The near-critical Ising model with i.i.d. weights

The planar Ising model is one of the most studied models in statistical mechanics. Around its critical point, it exhibits a sharp phase transition and has a drastically changing behavior from a disordered to an ordered phase. Exactly at the critical point the model turns out to be exactly conformally invariant, and the works following the breakthroughs of Smirnov allo to describe its scaling limit. In this talk, we will review all conformal invariance statements for the critical Ising model for a board class of graphs, and then look at some near-critical i.i.d. deformation of weights, which keeps intact the conformally invariant scaling limit, while the associated deterministic model would be off-critical.