Cardy embedding of random planar maps and a KPZ formula for mated trees

Abstract

The Schramm-Loewner evolution (SLE) is a random fractal curve which describes the scaling limit of interfaces in a wide range of statistical physics models. Liouville quantum gravity (LQG) is a random fractal surface which arises as the scaling limit of discrete surfaces known as random planar maps (RPM). First, we study Hausdorff dimensions for SLE curves. We prove a KPZ-type formula which relates the Hausdorff dimension of an arbitrary subset of an SLE curve to the Hausdorff dimension of a time set for a two-dimensional correlated Brownian motion. Using our formula, we obtain new and simple proofs for a number of SLE Hausdorff dimensions, and we prove an explicit formula which says how much the Hausdorff dimension of a deterministic set increases upon being conformally mapped to an SLE curve. This is joint work with Gwynne and Miller. Then we introduce a mating-of-trees construction of SLE in Euclidean geometry in collaboration with Sun. This is the Euclidean counterpart to the mating-of-trees construction of SLE in an LQG environment by Duplantier, Miller, and Sheffield, which plays an essential role throughout the thesis. Finally, we prove scaling limit results for uniformly sampled RPM known as triangulations. In a joint work with Bernardi and Sun we show that a number of observables associated with critical site percolation on the triangulation converge jointly in law to the associated observables of SLE6 on an independent [square root of] 8/3-LQG surface. In a joint work with Sun we use this and other results to prove convergence of the triangulation under a discrete conformal embedding which we call the Cardy embedding. The conformally embedded triangulation induces an area measure and a metric on the complex plane, and we show that this measure and metric converge jointly in the scaling limit to an instance of the [square root of] 8/3-LQG disk (equivalently, to an instance of the conformally embedded Brownian disk).