In the domain of quantum gravity, people often consider random metrics on surfaces, defined as Riemannian ones with the factor being an exponent of the Gaussian Free Field. Though, GFF is only a distribution, not a function, and its exponent is not well-defined. This leaves open the problem of giving a mathematical sense to this definition (or, to be more precise, of showing rigorously that one of the known regularization procedures indeed converges).
In a joint work with M. Khristoforov and M. Triestino, we approach a « baby version » of this problem, constructing random metrics on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.). This situation is still accessible due to the graph structure, but already shares with the planar case the complexity of high non-uniqueness of candidates for the geodesic lines. The behavior of some (pivotal, bridge-type) graphs seems to be a good model for the behavior of the full planar case.