The level sets of the Gaussian free field give rise to natural long-range percolation models whose critical behaviour is particularly well understood. We will give precise meaning to studying this model in dimensions between two and three, with dimension two playing the role of the lower critical dimension.
This setting unveils a rich family of delicate scaling laws that fall outside the conventional scaling theory. Despite this anomalous behaviour, we show that the classical critical scaling relations nevertheless remain valid in a generalised sense.
Based on joint work with Pierre-Francois Rodriguez.