What is the most natural way of choosing a random surface (two-dimensional Riemannian
manifold)? If we are given a finite set 𝑋, the easiest way to choose a random element of 𝑋 is …

There is a simple way to" glue together" a coupled pair of continuum random trees (CRTs) to
produce a topological sphere. The sphere comes equipped with a measure and a space …

We show that for each γ ∈ (0, 2) γ∈(0, 2), there is a unique metric (ie, distance function)
associated with γ γ-Liouville quantum gravity (LQG). More precisely, we show that for the …

We survey the theory and applications of mating-of-trees bijections for random planar maps
and their continuum analog: the mating-of-trees theorem of Duplantier, Miller, and Sheffield …

We consider an embedding of planar maps into an equilateral triangle $\Delta $ which we
call the Cardy embedding. The embedding is a discrete approximation of a conformal map …

We prove that for each γ ∈ (0, 2) γ∈(0, 2), there is an exponent d_ γ> 2 d γ> 2, the “fractal
dimension of γ γ-Liouville quantum gravity (LQG)”, which describes the ball volume growth …

Abstract For γ ∈ (0, 2) γ∈(0, 2), we define a weak γ γ-Liouville quantum gravity (LQG) metric
to be a function h ↦ D_h h↦ D h which takes in an instance of the planar Gaussian free field …

We demonstrate that the conformal loop ensemble (CLE) has a rich integrable structure by
establishing exact formulas for two CLE observables. The first describes the joint moments …

We give bijections between bipolar-oriented (acyclic with unique source and sink) planar
maps and certain random walks, which show that the uniformly random bipolar-oriented …

Let $\gamma\in (0, 2) $, let $ h $ be the planar Gaussian free field, and let $ D_h $ be the
associated $\gamma $-Liouville quantum gravity (LQG) metric. We prove that for any …