Related papers: Supercritical Liouville quantum gravity and CLE$_4…
Consider a critical ($\gamma=2$) Liouville quantum gravity (LQG) disk together with an independent conformal loop ensemble (CLE) with parameter $\kappa=4$. We show that the critical LQG surfaces parametrized by the regions enclosed by the…
We study Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which has background charge $Q \in (0,2)$ and central charge $\mathbf{c}_{\mathrm{L}} = 1+6Q^2 \in (1,25)$. Recent works have shown how to define…
There are many deep and useful theorems relating Schramm-Loewner evolution (SLE$_\kappa$) and Liouville quantum gravity ($\gamma$-LQG) in the case when the parameters satisfy $\kappa \in \{\gamma^2, 16/\gamma^2\}$. Roughly speaking, these…
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,…
The purpose of this article is threefold. First, we show that when one explores a conformal loop ensemble of parameter $\kappa=4$ ($\mathrm{CLE}_4$) on an independent $2$-Liouville quantum gravity ($2$-LQG) disk, the surfaces which are cut…
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…
We consider the Schramm-Loewner evolution (SLE$_\kappa$) with $\kappa=4$, the critical value of $\kappa > 0$ at or below which SLE$_\kappa$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$…
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,…
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 prove that SLE$_\kappa$ for $\kappa \in (4,8)$ on an independent $\gamma=4/\sqrt{\kappa}$-Liouville quantum gravity (LQG) surface is uniquely characterized by the form of its LQG boundary length process and the form of the conditional…
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…
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…
We show that when one draws a simple conformal loop ensemble (CLE$_\kappa$ for $\kappa \in (8/3,4)$) on an independent $\sqrt{\kappa}$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum…
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…
We study the relationship between certain SLE$_\kappa(\rho)$ processes, which are variants of the Schramm-Loewner evolution with parameter $\kappa$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG).…
We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{\kappa'}$ for $\kappa'$ in $(4,8)$ that is drawn on an independent $\gamma$-LQG surface for…
We show that for each ${\mathbf c}_{\mathrm M} \in [1,25)$, there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge ${\mathbf c}_{\mathrm M}$. An earlier series of works by…
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 give a simple set of geometric conditions on curves $\eta$, $\tilde{\eta}$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $\varphi \colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $\eta$ with $\varphi(\eta)…
We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of any type of SLE$_\kappa$. These hypotheses…