SLE and CLE have led to tremendous progress in the study of critical scaling limits in two dimensions. In light of this, it is tempting to hope that a similar unified picture might one day emerge in three dimensions.
This talk will introduce our recent result that for the cable graph of Z^d with d=3,4,5, clusters of the critical Gaussian free field level-set (or equivalently, clusters of the critical loop soup) admit a scaling limit. The description of these limits might offer a glimpse into the unified picture in three dimensions.
This is a joint work with Jian Ding (Peking University).