Scaling Limit of Critical Loop Soup Clusters in Three, Four, and Five Dimensions

Abstract

We prove that on the metric graph of ${\mathbb{Z}}^d$ for $d\in \{3,4,5\}$, clusters of the critical loop soup (or equivalently, clusters of the critical Gaussian free field level-set) admit a scaling limit. To the best of our knowledge, this presents the first scaling limit result for a percolation model on a three-dimensional lattice.