible talk, starting from the description of the nearest-neighbour Ising model - which is one of the most studied model in statistical mechanics - we will introduce the concepts of infinite-volume measures, Gibbs states, and Dobrushin states.
In dimension 2, the Aizenmann-Higuchi theorem states that all infinite-volume measures are convex combinations of \mu^+ and \mu^- and are therefore all translation invariant, whereas in dimension 3 and more, there exists some infinite-volume measures that are not translation invariant (Dobrushin states).
We will discuss how the question of the existence of such Dobrushin states can be treated on the specific example of 2D long-range Ising models.