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 demonstrate how to obtain integrability results for the Schramm‐Loewner evolution
(SLE) from Liouville conformal field theory (LCFT) and the mating‐of‐trees framework for …

Two-pointed quantum disks with a weight parameter W> 0 are a family of finite-area random
surfaces that arise naturally in Liouville quantum gravity. In this paper we show that …

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT)
typically travels graph distance at most n^ 1/4+ o_n (1) n 1/4+ on (1) in n units of time …

We set the foundation for a series of works aimed at proving strong relations between
uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a …

We obtain exact formulae for three basic quantities in random conformal geometry that
depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian …

We study (discretized)“random surfaces,” which are random functions from [special
characters omitted](or large subsets of [special characters omitted]) to either [special …

Let $ Q $ be a free Boltzmann quadrangulation with simple boundary decorated by a critical
($ p= 3/4$) face percolation configuration. We prove that the chordal percolation exploration …

We provide a unified approach to the three main non-compact models of random geometry,
namely the Brownian plane, the infinite-volume Brownian disk, and the Brownian half-plane …