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The seminal work of Sheffield showed that when random surfaces called Liouville quantum gravity (LQG) are conformally welded, the resulting interface is Schramm-Loewner evolution (SLE). This has been proved for a variety of configurations,…
For $\gamma \in (0,2)$, the quantum disk and $\gamma$-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar…
SLE curves describe the scaling limit of interfaces from many 2D lattice models. Heuristically speaking, the SLE partition function is the continuum counterpart of the partition function of the corresponding discrete model. It is well known…
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 conformally welding two quantum disks according to their…
We demonstrate how to obtain integrable results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact…
We consider the $\gamma$-Liouville quantum gravity (LQG) model for $\gamma \in (0,2)$, formally described by $e^{\gamma h}$ where $h$ is a Gaussian free field on a planar domain $D$. Sheffield showed that when a certain type of LQG surface,…
Originating in theoretical physics, Liouville quantum gravity (LQG) has been an important topic in probability theory and mathematical physics in the past two decades. In this proceeding, we review two aspects of this topic. The first is…
We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of…
Sheffield showed that conformally welding a $\gamma$-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter $\kappa = \gamma^2$ as the interface, and Duplantier-Miller-Sheffield proved…
Two-pointed quantum disks with a weight parameter $W>0$ is a canonical family of finite-volume random surfaces in Liouville quantum gravity. We extend the conformal welding of quantum disks in [AHS23] to the non-simple regime, and give a…
Consider two critical Liouville quantum gravity surfaces (i.e., $\gamma$-LQG for $\gamma=2$), each with the topology of $\mathbb{H}$ and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces,…
We prove that the SLE$_\kappa$ loop measure arises naturally from the conformal welding of two $\gamma$-Liouville quantum gravity (LQG) disks for $\gamma^2 = \kappa \in (0,4)$. The proof relies on our companion work on conformal welding of…
We showed that the SLE bubble measure recently constructed by Zhan arises naturally from the conformal welding of two Liouville quantum gravity (LQG) disks. The proof relies on (1) a "quantum version" of the limiting construction of the SLE…
Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The…
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-filling curve (which describes the "interface" between the…
We study Liouville quantum gravity (LQG) surfaces whose law has been reweighted according to nesting statistics for a conformal loop ensemble (CLE) relative to $n\in \mathbb{N}_0$ marked points $z_1,\dots,z_n$. The idea is to consider a…
We study the boundary correlation functions in Liouville theory and in solvable statistical models of 2D quantum gravity. In Liouville theory we derive functional identities for all fundamental boundary structure constants, similar to the…
We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent…
A Liouville quantum gravity (LQG) surface is a natural random two-dimensional surface, initially formulated as a random measure space and later as a random metric space. We show that the LQG measure can be recovered as the Minkowski measure…
We study a two-dimensional conformal field theory coupled to quantum gravity on a disk. Using the continuum Liouville field approach, we compute three-point correlation functions of boundary operators. The structure of momentum…