Critical long-range percolation
It is conjectured that many models of statistical mechanics have a rich, fractal-like behaviour at and near their points of phase transition, with power-law scaling governed by critical exponents that are expected to depend on the dimension but not on the small-scale details of the model such as the choice of lattice. This is now reasonably well understood in two dimensions and in high dimensions, but remains poorly understood in intermediate dimensions (e.g. $d=3$). I will describe a new technique for studying models with long-range interactions which leads to some striking new results some of which have surprisingly easy proofs.