A Conformal Loop Ensemble ($\mathrm{CLE}_\kappa$) is a canonical family of random collections of non-crossing loops in a two-dimensional planar domain, indexed by a parameter $\kappa \in (8/3, 8)$. CLEs describe the scaling limits of interfaces in critical statistical mechanics models on planar lattices and random planar maps such as the critical Ising model ($\kappa = 3$), the critical percolation ($\kappa = 6$), and the FK-Ising model ($\kappa = 16/3$). In this talk, I will consider CLE with the parameter $\kappa = 4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that there exists a unique metric on the set of $\mathrm{CLE}_4$ loops which is conformally invariant, local, geodesic metric, and it is a measurable function of the $\mathrm{CLE}_4$. The metric is constructed as the limit of the (appropriately) renormalized graph metric on the loops of $\mathrm{CLE}_\kappa$ as $\kappa \downarrow 4$, where we say that there is an edge between two vertices corresponding to two distinct $\mathrm{CLE}_\kappa$ loops if and only if the loops intersect each other. We also establish properties of the geodesics by showing that any geodesic between two loops is supported on the $\mathrm{CLE}_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Based on joint work with Emmanuel Kammerer, Jason Miller, and Yi Tian.
