We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $δ$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits $0$ is jointly continuous in $x$ and $δ$, provided $δ\le 0$. As an application, we show that the SLE($κ$) welding homeomorphism is continuous in $κ$ for $κ\in [0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE$_κ$ in $κ$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.
The backward chordal Schramm-Loewner Evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated solution to the welding problem (which is a natural renormalized limit of the finite time Loewner traces) is reversible. The proofs rely on an analysis of the action of analytic circle diffeomorphisms on the space of hulls, and on the coupling techniques of the second author.
We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $φ\colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $φ(η) = \tildeη$ then $φ$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $φ$ is a h…
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 Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.
We prove that the SLE$_κ$ loop measure arises naturally from the conformal welding of two $γ$-Liouville quantum gravity (LQG) disks for $γ^2 = κ\in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE$_{8/3}$ loop. Our result is also a key input to recent work of the first and third coauthors on …
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 formula for the law of a conformal derivative of a classical variant of SLE called $\mathrm{SLE}_κ(ρ_-;ρ_+)$. Our proof is built on two connections between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embeddin…
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 boundary lengths gives another quantum disk decorated with an independent chordal $\mathrm{SLE}_κ(ρ_-;ρ_+)$ curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees…
We consider the Schramm-Loewner evolution (SLE$_κ$) with $κ=4$, the critical value of $κ> 0$ at or below which SLE$_κ$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable, answering a question posed by Sheffield. Such curves arise as the conformal welding of a pair of independent critical ($γ=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\m…
We prove that, for $κ\le 4$, backward chordal SLE$_κ$ admits backward chordal SLE$_κ(-4,-4)$ decomposition for the capacity parametrization. This means that, for any bounded measurable subset $U\subset Q_4:={\mathbb R}_+\times{\mathbb R}_-$, if we integrate the laws of extended backward chordal SLE$_κ(-4,-4)$ with different pairs of force points $(x,y)$ against some suitable density function $G(x,y)$ restricted to $U$, then we get a measure, which is absolutely continuous with respect to the law of backward chordal SLE$_κ$, and the Radon-Nikodym derivative is a constant depending on $κ$ times …
A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with $N+1$ marked boundary points (MBPs) with $N\in\mathbb{Z}_{\ge 0}$. We propose the conformal welding problem for it in the case of $N\in\mathbb{Z}_{\ge 1}$. If $N=1$, it is reduced to the problem introduced by Sheffield, who solved it by coupling the QS with the Schramm--Loewner evolution (SLE). When $N \ge 3$, there natu…
We prove that the welding homeomomorphism of SLE$_4$ is almost surely $\log$-regular. In a previous version of this work, we had erroneously deduced its removability from this property. Nevertheless, the $\log$-regularity does provide some information and could lead to future developments.
We define radial exploration processes from $a$ to $b$ and from $b$ to $a$ in a domain $D$ of hexagons where $a$ is a boundary point and $b$ is an interior point. We prove the reversibility: the time-reversal of the process from $b$ to $a$ has the same distribution as the process from $a$ to $b$. We show the scaling limit of such an exploration process is a radial SLE$_6$ in $D$. As a consequence, the distribution of the last hitting point with the boundary of any radial SLE$_6$ is harmonic measure. We also prove the scaling limit of a similar exploration process defined in the full complex pl…
We establish existence and uniqueness for Gaussian free field flow lines started at {\em interior} points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at {\em boundary} points and use Gaussian free field machinery to determine which chordal \SLE_κ(ρ_1; ρ_2) processes are time-reversible when κ< 8. Here we extend these results to whole-plane \SLE_κ(ρ) and establish continuity and transience of these paths. In particular, we ex…
