Title: The Minkowski content measure for the Liouville quantum gravity metric.
Abstract: A Liouville quantum gravity (LQG) surface is a "canonical" random two-dimensional Riemannian manifold that is conjectured to be the scaling limit of a wide variety of random planar graph models. LQG was formulated initially as a random measure space and, more recently, as a random metric space. In this talk, I will explain how the LQG measure can be recovered as the Minkowski content measure with respect to the LQG metric, thereby providing a direct connection between the two formulations for the first time. Our primary tool is the mating-of-trees theory of Duplantier, Miller, and Sheffield, which says that an LQG surface is an infinitely divisible metric measure space when explored by an independent space-filling Schramm–Loewner evolution (SLE) curve. This is joint work with Ewain Gwynne (University of Chicago).
