# Quantum triangles and imaginary geometry flow lines

Morris Ang Xin Sun Pu Yu

###### Abstract

We define a three-parameter family of random surfaces in Liouville quantum
gravity (LQG) which can be viewed as the quantum version of triangles. These
quantum triangles are natural in two senses. First, by our definition they
produce the boundary three-point correlation functions of Liouville conformal
field theory on the disk. Second, it turns out that the laws of the triangles
bounded by flow lines in imaginary geometry coupled with LQG are given by
these quantum triangles. In this paper we demonstrate the second point for
boundary flow lines on a quantum disk. Our method has the potential to prove
general conformal welding results with quantum triangles glued in an arbitrary
way. Quantum triangles play a basic role in understanding the integrability of
SLE and LQG via conformal welding. In this paper, we deduce integrability
results for chordal SLE with three force points, using the conformal welding
of a quantum triangle and a two-pointed quantum disk. Further applications
will be explored in subsequent works.

###### Contents

  1. 1 Introduction
  2. 2 Quantum triangles: definition and basic properties
  3. 3 Imaginary geometry and $\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$
  4. 4 Conformal welding of $\mathrm{QT}(W,W,2)$ and two thin quantum disks
  5. 5 Proof of Theorem 1.2 for a restricted range
  6. 6 Proof of Theorem 1.2 in the full range
  7. 7 Applications to $\mathrm{SLE}$

##  1 Introduction

Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) are
central subjects in random conformal geometry as canonical theories for random
curves and surfaces, respectively. Starting from [She16], a key tool to study
SLE and LQG is their coupling, where SLE curves arise as the interfaces of LQG
surfaces under conformal welding. This leads to the mating-of-trees theory
[DMS21], which is fundamental in connecting LQG and the scaling limits of
random planar maps decorated with statistical physics models; see the textbook
[BP21] and the survey [GHS19]. More recently, conformal welding was used to
study the integrability of SLE and LQG [AHS21, ARS21, AS21].

In most conformal welding results established so far, the SLE curves cut the
LQG surfaces into smaller surfaces with two boundary marked points. The
infinite-area version of these two-pointed marked surfaces are called quantum
wedges, while the finite-area variants are called two-pointed quantum disks.
As shown in [DMS21, AHS20], when these surfaces are welded together, the law
of the SLE interfaces are a collection of flow lines in the sense of imaginary
geometry [MS16a, MS17], which is a canonical framework to couple multiple SLE
curves. Two-pointed quantum disks also plays a basic role in the Liouville
conformal field theory (LCFT) as they determine the reflection coefficient for
LCFT on the disk [HRV18, RZ22, AHS21].

In this paper we define a three-parameter family of LQG surfaces with three
boundary marked points, which we call quantum triangles. They are defined to
produce the boundary three-point correlation functions of LCFT on the disk.
When two of the parameters are equal, they reduce to a two-parameter family of
quantum surfaces defined in [AHS21]. The main goal of our paper is to
demonstrate that the law of the triangular surfaces cut out by imaginary
geometry flow lines on a LQG disk with multiple boundary marked points are
given by quantum triangles; see Theorem 1.3. Based on our work, a general
result with quantum triangles conformally welded in an arbitrary way will be
proved by the first and the third authors in a subsequent work. Quantum
triangles enrich the applications of conformal welding to SLE and LQG. In this
paper, we deduce integrablity results for chordal SLE with three force points.
Further applications will be discussed in Section 1.5.

We will give a brief description of quantum triangles in Section 1.1 with the
precise definition postponed to Section 2. Then in Section 1.2 we state a key
result (Theorem 1.2) saying that the conformal welding of a quantum triangle
and a two-pointed quantum disk gives another quantum triangle, which is proved
in Sections 4—6. The proof includes several novel techniques for proving
general conformal welding results. In particular, we give a Markovian
characterization of the Liouville fields defining quantum triangles, which
explains their ubiquity. As a corollary of Theorem 1.2, we state the
aforementioned Theorem 1.3 in Section 1.3 with more details on imaginary
geometry provided in Section 3. We present some applications of Theorem 1.2 to
SLE in Section 1.4, whose proofs are given in Section 7. In Section 1.5, we
discuss some perspectives and related works.

###  1.1 Definition of the quantum triangle

Fix $\gamma\in(0,2)$. A quantum surface in $\gamma$-LQG is a surface with an
area measure and a metric structure induced by a variant of Gaussian free
field (GFF). The area is defined in [DS11] and the metric is defined in
[DDDF20, GM21]. A quantum surface with the disk topology can be represented as
a pair $(D,h)$ where $D$ is a simply connected domain and $h$ is a variant of
GFF. For such surfaces there is also a notion of $\gamma$-LQG length measure
on the disk boundary [DS11]. Two pairs $(D,h)$ and $(D^{\prime},h^{\prime})$
represent the same quantum surface if there is a conformal map between $D$ and
$D^{\prime}$ preserving the geometry. A particular pair $(D,h)$ is called a
(conformal) embedding of the quantum surface.

For $W>0$, the two-pointed quantum disk of weight $W$ is a quantum surface
with two boundary marked points introduced in [DMS21, AHS20], which has finite
quantum area and length. It has two regimes: thick (i.e.
$W\geq\frac{\gamma^{2}}{2}$) and thin (i.e. $W\in(0,\frac{\gamma^{2}}{2})$).
For $W\geq\gamma^{2}/2$, the two-pointed quantum disk has the disk topology
with two boundary marked points. The field near the two marked points has a
$\beta$-log singularity where $\beta$ and $W$ are related by

$\beta=\gamma+\frac{2-W}{\gamma},\quad\textrm{i.e.}\quad
W=\gamma(Q+\frac{\gamma}{2}-\beta).$ (1.1)

For $W\in(0,\gamma^{2}/2)$, the weight-$W$ two-pointed quantum disk has the
topology of an ordered collection of disks, each of which has two boundary
marked points. There is a canonical law $\mathcal{M}_{2}^{\textup{disk}}(W)$
for the weight-$W$ two-pointed quantum disk, which has no constraint on the
total area and boundary lengths. Other variants with fixed area and/or length
can be obtained from $\mathcal{M}_{2}^{\textup{disk}}(W)$ by conditioning. We
also write $\mathcal{M}_{2}^{\textup{disk}}(2)$ as $\mathrm{QD}_{0,2}$. A
sample from $\mathrm{QD}_{0,2}$ is known as the quantum disk with two typical
boundary points, because in this case the two marked points are simply
distributed according to the $\gamma$-LQG boundary length measure. This
special case arises naturally as scaling limits of random planar maps. For
example, when $\gamma=\sqrt{8/3}$, $\mathrm{QD}_{0,2}$ is the law of the LQG
realization of the Brownian disk with two boundary marked points, with free
area and boundary length [MS20, MS21]. This is the scaling limit of
triangulation or quadrangulations sampled from the critical Boltzmann measure
[BM17, GM19]. In general, $\mathcal{M}_{2}^{\textup{disk}}(W)$ is an infinite
measure. For $W\in(0,\frac{\gamma^{2}}{2})$, the ordered collections of disks
in $\mathcal{M}_{2}^{\textup{disk}}(W)$ can be obtained from an initial
segment of the Poisson point process with intensity measure
$\mathcal{M}_{2}^{\textup{disk}}(\gamma^{2}-W)$. We will recall the precise
definition of $\mathcal{M}_{2}^{\textup{disk}}(W)$ in Section 2.

Two-pointed quantum disks are intimately related to Liouville conformal field
theory on the disk [HRV18]. This relation is most transparent when we
parameterize a quantum disk by a strip. Let $\mathcal{S}$ be the horizontal
strip $\mathbb{R}\times(0,\pi)$. For $W>\frac{\gamma^{2}}{2}$, let
$(\mathcal{S},\phi,+\infty,-\infty)$ be an embedding of a sample from
$\mathcal{M}_{2}^{\textup{disk}}(W)$. Let $\beta=\gamma+\frac{2-W}{\gamma}<Q$
as in (1.1). By [AHS21], if we independently sample $T$ from the Lebesgue
measure on $\mathbb{R}$, then the law of the field
$\tilde{\phi}:=\phi(\cdot+T)$ is
$\frac{\gamma}{2(Q-\beta)^{2}}\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty)}$,
where $\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty)}$ is the Liouville field
on $\mathcal{S}$ with $\beta$ insertions at $\pm\infty$. See Section 2 for the
definition of Liouville fields with insertions.

We now describe our main quantum surfaces of interest, the quantum triangles.
We first recall a special case that is already considered in [AHS21] and
played a crucial rule there. For $\beta,\beta_{3}<Q$, the Liouville field
measure $\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty),(0,\beta_{3})}$ is
formally defined by
$\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty),(\beta_{3},0)}(d\phi)=e^{\beta_{3}\phi(0)}\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty)}(d\phi)$,
and can be made rigorous by regularization. Let $W,W_{3}>\frac{\gamma^{2}}{2}$
be determined by $\beta,\beta_{3}$ as in (1.1), respectively. Sample $\phi$
from
$\frac{1}{(Q-\beta)^{2}(Q-\beta_{3})}\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty),(0,\beta_{3})}$
and let $\mathrm{QT}(W,W,W_{3})$ be the law of the three-pointed quantum
surface $(\mathcal{S},\phi,\pm\infty,0)$. We call a sample from
$\mathrm{QT}(W,W,W_{3})$ a quantum triangle of weight $(W,W,W_{3})$. Up to a
multiplicative constant, the measure $\mathrm{QT}(W,W,W_{3})$ agrees with
$\mathcal{M}_{2,\bullet}^{\textup{disk}}(W;\beta_{3})$ defined in [AHS21];
also see Definition 2.15. For $W=\frac{\gamma^{2}}{2}$, we define
$\mathrm{QT}(\frac{\gamma^{2}}{2},\frac{\gamma^{2}}{2},W_{3})$ as the
$W\downarrow\frac{\gamma^{2}}{2}$ limit of $\mathrm{QT}(W,W,W_{3})$. For
$W\in(0,\frac{\gamma^{2}}{2})$, following the definition of
$\mathcal{M}_{2,\bullet}^{\textup{disk}}(W;\beta_{3})$ in [AHS21], we let
$\mathrm{QT}(W,W,W_{3})$ be the law of the three-pointed surface obtained by
attaching an independent weight-$W$ two pointed disk at a quantum triangle of
weight $(\gamma^{2}-W,\gamma^{2}-W,W_{3})$.

For $W_{1},W_{2},W_{3}>0$, we define $\mathrm{QT}(W_{1},W_{2},W_{3})$ as
follows. For $W_{1},W_{2},W_{3}>\frac{\gamma^{2}}{2}$, set
$\beta_{i}=\gamma+\frac{2-W_{i}}{\gamma}<Q$ and let
$\textup{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},0)}$
be the Liouville field on $\mathcal{S}$ with insertion
$\beta_{1},\beta_{2},\beta_{3}$ at $+\infty,-\infty$ and $0$, respectively.
Sample $\phi$ from

$\frac{1}{(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}\textup{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},0)}.$

We define $\textup{QT}(W_{1},W_{2},W_{3})$ to be the law of the 3-pointed
quantum surface $(\mathcal{S},\phi,+\infty,-\infty,0)$. We call a sample from
$\mathrm{QT}(W,W,W_{3})$ a quantum triangle of weight $(W_{1},W_{2},W_{3})$.
Taking the limit $W_{i}\downarrow\frac{\gamma^{2}}{2}$, we can extend the
definition of $\textup{QT}(W_{1},W_{2},W_{3})$ to
$W_{1},W_{2},W_{3}\geq\frac{\gamma^{2}}{2}$; see Section 2.5. In this regime a
quantum triangle has the disk topology. When
$W_{1}\in(0,\frac{\gamma^{2}}{2})$ and $W_{2},W_{3}\geq\frac{\gamma^{2}}{2}$,
we define $\mathrm{QT}(W_{1},W_{2},W_{3})$ by attaching an independent
weight-$W_{1}$ two pointed disk at a quantum triangle of weight
$(\gamma^{2}-W_{1},W_{2},W_{3})$. Using this method we extend the definition
of $\textup{QT}(W_{1},W_{2},W_{3})$ to $W_{1},W_{2},W_{3}>0$. We call the
three marked points vertices of a quantum triangle and $W_{i}$ ($i=1,2,3$) is
called the weight of the corresponding vertex. Given a sample of
$\mathrm{QT}(W_{1},W_{2},W_{3})$, the geometry near the vertex of weight
$W_{i}$ looks like the neighborhood of a marked point on a weight-$W_{i}$
quantum disk. We say a vertex is thick if its weight
$W\geq\frac{\gamma^{2}}{2}$. We call it thin if
$W\in(0,\frac{\gamma^{2}}{2})$. See Figure 1 for an illustration.

Figure 1: A sample of $\mathrm{QT}(W_{1},W_{2},W_{3})$ with a thick vertex
$a_{2}$ and two thin vertices $a_{1},a_{3}$, i.e.
$W_{2}\geq\frac{\gamma^{2}}{2}$ and $W_{1},W_{3}<\frac{\gamma^{2}}{2}$. The
yellow surface is a quantum triangle with thick vertices
$\tilde{a}_{1},a_{2},\tilde{a}_{3}$. The two thin two-pointed quantum disks
(colored green) are concatenated with the yellow triangle at $\tilde{a}_{1}$
and $\tilde{a}_{3}$.

###  1.2 Conformal welding of a quantum triangle and a 2-pointed quantum disks

We first recall the conformal welding result for two-pointed quantum disk
proved in [AHS20] based on its infinite-area variant in [DMS21]. For $W>0$,
define $\mathcal{M}_{2}^{\textup{disk}}(W;\ell,r)$ via the disintegration
$\mathcal{M}_{2}^{\textup{disk}}(W)=\iint_{0}^{\infty}\mathcal{M}_{2}^{\textup{disk}}(W;\ell,r)d\ell
dr$, where $\mathcal{M}_{2}^{\textup{disk}}(W;\ell,r)$ is supported on
surfaces with left boundary length $\ell$ and right boundary length $r$. Given
a pair of quantum surfaces sampled from
$\mathcal{M}_{2}^{\textup{disk}}(W_{1};\ell_{1},\ell)\times\mathcal{M}_{2}^{\textup{disk}}(W_{2};\ell,\ell_{2})$,
we can conformally weld them together along the boundary with length $\ell$ to
obtain a quantum surface decorated with a curve. We denote its law by
$\mathrm{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W_{1};\ell_{1},\ell),\mathcal{M}_{2}^{\textup{disk}}(W_{2};\ell,\ell_{2}))$.
For $\kappa>0$, $\rho_{-}>-2$ and $\rho_{+}>-2$, chordal
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+})$ is a classical variant of SLEκ
curve on simply connected domain between two boundary points, which will be
recalled in Section 3.1. Fix $W_{1},W_{2}>0$, the conformal welding result for
$\mathcal{M}_{2}^{\textup{disk}}(W_{1})$ and
$\mathcal{M}_{2}^{\textup{disk}}(W_{2})$ says the following. Let $(D,h,a,b)$
be an embedding of a two-pointed quantum disk sampled from
$\mathcal{M}_{2}^{\textup{disk}}(W_{1}+W_{2})$ with $a,b$ being the two
boundary marked points. Let $\eta$ be a
$\mathrm{SLE}_{\kappa}(\rho^{-};\rho^{+})$ curve on $D$ from $a$ to $b$
independent of $h$, where

$\kappa=\gamma^{2}\in(0,4);\quad\textrm{and}\quad\rho_{-}=W_{1}-2>-2;\quad\textrm{and}\quad\rho_{+}=W_{2}-2>-2.$
(1.2)

We write
$\mathcal{M}_{2}^{\textup{disk}}(W_{1}+W_{2})\otimes{\mathrm{SLE}}_{\kappa}(W_{1}-2;W_{2}-2)$
as the law of the curve-decorated surface $(D,h,\eta,a,b)$. Then there is a
constant $c>0$ such that

$\mathcal{M}_{2}^{\textup{disk}}(W_{1}+W_{2})\otimes{\mathrm{SLE}}_{\kappa}(W_{1}-2;W_{2}-2)=c\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W_{1}),\mathcal{M}_{2}^{\textup{disk}}(W_{2})),$
(1.3)

where
$\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W_{1}),\mathcal{M}_{2}^{\textup{disk}}(W_{2})):=\iiint_{0}^{\infty}\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W_{1};\ell,\ell_{1}),\mathcal{M}_{2}^{\textup{disk}}(W_{2};\ell_{1},\ell_{2}))d\ell
d\ell_{1}d\ell_{2}$ is called the conformal welding of
$\mathcal{M}_{2}^{\textup{disk}}(W_{1})$ and
$\mathcal{M}_{2}^{\textup{disk}}(W_{2})$.

The bulk of our paper is devoted to proving that the conformal welding of a
quantum triangle and a two-pointed quantum disk gives another quantum triangle
with an SLE curve whose law is explicit. Similarly as in (1.3), we define
$\mathrm{QT}(W_{1},W_{2},W_{3};\ell_{1},\ell_{2},\ell_{3})$ via the
disintegration
$\mathrm{QT}(W_{1},W_{2},W_{3})=\int\mathrm{QT}(W_{1},W_{2},W_{3};\ell_{1},\ell_{2},\ell_{3})d\ell_{1}d\ell_{2}d\ell_{3}$.
Here $\ell_{i}$ is the length between the weight-$W_{i}$ and weight-$W_{i+1}$
vertices where $i=1,2,3$ and $3+1$ is identified with $1$. Fix
$W,W_{1},W_{2},W_{3}>0$, given a pair of quantum surfaces sampled from
$\mathcal{M}_{2}^{\textup{disk}}(W;\ell_{1},\ell)\times\mathrm{QT}(W_{1},W_{2},W_{3};\ell,\ell_{2},\ell_{3})$,
we conformally weld them together along the boundary with length $\ell$ to
obtain a quantum surface decorated with a curve and three marked points, whose
law is denoted by
$\mathrm{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W;\ell,\ell_{1}),\mathrm{QT}(W_{1},W_{2},W_{3};\ell,\ell_{2},\ell_{3}))$.
We define the conformal welding of $\mathcal{M}_{2}^{\textup{disk}}(W)$ and
$\mathrm{QT}(W_{1},W_{2},W_{3})$ by

$\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W),\mathrm{QT}(W_{1},W_{2},W_{3})):=\iiiint_{0}^{\infty}\mathrm{Weld}({\mathcal{M}}^{\mathrm{disk}}_{2}(W;\ell_{1},\ell),\mathrm{QT}(W_{1},W_{2},W_{3};\ell,\ell_{2},\ell_{3}))d\ell
d\ell_{1}d\ell_{2}d\ell_{3}.$ (1.4)

Similar to (1.3), the law of the three pointed quantum surface for
$\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W),\mathrm{QT}(W_{1},W_{2},W_{3}))$
is proportional to $\mathrm{QT}(W+W_{1},W+W_{2},W_{3})$. To describe the law
of the SLE interface, we need chordal $\mathrm{SLE}_{\kappa}$ with multiple
boundary forces points, which is a more general variant of chordal
$\mathrm{SLE}$ that arises in imaginary geometry [MS16a]. We let
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ be the law of a chordal
$\mathrm{SLE}_{\kappa}$ on the upper half plane $\mathbb{H}$ from $0$ to
$\infty$ with forces points at $0^{-},0^{+},1$, whose weight are
$\rho_{-},\rho_{+},\rho_{1}$ respectively. We will recall its definition in
Section 3, for now it is sufficient to know that it is a random simple curve
on $\mathbb{H}$ from $0$ to $\infty$, with an additional boundary marked
points $0^{-},0^{+},1$ called force points, each of which is labeled by a
number called weight. (This is not to be confused with the weight for a vertex
of a quantum triangle). Our previous notion of chordal
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+})$ on $\mathbb{H}$ from $0$ to
$\infty$ is the special case where $\rho_{1}=0$.

Our first welding result (Theorem 1.1) says that when $W_{1},W_{2},W_{3}$
satisfies $W_{1}+2=W_{2}+W_{3}$, the interface in
$\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W),\mathrm{QT}(W_{1},W_{2},W_{3}))$
is a chordal $\mathrm{SLE}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2})$ curve if
$W+W_{1},W+W_{2},W_{3}$ are all thick weights (namely
$\geq\frac{\gamma^{2}}{2}$); and if some of $W+W_{1},W+W_{2},W_{3}$ are thin,
the analogous result holds after natural modifications. Let us first assume
$W+W_{1},W+W_{2},W_{3}$ are all thick so that a sample from
$\mathrm{QT}(W+W_{1},W+W_{2},W_{3})$ can be embedded as
$(\mathbb{H},h,0,\infty,1)$. Sample $\eta$ from
${\mathrm{SLE}}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2})$ independently from $h$. We
write
$\mathrm{QT}(W+W_{1},W+W_{2},W_{3})\otimes{\mathrm{SLE}}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2})$
as the law of the curve-decorated surface $(\mathbb{H},h,\eta,0,\infty,1)$.
Now if $W_{3}\in(0,\frac{\gamma^{2}}{2})$ instead, then a sample of
$\mathrm{QT}(W+W_{1},W+W_{2},W_{3})$ can be obtained by attaching a weight
$W_{3}$ two-pointed quantum disk to a quantum triangle of weight
$(W+W_{1},W+W_{2},\gamma^{2}-W_{3})$ at the weight $(\gamma^{2}-W_{3})$
vertex. We now embed the weight $(W+W_{1},W+W_{2},\gamma^{2}-W_{3})$ triangle
to $(\mathbb{H},0,+\infty,1)$ and run an independent
${\mathrm{SLE}}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2})$ curve from $0$ to $\infty$.
We still write
$\mathrm{QT}(W+W_{1},W+W_{2},W_{3})\otimes{\mathrm{SLE}}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2})$
as the law of the resulting curve-decorated surface with the two-pointed
quantum disk attached. We will give the precise definition of this law for the
case when $W+W_{1}$ or $W+W_{2}$ is thin in Section 6. See Figure 2 for
illustrations of various cases.

######  Theorem 1.1.

Suppose $W,W_{1},W_{2},W_{3}>0$ with $W_{1}+2=W_{2}+W_{3}$. Then there exists
some constant $c=c_{W,W_{1},W_{2}}\in(0,\infty)$ such that

$\mathrm{QT}(W+W_{1},W+W_{2},W_{3})\otimes{\mathrm{SLE}}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2})=c\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W),\mathrm{QT}(W_{1},W_{2},W_{3})).$
(1.5)

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Figure 2: Illustration of some topological scenarios in Theorem 1.1. (a)
$W\in(0,\frac{\gamma^{2}}{2})$ and
$W_{1},W_{2},W_{3}\geq\frac{\gamma^{2}}{2}$; (b)
$W,W_{1},W_{2}\geq\frac{\gamma^{2}}{2}$ and
$W_{3}\in(0,\frac{\gamma^{2}}{2})$; (c) $W+W_{1}\in(0,\frac{\gamma^{2}}{2})$
and $W_{2},W_{3}\geq\frac{\gamma^{2}}{2}$; (d)
$W,W_{3}\geq\frac{\gamma^{2}}{2}$ and
$W_{1},W_{2}\in(0,\frac{\gamma^{2}}{2})$.

As we will see in Theorem 1.3, quantum triangles whose weight satisfy
$W_{1}-W_{2}=W_{3}-2$ are those that will appear naturally in imaginary
geometry on quantum disk with boundary typical points. The conformal welding
result for $W_{1}-W_{2}\neq W_{3}-2$ can be easily deduced from Theorem 1.1
following arguments in [AHS21]. Suppose $\eta$ is a curve from 0 to $\infty$
on $\mathbb{H}$ that does not touch $1$. Let $D_{\eta}$ be the component of
$\mathbb{H}\backslash\eta$ containing $1$, and $\psi_{\eta}$ is the unique
conformal map from the component $D_{\eta}$ to $\mathbb{H}$ fixing 1 and
sending the first (resp. last) point on $\partial D_{\eta}$ hit by $\eta$ to 0
(resp. $\infty$). Define the measure
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)$ on
curves from 0 to $\infty$ on $\mathbb{H}$ as follows.

$\frac{d\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)}{d{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})}(\eta)=\psi^{\prime}_{\eta}(1)^{\alpha}.$
(1.6)

Then we have the following extension of Theorem 1.1.

######  Theorem 1.2.

Suppose $W,W_{1},W_{2},W_{3}>0$. Set

$\alpha=\frac{W_{3}+W_{2}-W_{1}-2}{4\kappa}(W_{3}+W_{1}+2-W_{2}-\kappa).$
(1.7)

Then with the same constant $c=c_{W,W_{1},W_{2}}\in(0,\infty)$ as in Theorem
1.1, we have

$\mathrm{QT}(W+W_{1},W+W_{2},W_{3})\otimes\widetilde{\mathrm{SLE}}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2};\alpha)=c\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W),\mathrm{QT}(W_{1},W_{2},W_{3})).$
(1.8)

We will give the precise definition of
$\mathrm{QT}(W+W_{1},W+W_{2},W_{3})\otimes\widetilde{\mathrm{SLE}}_{\kappa}(W-2;W_{2}-2,W_{1}-W_{2};\alpha)$
in Section 6, which again requires a proper interpretation when some of
$W+W_{1},W+W_{2},W_{3}$ are thin.

The proof of Theorems 1.1 and 1.2 is divided into three steps that are carried
out in Sections 4—6, respectively. The first step (Theorem 4.1) intuitively
says the following. Suppose $W_{1}>\gamma^{2}/2$ and
$W_{2}\in(0,\gamma^{2}/2)$ in the welding equation (1.3) for
$\mathcal{M}_{2}^{\textup{disk}}(W_{1})$ and
$\mathcal{M}_{2}^{\textup{disk}}(W_{2})$, if a cut point is added to the
weight-$W_{2}$ disk so that it is split into two independent copies of
$\mathcal{M}_{2}^{\textup{disk}}(W_{2})$, then the addition of the third point
create a quantum triangles with weights compatible with Theorem 1.2. Theorem
4.1 is proved via a limiting procedure based on results from [AHS21]. Quantum
triangles that we are able to identify in this step all have two vertices of
equal weight.

The second and third steps require essential new techniques for proving
welding results. First of all, there is no existing mechanism to identify the
law of a quantum surface obtained from welding that has three boundary marked
points of three different log singularities. In Step 2 (Section 5), we provide
a Markovian characterizations of the three-pointed Liouville field that allows
us to identify the law of quantum triangles after welding
$\mathcal{M}_{2}^{\textup{disk}}(W)$ and $\mathrm{QT}(W_{1},W_{2},W_{3})$ as
in Theorem 1.1. This proves Theorems 1.1 and 1.2 in a restricted range of
weights. The range constraint is removed in Step 3 (Section 6). For this
purpose it is crucial to work under the setting of Theorem 1.2, because we
need the freedom to perform conformal welding along different edges of the
same quantum triangle where the condition $W_{1}+2=W_{2}+W_{3}$ in Theorem 1.1
cannot be satisfied simultaneously for every welding. Techniques in Sections 5
and 6 are quite robust and will play a crucial role in the subsequent work
[AY] proving more general welding results for quantum triangles; see Section
1.5.

###  1.3 Imaginary geometry on a quantum disk with multiple boundary points

When welding multiple two-pointed quantum disks, the interfaces are a set of
flow lines in imaginary geometry. We briefly recall the flow line construction
from [MS16a]. For $\kappa\in(0,4)$ and $\rho_{-},\rho_{+}>-2$, set

$\chi=\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}\quad\textrm{and}\quad\lambda=\frac{\pi}{\sqrt{\kappa}};\quad\quad\quad\lambda_{-}=-\lambda(1+\rho_{-})\quad\textrm{and}\quad\lambda_{+}=\lambda(1+\rho_{+}).$
(1.9)

Let $\mathfrak{h}$ be a GFF on the upper half plane $\mathbb{H}$ with
Dirichlet boundary condition such that the boundary value is $\lambda_{-}$
between $(-\infty,0)$ and $\lambda_{+}$ between $(0,\infty)$. Then there
exists a coupling between $\mathfrak{h}$ and an
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+})$ curve $\eta$ on $\mathbb{H}$ from 0
to $\infty$, under which $\eta$ is determined by $\mathfrak{h}$. Although
$\mathfrak{h}$ is only a generalized function, the curve $\eta$ can be
interpreted as the flow line from 0 to $\infty$ of the random vector field
$e^{\frac{i\mathfrak{h}}{\chi}}$. For
$\theta\in(-\frac{\lambda+\lambda_{+}}{\chi},\frac{\lambda-\lambda_{-}}{\chi})$,
we can also consider the flow line of $e^{\frac{i\mathfrak{h}}{\chi}+\theta}$
which is the chordal
$\mathrm{SLE}_{\kappa}(-\frac{\lambda_{-}+\chi\theta}{\lambda}-1;\frac{\lambda_{+}+\chi\theta}{\lambda}-1)$
determined by $\mathfrak{h}+\chi\theta$ via (1.9) with
$(\lambda_{-},\lambda_{+})$ replaced by
$(\lambda_{-}+\chi\theta,\lambda_{+}+\chi\theta)$. Varying $\theta$, we have
multiple SLE curves between $0$ and $\infty$ with force points at $0^{-}$ and
$0^{+}$coupled together. As generalization of (1.3), the conformal welding
result from [AHS20] for multiple two-pointed quantum disks can be stated as
follows. Fix $\gamma\in(0,2)$ and $\kappa=\gamma^{2}$. Consider
$W=\sum_{i=0}^{n}W_{i}$ with $W_{i}>0$. Let $(\mathbb{H},h,0,\infty)$ be an
embedding of a sample from $\mathcal{M}^{\mathrm{disk}}_{2}(W)$. Let
$\mathfrak{h}$ be a GFF with boundary condition $\lambda_{-}=-(W-2)\lambda$
and $\lambda_{+}=0$. Let $\theta_{1}>\theta_{2}>\cdots>\theta_{n}$ be defined
by

$W_{i}=\frac{(\theta_{i}-\theta_{i+1})}{\lambda}\quad\textrm{for
}i<n\quad\textrm{and}\quad W_{n}=1+\frac{\theta_{n}\chi}{\lambda}.$ (1.10)

For $1\leq i\leq n$, let $\eta_{i}$ be the flow line of
$e^{\frac{i\mathfrak{h}}{\chi}+\theta_{i}}$ from $0$ to $\infty$. Then the law
of the decorated quantum surface $(\mathbb{H},h,\eta_{1},\cdots,\eta_{n})$ is
given by the conformal welding of
$\mathcal{M}_{2}^{\textup{disk}}(W_{1}),\cdots,\mathcal{M}_{2}^{\textup{disk}}(W_{n})$
in that order.

Figure 3: An illustration of Theorem 1.3 where $n=3$. The three flow lines cut
the quantum disk $\mathrm{QD}_{0,4}$ into four parts: a weight $W_{0}$ and a
weight $W_{3}$ quantum disk, a weight $(W^{1}_{1},W_{1}^{2},W_{1}^{3})$ thick
quantum triange and a weight $(W_{2}^{1},W_{2}^{2},W_{2}^{3})$ thin triangle.

In the framework of imaginary geometry in [MS16a], it is possible to emanate
flow lines from different boundary points with the same target point.
Unfortunately the conformal welding result for two-pointed disk falls short of
producing this rich picture. Thanks to the introduction of quantum triangles,
this can now be achieved as in Theorem 1.3. Although our result can be stated
more generally, we restrict ourselves to the following neat setting to make
the point. We consider a quantum disk with more than two typical boundary
marked points. For $n\geq 2$, a quantum disk with $n+1$ quantum typical can
sampled as follows. First sample a two-pointed quantum disk from the tilted
measure $L^{n-1}\mathrm{QD}_{0,2}$ where $L$ is the total quantum length of a
sample from $\mathrm{QD}_{0,2}$; then sample $n-1$ additional marked points
independently according to the boundary length measure. Moreover, we consider
the imaginary geometry whose field has zero boundary condition, namely on
$\mathbb{H}$ the boundary value of $\mathfrak{h}$ is
$\lambda_{-}=\lambda_{+}=0$ on $\mathbb{R}$, hence the origin is not special
anymore.

######  Theorem 1.3.

Fix $\gamma\in(0,2)$, $\kappa=\gamma^{2}$, $\lambda=\frac{\pi}{\gamma}$ and
$\chi=\frac{2}{\gamma}-\frac{\gamma}{2}$. For $n\geq 1$, let
$(\mathbb{H},h,\infty,z_{1},\cdots,z_{n})$ be an embedding of a quantum disk
with $n+1$ boundary marked points and $z_{1}<...<z_{n}$. Let $\mathfrak{h}$ be
a zero-boundary Gaussian free field on $\mathbb{H}$ independent of
$h,z_{1},\cdots,z_{n}$. Fix
$\frac{\lambda}{\chi}>\theta_{1}{\color[rgb]{0,0,1}{>}}\theta_{2}>\cdots{\color[rgb]{0,0,1}{>}}\theta_{n}>-\frac{\lambda}{\chi}$.
For $1\leq i\leq n$, let $\eta_{i}$ be the flow line of
$e^{\frac{i\mathfrak{h}}{\chi}+\theta_{i}}$ starting from $z_{i}$. Then the
law of the decorated quantum surface $(\mathbb{H},h,\eta_{1},\cdots,\eta_{n})$
is given by the conformal welding of

${\mathcal{M}}^{\mathrm{disk}}_{2}(W_{0}),\mathrm{QT}(W^{1}_{1},W_{1}^{2},W_{1}^{3}),\cdots,\mathrm{QT}(W_{n-1}^{1},W_{n-1}^{2},W_{n-1}^{3}),{\mathcal{M}}^{\mathrm{disk}}_{2}(W_{n})\quad\textrm{in
that order},$

with $W_{0}=1-\frac{\theta_{1}\chi}{\lambda}$,
$W_{i}^{1}=\frac{(\theta_{i}-\theta_{i+1})\chi}{\lambda}$,
$W_{i}^{2}=1+\frac{\theta_{i}\chi}{\lambda}$,
$W_{i}^{3}=1-\frac{\theta_{i+1}\chi}{\lambda}$ for $i=1,...,n-1$ and
${W}_{n}=1+\frac{\theta_{n}\chi}{\lambda}$.

Every quantum triangle appearing in Theorem 1.3 satisfies the weight
constraint in Theorem 1.1. As we will show in Section 6.6, Theorem 1.3 is an
easy consequence of Theorem 1.1. By a limiting argument, it is also possible
to allow $\theta_{i}=\theta_{i+1}$, in which case $\eta_{i}$ and $\eta_{i+1}$
will merge before hitting the target. The result can also be refined by
allowing $z_{i}=z_{i+1}$. We will not carry out these extensions explicitly.

###  1.4 Applications of Theorem 1.2 to
$\mathrm{SLE}_{\kappa}(\rho^{-};\rho^{+};\rho_{1})$

As demonstrated in [AHS21], conformal welding results such as Theorem 1.2 can
be used to derive the law of the conformal derivative $\psi^{\prime}(1)$ in
(1.6), which is Theorem 1.4 below. Define the function

$F(x,\kappa,\rho_{-},\rho_{+},\rho_{1}):=\frac{\Gamma_{\frac{\sqrt{\kappa}}{2}}(\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}+\frac{\rho_{+}}{\sqrt{\kappa}}+\frac{x}{2})\Gamma_{\frac{\sqrt{\kappa}}{2}}(\frac{4}{\sqrt{\kappa}}+\frac{\rho_{+}+\rho_{1}}{\sqrt{\kappa}}-\frac{x}{2})}{\Gamma_{\frac{\sqrt{\kappa}}{2}}(\frac{4}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}+\frac{\rho_{+}+\rho_{-}}{\sqrt{\kappa}}+\frac{x}{2})\Gamma_{\frac{\sqrt{\kappa}}{2}}(\frac{6}{\sqrt{\kappa}}+\frac{\rho_{-}+\rho_{+}+\rho_{1}}{\sqrt{\kappa}}-\frac{x}{2})}.$
(1.11)

where $\Gamma_{b}(z)$ is the double gamma function that appears frequently in
LCFT; see (2.14) for the definition.

######  Theorem 1.4.

Fix $\kappa\in(0,4)$, $\rho_{-},\rho_{+}>-2$ and $\rho_{1}>-2-\rho_{+}$. Let
$\alpha_{0}=\frac{1}{\kappa}(\rho_{+}+2)(\rho_{+}+\rho_{1}+4-\frac{\kappa}{2})$.
For any $\alpha<\alpha_{0}$, let $\beta$ be a solution to

$\frac{\sqrt{\kappa}(\sqrt{\kappa}-\beta)-\rho_{1}}{4\kappa}\big{(}4-\rho_{1}-\sqrt{\kappa}\beta\big{)}=\alpha.$
(1.12)

Let $\eta$ be an SLE${}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ on $\mathbb{H}$
from 0 to $\infty$ with force points at $0^{-},0^{+},1$, and $\psi$ be as in
(1.6). Then

$\mathbb{E}[\psi_{\eta}^{\prime}(1)^{\alpha}]=\frac{F(\beta+\frac{\rho_{1}}{\sqrt{\kappa}},\kappa,\rho_{-},\rho_{+},\rho_{1})}{F(\sqrt{\kappa},\kappa,\rho_{-},\rho_{+},\rho_{1})}\quad\textrm{for
}\alpha<\alpha_{0}.$ (1.13)

Moreover, if $\alpha\geq\alpha_{0}$ then
$\mathbb{E}[\psi_{\eta}^{\prime}(1)^{\alpha}]=\infty$.

Theorem 1.4 generalizes the main result in [AHS21], which corresponds to the
case $\rho_{1}=0$. The result in [AHS21] is stated for all $\kappa>0$. Our
Theorem 1.4 can also be extended similarly using the same argument based on
SLE duality. By the definition of the measure
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)$ in
(1.13), $\mathbb{E}[\psi_{\eta}^{\prime}(1)^{\alpha}]$ equals its total mass,
which can be computed from the conformal welding identity (1.5) combined with
the integrability of boundary LCFT [RZ22]. See Section 7.2 for its proof. See
the introduction of [AHS21] for a literature review of integrability results
for SLE. Theorem 1.2 also makes the following reversibility of
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ transparent.

######  Theorem 1.5.

Fix $\rho_{+}>-2$, $\rho_{-}>-2$, and $\rho_{1}>-2-\rho_{+}$. Let $\eta$ be an
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ curve in $\mathbb{H}$ from
$0$ to $\infty$ with force located at $0^{-},0^{+}$ and 1. Let $\bar{\eta}$ be
the image of the time reversal of $\eta$ under $z\mapsto\frac{1}{\bar{z}}$.
Then the law of $\bar{\eta}$ is the probability measure proportional to
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+}+\rho_{1},-\rho_{1};\frac{\rho_{1}(4-\kappa)}{2\kappa})$.

For $\rho_{-}=0$, Theorem 1.5 follows from the main result in [Zha22]. Based
on this we prove the $\rho_{-}\neq 0$ case in Section 3.1 using imaginary
geometry. Although the proof does not use LQG, we first guessed the statement
of Theorems 1.1 and 1.2 and then use them to guess the statement Theorem 1.5
before proving it. Indeed, if
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ is the interface of a
sample from
$\mathrm{Weld}({\mathcal{M}}^{\mathrm{disk}}_{2}(W),\mathrm{QT}(W_{1},W_{2},W_{3}))$
from the weight $W+W_{1}$ vertex to the weight $W+W_{2}$ vertex as in Theorem
1.1, then by Theorem 1.2, the law of the interface from the weight $W+W_{2}$
vertex to the weight $W+W_{1}$ vertex in
$\mathrm{Weld}({\mathcal{M}}^{\mathrm{disk}}_{2}(W),\mathrm{QT}(W_{2},W_{1},W_{3}))$
is
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+}+\rho_{1},-\rho_{1};\frac{\rho_{1}(4-\kappa)}{2\kappa})$
with $\alpha=\frac{\rho_{1}(4-\kappa)}{2\kappa}$. Once proved, Theorem 1.5 is
in turn used as a tool to prove Theorems 1.1 and 1.2 in the full range of
parameters.

As another application of Theorem 1.2, let $(\eta_{1},\eta_{2})$ be the two
interfaces in the conformal welding of a two-pointed quantum disk, a quantum
triangle, another two-pointed quantum disk, in that order. Then for $i=1,2$
the marginal law of $\eta_{i}$ and the conditional law of $\eta_{3-i}$ are
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)$ curves
with various parameters. This is an instance of commutation relation for in
the spirit of [Dub07, Zha08]. See Section 7 for the precise statement and its
proof.

###  1.5 Perspectives and related work

We describe a few subsequent works and future directions concerning quantum
triangles and their various applications.

  * •

(Integrability of quantum triangles.) Let $A$ be the area of a sample of
$\mathrm{QT}(W_{1},W_{2},W_{3})$ and $L_{1},L_{2},L_{3}$ be the three boundary
lengths. Then
$(\mu,\mu_{1},\mu_{2},\mu_{3})\mapsto\mathrm{QT}(W_{1},W_{2},W_{3})[e^{-\mu
A-\sum_{i=1}^{3}\mu_{i}L_{i}}]$ gives the boundary three-point structure
constant of Liouville conformal field theory. For $\mu=0$ an exact formula was
obtained in [RZ22] and is used in our proof of Theorem 1.4. With Remy and Zhu,
the first and the second authors of this paper will prove the conjecture of
Ponsot and Teschner [PT02] that the exact expression for $\mu>0$ is given by
the Virasoro fusion kernel.

  * •

(Integrability of imaginary geometry coupled with LQG.) The aforementioned
integrability of quantum triangles, and the welding results in this paper, and
the mating of trees theory [DMS21] can together be used to study the
integrablity of imaginary geometry coupled with LQG. For example, a class of
permutons (i.e. scaling limit of permutation) called the skew Brownian
permutons were recently introduced in [Bor21], with the Baxter permuton [BM22]
as a special case. As shown in [BHSY22, Proposition 1.14], the expected
portion of inversions for these permutons is related to a natural quantity in
imaginary geometry coupled with LQG. In a subsequent work we will derive an
exact expression for this quantity. See [BGS22] for other applications of
SLE/LQG to permutons.

  * •

(Reversibility of SLE.) As explained in Section 1.4 conformal welding of
quantum triangles is closely related to the reversal property of
$\mathrm{SLE}_{\kappa}(\rho^{-};\rho^{+};\rho_{1})$. Recently Zhan [Zha22] and
the third named author [Yu22] gave a description of the law of the time
reversal of chordal SLE curves with multiple force points. We believe that the
conformal welding of multiple quantum disks and quantum triangles can provide
an alternative and more robust approach to such results, which can extend to
other cases such as the time reversal of radial $\mathrm{SLE}$ with multiple
force points.

  * •

(Extensions to $\kappa\in(4,8)$ and integrability of non-simple CLE.) Our
conformal welding results have nontrivial extension to $\mathrm{SLE}_{\kappa}$
curves with $\kappa\in(4,8)$, which corresponds to counter flow lines in
imaginary geometry [MS16a]. These results will be used to study the
integrability of conformal loop ensemble (CLE) with $\kappa\in(4,8)$ where the
loops are non-simple. In particular, we aim at extending results in [AS21,
ARSZ22] for simple CLE, and deriving exact results specific to the non-simple
regime such as the probability that an outermost loop of a CLE on the disk
touches the boundary.

  * •

(Interior flow lines.) Imaginary geometry with interior flow lines was
developed in [MS17]. The first and the third named authors will prove the
counterpart of Theorems 1.1—1.3 in that setting and plan to use them to study
properties of radial and whole plane SLE. Both results and techniques in this
paper will play a crucial role.

  * •

(Quantum triangulation) Given a triangulation of any surface, we can
conformally weld quantum triangles following the topological prescription. The
first and third named authors will prove that conditioning on the conformal
structure of the resulting Riemann surface, the field is a Liouville field on
that surface. If the resulting surface is non-simple, then the conformal
structure (i.e. modulus) of surface itself is random. It is an interesting
challenge to understand the random moduli and the SLE interfaces in this
setting.

Acknowledgements. We thank Nina Holden and Scott Sheffield for helpful
discussions at the early stage of the project. We thank Dapeng Zhan for
explaining his work [Zha22]. M.A. and P.Y. were partially supported by NSF
grant DMS-1712862. M.A. was partially supported by the Simons Foundation as a
Junior Fellow at the Simons Society of Fellows. X.S. was partially supported
by the NSF grant DMS-2027986, the NSF Career award 2046514, and by a
fellowship from the Institute for Advanced Study (IAS) during 2022-2023. P.Y.
thanks IAS for hosting his visit during Fall 2022.

##  2 Quantum triangles: definition and basic properties

In this section we recall some preliminaries. In Section 2.1, we start with
the definition of the Gaussian free field (GFF) and review the definition of
quantum surfaces. In Section 2.2 and Section 2.3, we relate marked quantum
disks and Liouville CFT and establish the precise definition of the quantum
triangle. In Section 2.4, we consider the quantum triangles with fixed
boundary lengths. Finally in Section 2.5, we define quantum triangles with
weight $\frac{\gamma^{2}}{2}$ vertices by a limiting procedure.

In this paper we work with non-probability measures and extend the terminology
of ordinary probability to this setting. For a finite or $\sigma$-finite
measure space $(\Omega,\mathcal{F},M)$, we say $X$ is a random variable if $X$
is an $\mathcal{F}$-measurable function with its law defined via the push-
forward measure $M_{X}=X_{*}M$. In this case, we say $X$ is sampled from
$M_{X}$ and write $M_{X}[f]$ for $\int f(x)M_{X}(dx)$. Weighting the law of
$X$ by $f(X)$ corresponds to working with the measure $d\tilde{M}_{X}$ with
Radon-Nikodym derivative $\frac{d\tilde{M}_{X}}{dM_{X}}=f$, and conditioning
on some event $E\in\mathcal{F}$ (with $0<M[E]<\infty$) refers to the
probability measure $\frac{M[E\cap\cdot]}{M[E]}$ over the space
$(E,\mathcal{F}_{E})$ with $\mathcal{F}_{E}=\\{A\cap E:A\in\mathcal{F}\\}$.
For a finite measure $M$ we write $M^{\\#}=M/|M|$ for the probability measure
proportional to $M$. We also fix the notation $|z|_{+}:=\max\\{|z|,1\\}$.

###  2.1 The Gaussian free field and quantum surfaces

Let $D\subset\mathbb{C}$ be a domain with $\partial
D=\partial^{D}\cup\partial^{F}$, $\partial^{D}\cap\partial^{F}=\emptyset$. We
construct the GFF on $D$ with Dirichlet boundary conditions on $\partial^{D}$
and free boundary conditions on $\partial^{F}$ as follows. Consider the space
of smooth functions on $D$ with finite Dirichlet energy and zero value near
$\partial^{D}$, and let $H(D)$ be its closure with respect to the inner
product $(f,g)_{\nabla}=\int_{D}(\nabla f\cdot\nabla g)\ dx\ dy$. Then our GFF
is defined by

$h=\sum_{n=1}^{\infty}\xi_{n}f_{n}$ (2.1)

where $(\xi_{n})_{n\geq 1}$ is a collection of i.i.d. standard Gaussians and
$(f_{n})_{n\geq 1}$ is an orthonormal basis of $H(D)$. One can show that the
sum (2.1) a.s. converges to a random distribution independent of the choice of
the basis $(f_{n})_{n\geq 1}$. Note that if $\partial^{D}$ is harmonically
trivial, then elements in $H(D)$ should be understood as smooth functions
modulo global additive constants, and the resulting $h$ is a distribution
modulo an additive (random) global constant. For $D=\mathcal{S}$, the
horizontal strip $\mathbb{R}\times(0,\pi)$, we fix the constant by requiring
every function in $H(\mathcal{S})$ has mean value zero on
$\\{0\\}\times[0,i\pi]$, while for $D=\mathbb{H}$, the upper half plane
$\\{z:\text{Im}z>0\\}$, every function in $H(\mathbb{H})$ should have zero
average value on the semicircle $\\{e^{i\theta}:\theta\in(0,\pi)\\}$, and we
denote the corresponding laws of $h$ by $P_{\mathcal{S}}$ and
$P_{\mathbb{H}}$, and the samples from $P_{\mathcal{S}}$ and $P_{\mathbb{H}}$
are referred as $h_{\mathcal{S}}$ and $h_{\mathbb{H}}$. See [DMS21, Section
4.1.4] for more details.

For $h_{\mathcal{S}}$ and $h_{\mathbb{H}}$, the covariance kernels
$G_{D}(z,w):=\mathbb{E}[h_{D}(z)h_{D}(w)]$ are given by

$\begin{split}&G_{\mathcal{S}}(z,w)=-\log|e^{z}-e^{w}|-\log|e^{z}-e^{\bar{w}}|+2\max\\{\text{Re}z,0\\}+2\max\\{\text{Re}w,0\\},\\\
&G_{\mathbb{H}}(z,w)=G_{\mathcal{S}}(e^{z},e^{w})=-\log|z-w|-\log|z-\bar{w}|+2\log|z|_{+}+2\log|w|_{+}.\end{split}$
(2.2)

Note that the notion $h_{D}(z)$ is defined by first taking the circle average
$h_{D,\varepsilon}(z)$ of $h_{D}$ over $\partial B(z,\varepsilon)$ and then
sending $\varepsilon\to 0$.

One important fact is the radial-lateral decomposition of $h_{\mathcal{S}}$.
Consider the subspace $H_{1}(\mathcal{S})\subset H(\mathcal{S})$ (resp.
$H_{2}(\mathcal{S})\subset H(\mathcal{S})$) of functions with constant value
(resp. mean zero) on $[t,t+i\pi]:=\\{t\\}\times(0,\pi)$ for every $t>0$. Then
we have the orthogonal decomposition $H(\mathcal{S})=H_{1}(\mathcal{S})\oplus
H_{2}(\mathcal{S})$, and we can write

$h_{\mathcal{S}}=h_{\mathcal{S}}^{1}+h_{\mathcal{S}}^{2}$ (2.3)

by gathering the corresponding orthonormal bases of $H_{1}(\mathcal{S})$ and
$H_{2}(\mathcal{S})$. Moreover, the common values
$\\{h_{\mathcal{S}}^{1}(t)\\}_{t\in\mathbb{R}}$ agrees with the law of
$\\{B_{2t}\\}_{t\in\mathbb{R}}$ where $\\{B_{t}\\}_{t\in\mathbb{R}}$ is the
standard two-sided Brownian motion, while
$h_{\mathcal{S}}^{1},h_{\mathcal{S}}^{2}$ are independent. See [DMS21, Section
4.1.6] for more details.

Another important result is the Markov property of the GFF, which we state
below.

######  Proposition 2.1 (Markov Property of GFF).

Let $D\subset\mathbb{C}$ be a domain with $\partial
D=\partial^{D}\cup\partial^{F}$, $\partial^{D}\cap\partial^{F}=\emptyset$, and
$U\subset D$ open. Let $h$ be the GFF on $D$ with Dirichlet (resp. free)
boundary conditions on $\partial^{D}$ (resp. $\partial^{F}$). Then we can
write $h=h_{1}+h_{2}$ where:

  1. 1.

$h_{1}$ and $h_{2}$ are independent;

  2. 2.

$h_{1}$ is a GFF on $U$ with Dirichlet boundary condition on $\partial
U\backslash\partial^{F}$ and free on $\partial U\cap\partial^{F}$;

  3. 3.

$h_{2}$ is the same as $h$ outside $U$ and harmonic inside $U$.

Note that if $\partial^{D}=\emptyset$ (i.e., $h$ is free) then $h_{2}$ is
defined modulo constant. See [DMS21, Section 4.1.5] for more details. The
above property can also be extended to random sets. We say that a (random)
closed set $A\subset D$ containing $\partial D$ is local, if one can find a
law on pairs $(A,h_{2})$ such that $h_{2}|_{D\backslash A}$ is harmonic, while
given $(A,h_{2})$, we have $h=h_{1}+h_{2}$ where $h_{1}$ is an instance of
zero boundary GFF on $D\backslash A$.

Now we turn to Liouville quantum gravity and the quantum surfaces. Throughout
this paper, we fix the LQG coupling constant $\gamma\in(0,2)$ and set

$Q=\frac{2}{\gamma}+\frac{\gamma}{2},\qquad\qquad\kappa=\gamma^{2}.$

For two tuples $(D,h,z_{1},...,z_{m})$ and
$(\tilde{D},\tilde{h},\tilde{z}_{1},...,\tilde{z}_{m})$, where $D$ and
$\tilde{D}$ are simply connected domains on $\mathbb{C}$ with
$(z_{1},...,z_{m})$ and $(\tilde{z}_{1},...,\tilde{z}_{m})$ being $m$ marked
points on the bulk and the boundary of $D$ and $\tilde{D}$, and $h$ (resp.
$\tilde{h}$) a distribution on $D$ (resp. $\tilde{D}$), we say

$(D,h,z_{1},...,z_{m})\sim_{\gamma}(\tilde{D},\tilde{h},\tilde{z}_{1},...,\tilde{z}_{m})$
(2.4)

if one can find a conformal mapping $\psi:{D}\to\tilde{D}$ such that
$\psi({z}_{j})=\tilde{z}_{j}$ for each $j$ and
$\tilde{h}=\psi\bullet_{\gamma}h:=h\circ\psi^{-1}+Q\log|(\psi^{-1})^{\prime}|$,
and we call each tuple $(D,h,z_{1},...,z_{m})$ modulo the equivalence relation
$\bullet_{\gamma}$ a $\gamma$-quantum surface.

For a $\gamma$-quantum surface $(D,h,z_{1},...,z_{m})$, its quantum area
measure $\mu_{h}$ is defined by taking the weak limit $\varepsilon\to 0$ of
$\mu_{h_{\varepsilon}}:=\varepsilon^{\frac{\gamma^{2}}{2}}e^{\gamma
h_{\varepsilon}(z)}d^{2}z$, where $d^{2}z$ is the Lebesgue area and
$h_{\varepsilon}(z)$ is the circle average of $h$ over $\partial
B(z,\varepsilon)$. When $D=\mathbb{H}$, we can also define the quantum
boundary length measure $\nu_{h}:=\lim_{\varepsilon\to
0}\varepsilon^{\frac{\gamma^{2}}{4}}e^{\frac{\gamma}{2}h_{\varepsilon}(x)}dx$
where $h_{\varepsilon}(x)$ is the average of $h$ over the semicircle
$\\{x+\varepsilon e^{i\theta}:\theta\in(0,\pi)\\}$. It has been shown in
[DS11, SW16] that all these weak limits are well-defined for the GFF and its
variants we are considering in this paper, while $\mu_{h}$ and $\nu_{h}$ could
be conformally extended to other domains using the relation
$\bullet_{\gamma}$.

Next we present the definition of weight $W$ (thick) quantum disk, introduced
in [DMS21, Section 4.5].

######  Definition 2.2.

Fix $W\geq\frac{\gamma^{2}}{2}$ and let $\beta=\gamma+\frac{2-W}{\gamma}<Q$.
Let $\psi=\psi_{1}+\psi_{2}$ where $\psi_{1}$ and $\psi_{2}$ be independent
distributions on $\mathcal{S}$ such that:

  1. 1.

$\psi_{1}$ has the same law as

$X_{t}:=\left\\{\begin{array}[]{rcl}B_{2t}-(Q-\beta)t+\mathbf{c}&\mbox{for}&t\geq
0\\\ \tilde{B}_{-2t}+(Q-\beta)t+\mathbf{c}&\mbox{for}&t<0\end{array}\right.$
(2.5)

where $(B_{t})_{t\geq 0}$ and $(\tilde{B}_{t})_{t\geq 0}$ are standard
Brownian motions conditioned on $B_{2t}-(Q-\beta)t<0$ and
$\tilde{B}_{2t}-(Q-\beta)t<0$ for all $t>0$, and $\mathbf{c}$ is independently
sampled from $\frac{\gamma}{2}e^{(\beta-Q)c}dc$;

  2. 2.

$\psi_{2}$ has the same law as $h_{\mathcal{S}}^{2}$ described in (2.3).

Let $\mathcal{M}^{\textup{disk}}_{2}(W)$ be infinite measure describing the
law of $(\mathcal{S},\psi,-\infty,+\infty)/\sim_{\gamma}$. We call a sample
from $\mathcal{M}^{\textup{disk}}_{2}(W)$ a (two-pointed) _quantum disk of
weight $W$_.

When $0<W<\frac{\gamma^{2}}{2}$, we can also define the thin quantum disk as a
concatenation of weight $\gamma^{2}-W$ (two-pointed) thick disks as in [AHS20,
Section 2].

######  Definition 2.3.

For $W\in(0,\frac{\gamma^{2}}{2})$, the infinite measure
$\mathcal{M}_{2}^{\textup{disk}}(W)$ on two-pointed beaded surfaces is defined
as follows. First sample $T$ from
$(1-\frac{2}{\gamma^{2}}W)^{-2}\textup{Leb}_{\mathbb{R}_{+}}$, then sample a
Poisson point process $\\{(u,\mathcal{D}_{u})\\}$ from the intensity measure
$\textbf{1}_{t\in[0,T]}dt\times\mathcal{M}_{2}^{\textup{disk}}(\gamma^{2}-W)$
and finally concatenate the disks $\\{\mathcal{D}_{u}\\}$ according to the
ordering induced by $u$. The total sum of the left (resp. right) boundary
lengths of all the $\mathcal{D}_{u}$’s is referred as the left (resp. right)
boundary length of the thin quantum disk.

We introduce the notion of embedding a thin quantum disk in the plane.
Although not mathematically essential for our arguments, it simplifies
exposition by letting us talk concretely about points and curves in the plane
rather than abstractly on quantum surfaces. We follow the treatment of
[DMS21].

A _(beaded) quantum surface_ is a tuple $(D,h,z_{1},\dots,z_{m})$ modulo the
equivalence relation (2.4), except that $D\subset\mathbb{C}$ is a closed set
such that each component of its interior together with its prime-end boundary
is homeomorphic to the closed disk, $h$ is defined as a distribution on each
such component, and $\psi:D\to\tilde{D}$ is any homeomorphism which is
conformal on each component of the interior of $D$ and sends
$\psi(z_{i})=\tilde{z}_{i}$ for each $i$. An _embedding_ of a beaded quantum
surface is any choice of representative $(D,h,z_{1},\dots,z_{m})$. It is easy
to see that a thin quantum disk is a beaded quantum surface.

###  2.2 Liouville conformal field theory and thick quantum triangles

In this section we review the theory of Liouville CFT and its relation with
quantum disks as established in [AHS21, Section 2]. We will recap the notion
of $\mathcal{M}_{2,\cdot}^{\textup{disk}}(W)$, the three-pointed quantum disks
and then give the definition of quantum triangles in terms of LCFT.

We start from the LCFT on the upper half plane. Recall that $P_{\mathcal{S}}$
and $P_{\mathbb{H}}$ are the probability measure induced by the GFF as in
(2.1) with our normalization.

######  Definition 2.4.

Let $(h,\mathbf{c})$ be sampled from $P_{\mathbb{H}}\times[e^{-Qc}dc]$ and
take $\phi=h-2Q\log|z|_{+}+\mathbf{c}$. We say $\phi$ is a Liouville field on
$\mathbb{H}$ and let $\textup{LF}_{\mathbb{H}}$ be its law.

######  Definition 2.5 (Liouville field with boundary insertions ).

Let $\beta_{i}\in\mathbb{R}$ and $s_{i}\in\partial\mathbb{H}\cup\\{\infty\\}$
for $i=1,...,m,$ where $m\geq 1$ and all the $s_{i}$’s are distinct. Also
assume $s_{i}\neq\infty$ for $i\geq 2$. We say $\phi$ is a Liouville Field on
$\mathbb{H}$ with insertions $\\{(\beta_{i},s_{i})\\}_{1\leq i\leq m}$ if
$\phi$ can be produced as follows by first sampling $(h,\mathbf{c})$ from
$C_{\mathbb{H}}^{(\beta_{i},s_{i})_{i}}P_{\mathbb{H}}\times[e^{(\frac{1}{2}\sum_{i=1}^{m}\beta_{i}-Q)c}dc]$
with

$C_{\mathbb{H}}^{(\beta_{i},s_{i})_{i}}=\left\\{\begin{array}[]{rcl}\prod_{i=1}^{m}|s_{i}|_{+}^{-\beta_{i}(Q-\frac{\beta_{i}}{2})}\exp(\frac{1}{4}\sum_{j=i+1}^{m}\beta_{i}\beta_{j}G_{\mathbb{H}}(s_{i},s_{j}))&\mbox{if}&s_{1}\neq\infty\\\
\prod_{i=2}^{m}|s_{i}|_{+}^{-\beta_{i}(Q-\frac{\beta_{i}}{2}-\beta_{1})}\exp(\frac{1}{4}\sum_{j=i+1}^{m}\beta_{i}\beta_{j}G_{\mathbb{H}}(s_{i},s_{j}))&\mbox{if}&s_{1}=\infty\end{array}\right.$

and then taking

$\phi(z)=h(z)-2Q\log|z|_{+}+\frac{1}{2}\sum_{i=1}^{m}\beta_{i}G_{\mathbb{H}}(s_{i},z)+\mathbf{c}$
(2.6)

with the convention $G_{\mathbb{H}}(\infty,z)=2\log|z|_{+}$. We write
$\textup{LF}_{\mathbb{H}}^{(\beta_{i},s_{i})_{i}}$ for the law of $\phi$.

The following lemma explains that adding a $\beta$-insertion point at
$s\in\partial\mathbb{H}$ is equal to weighting the law of Liouville field
$\phi$ by $e^{\frac{\beta}{2}\phi(s)}$ in some sense.

######  Lemma 2.6 (Lemma 2.6 of [AHS21]).

For $\beta,s\in\mathbb{R}$ such that $s\notin\\{s_{1},...,s_{m}\\}$, in the
sense of vague convergence of measures,

$\lim_{\varepsilon\to
0}\varepsilon^{\frac{\beta^{2}}{4}}e^{\frac{\beta}{2}\phi_{\varepsilon}(s)}\textup{LF}_{\mathbb{H}}^{(\beta_{i},s_{i})_{i}}=\textup{LF}_{\mathbb{H}}^{(\beta_{i},s_{i})_{i},(\beta,s)}.$
(2.7)

On the other hand, insertions at the infinity can also be handled via the
following approximation. For $\beta\in\mathbb{R}$, we use the shorthand

$\Delta_{\beta}:=\frac{\beta}{2}(Q-\frac{\beta}{2}).$ (2.8)

######  Lemma 2.7 (Lemma 2.9 of [AHS21]).

With the same notation as Lemma 2.6, in the topology of vague convergence of
measures,

$\lim_{r\to+\infty}r^{2\Delta_{\beta}}\textup{LF}_{\mathbb{H}}^{(\beta,r),(\beta_{i},s_{i})_{i}}=\textup{LF}_{\mathbb{H}}^{(\beta,\infty),(\beta_{i},s_{i})_{i}}.$
(2.9)

Sometimes it is also natural to work on Liouville fields on the strip
$\mathcal{S}$ with insertions at $\pm\infty$.

######  Definition 2.8.

Let $(h,\mathbf{c})$ be sampled from
$C_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},s_{3})}P_{\mathcal{S}}\times[e^{(\frac{\beta_{1}+\beta_{2}+\beta_{3}}{2}-Q)c}dc]$
with $\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$,
$s_{3}\in\partial\mathcal{S}$ and

$C_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},s_{3})}=e^{(-\Delta_{\beta_{3}}+\frac{(\beta_{1}+\beta_{2})\beta_{3}}{4})|\textup{Re}s_{3}|+\frac{(\beta_{1}-\beta_{2})\beta_{3}}{4}\textup{Re}s_{3}}.$

Let
$\phi(z)=h(z)+\frac{\beta_{1}+\beta_{2}-2Q}{2}|\textup{Re}z|+\frac{\beta_{1}-\beta_{2}}{2}\textup{Re}z+\frac{\beta_{3}}{2}G_{\mathcal{S}}(z,s_{3})+\mathbf{c}$.
We write
$\textup{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},s_{3})}$
for the law of $\phi$.

In general, the Liouville fields has nice compatibility with the notion of
quantum surfaces. To be more precise, for a measure $M$ on the space of
distributions on a domain $D$ and a conformal map $\psi:D\to\tilde{D}$, if we
let $\psi_{*}M$ be the push-forward of $M$ under the mapping
$\phi\mapsto\phi\circ\psi^{-1}+Q\log|(\psi^{-1})^{\prime}|$. Then under this
push-forward, the corresponding Liouville field measures only differs a
multiple constant. For instance,

######  Lemma 2.9.

For $\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$ and
$s_{3}\in\partial\mathcal{S}$, we have

$\textup{LF}_{\mathbb{H}}^{(\beta_{1},\infty),(\beta_{2},0),(\beta_{3},e^{s_{3}})}=e^{-\Delta_{\beta_{3}}\textup{Re}s_{3}}\exp_{*}\textup{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},s_{3})}.$
(2.10)

For a proof, one can directly compare the expressions of the corresponding
multiplicative constants and invoke the conformal invariance of the GFF and
the Green’s function (with the mapping $z\mapsto e^{z}$). We also have the
following

######  Lemma 2.10 (Proposition 2.7 of [AHS21]).

Fix $\beta_{i},s_{i}\in\mathbb{R}$ for $i=1,...,m$ with $s_{i}$’s being
distinct. Suppose $\psi:\mathbb{H}\to\mathbb{H}$ is conformal such that
$\psi(s_{i})\neq\infty$ for each $i$. Then
$\textup{LF}_{\mathbb{H}}=\psi_{*}\textup{LF}_{\mathbb{H}}$, and

$\textup{LF}_{\mathbb{H}}^{(\beta_{i},\psi(s_{i}))_{i}}=\prod_{i=1}^{m}|\psi^{\prime}(s_{i})|^{-\Delta_{\beta_{i}}}\psi_{*}\textup{LF}_{\mathbb{H}}^{(\beta_{i},s_{i})_{i}}.$
(2.11)

Using Lemma 2.7, the above result can also be extended to Liouville fields
with insertions at infinity.

######  Lemma 2.11.

Suppose $\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$ and
$\psi:\mathbb{H}\to\mathbb{H}$ being conformal with $\psi(0)=1$,
$\psi(1)=\infty$ and $\psi(\infty)=0$. Then

$\textup{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}=\psi_{*}\textup{LF}_{\mathbb{H}}^{(\beta_{1},\infty),(\beta_{2},0),(\beta_{3},1)}.$
(2.12)

###### Proof.

The proof is almost identical to that of [AHS21, Lemma 2.13]. Note
$\psi(z)=\frac{1}{1-z}$, and for $r>0$ set
$\psi_{r}(z):=\frac{z-r}{(r-1)z-r}$. Now $|\psi_{r}^{\prime}(0)|=1+o_{r}(1)$,
$|\psi_{r}^{\prime}(1)|=(1+o_{r}(1))r^{2}$ and
$|\psi_{r}^{\prime}(r)|=(1+o_{r}(1))r^{-2}$, by Lemma 2.10, as $r\to\infty$,

$\textup{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},r)}=(1+o_{r}(1))r^{-2\Delta_{\beta_{3}}+2\Delta_{\beta_{1}}}(\psi_{r})_{*}\textup{LF}_{\mathbb{H}}^{(\beta_{1},r),(\beta_{2},0),(\beta_{3},1)}.$
(2.13)

Since $\psi_{r}\to\psi$ in the topology of uniform convergence of analytic
functions and their derivatives on compact sets, we are done by multiplying
both sides of (2.13) by $r^{2\Delta_{\beta_{3}}}$ and applying Lemma 2.7. ∎

The _uniform embedding_ of two-pointed quantum disk in the strip gives a
Liouville field:

######  Theorem 2.12 (Theorem 2.22 of [AHS21]).

For $W>\frac{\gamma^{2}}{2}$ and $\beta=\gamma+\frac{2-W}{\gamma}$, if we
independently sample $T$ from $\textup{Leb}_{\mathbb{R}}$ and
$(\mathcal{S},\phi,+\infty,-\infty)$ from
$\mathcal{M}_{2}^{\textup{disk}}(W)$, then the law of
$\tilde{\phi}:=\phi(\cdot+T)$ is
$\frac{\gamma}{2(Q-\beta)^{2}}\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty)}$.

This result also leads to the notion of three-pointed quantum disks, where we
may first sample a surface from the quantum disk measure reweighted by the
left/right boundary length, and then sample a third marked point on
$\mathbb{R}$ from the quantum length measure.

######  Definition 2.13.

Fix $W>\frac{\gamma^{2}}{2}$. First sample
$(\mathcal{S},\phi,+\infty,-\infty)$ from
$\nu_{\phi}(\mathbb{R})\mathcal{M}_{2}^{\textup{disk}}(W)[d\phi]$ and then
sample $s\in\mathbb{R}$ according to the probability measure proportional to
$\nu_{\phi}|_{\mathbb{R}}$. We denote the law of the surface
$(\mathcal{S},\phi,+\infty,-\infty,s)/{\sim_{\gamma}}$ by
$\mathcal{M}_{2,\bullet}^{\textup{disk}}(W)$.

The definition above can be naturally extended to the case with the marked
point added on $\mathbb{R}+i\pi$. And we have the following relation between
$\mathcal{M}_{2,\cdot}^{\textup{disk}}(W)$ and Liouville fields.

######  Proposition 2.14 (Proposition 2.18 of [AHS21]).

For $W>\frac{\gamma^{2}}{2}$ and $\beta=\gamma+\frac{2-W}{\gamma}$, let $\phi$
be sampled from
$\frac{\gamma}{2(Q-\beta)^{2}}\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty),(\gamma,0)}$.
Then $(\mathcal{S},\phi,+\infty,-\infty,0)/{\sim_{\gamma}}$ has the same law
as $\mathcal{M}_{2,\bullet}^{\textup{disk}}(W)$.

This third added point, which is sampled from the quantum length measure, is
usually referred as quantum typical point, and results in a $\gamma$-insertion
to the Liouville field. This gives rise to the quantum disks with general
third insertion points, which could be defined via three-pointed Liouville
fields.

######  Definition 2.15.

Fix $W>\frac{\gamma^{2}}{2}$ and let $\alpha\in\mathbb{R}$. Set
$\mathcal{M}_{2,\cdot}^{\textup{disk}}(W;\alpha)$ to be the law of
$(\mathcal{S},\phi,+\infty,-\infty,0)/\sim_{\gamma}$ with $\phi$ sampled from
$\frac{\gamma}{2(Q-\beta)^{2}}\textup{LF}_{\mathcal{S}}^{(\beta,\pm\infty),(\alpha,0)}$.
We call the boundary arc between the two $\beta$-singularities with (resp. not
containing) the $\alpha$-singularity the marked (resp. unmarked) boundary arc.

One can also add a third boundary marked point for thin disks. Recall in
[AHS20, Proposition 4.4] this could be done by starting from first sampling a
thick disk from $\mathcal{M}_{2,\cdot}^{\textup{disk}}(\gamma^{2}-W)$ and then
gluing another two independent weight $W$ thin disks to the two endpoints.
Therefore this leads to

######  Definition 2.16.

For $W\in(0,\frac{\gamma^{2}}{2})$ and $\alpha\in\mathbb{R}$, suppose
$(S_{1},S_{2},S_{3})$ is sampled from

$(1-\frac{2}{\gamma^{2}}W)^{2}\mathcal{M}_{2}^{\textup{disk}}(W)\times\mathcal{M}_{2,\bullet}^{\textup{disk}}(\gamma^{2}-W;\alpha)\times\mathcal{M}_{2}^{\textup{disk}}(W)$

and $S$ is the concatenation of the three surfaces. Then we define the
infinite measure $\mathcal{M}_{2,\bullet}^{\textup{disk}}(W;\alpha)$ to be the
law of $S$.

So far we have studied three-pointed quantum surfaces in terms of LCFT
whenever two of the insertion points have the same $\alpha$ value. Indeed this
relation can be extended to three-pointed Liouville fields with different
insertion values, from which arises the notion of quantum triangles.

######  Definition 2.17 (Thick quantum triangles).

Fix $W_{1},W_{2},W_{3}>\frac{\gamma^{2}}{2}$. Set
$\beta_{i}=\gamma+\frac{2-W_{i}}{\gamma}<Q$ for $i=1,2,3$, and let $\phi$ be
sampled from
$\frac{1}{(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}\textup{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},0)}$.
Then we define the infinite measure $\textup{QT}(W_{1},W_{2},W_{3})$ to be the
law of $(\mathcal{S},\phi,+\infty,-\infty,0)/\sim_{\gamma}$.

We note that by Lemma 2.9 (where $s_{3}=0$) and Lemma 2.11, the measure
$\textup{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},0)}$
has invariance under the conformal mappings $\mathcal{S}\to\mathcal{S}$
rearranging $\\{+\infty,-\infty,0\\}$ and hence compatible with the relation
$\sim_{\gamma}$. To embed our quantum triangles onto other domains, we can
further apply the conformal transforms and use $\sim_{\gamma}$. Also as will
be explained in Section 2.5, the choice of the constant
$\frac{1}{(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}$ shall allow us to extend
the definition when some of the $W_{i}$ is the same as $\frac{\gamma^{2}}{2}$,
and the boundary length law is some sort of analytic.

###  2.3 Quantum triangles with thin vertices

Again recall that we can define thin quantum disks of weight
$W\in(0,\frac{\gamma^{2}}{2})$ via concatenation of weight $\gamma^{2}-W$
thick disks (Definition 2.3), and a triply-marked thin disk could also be
constructed by concatenating three-pointed weight $\gamma^{2}-W$ disks with
weight $W$ thin disks. We shall apply the same idea to construct quantum
triangles with thin vertices. See Figure 1 for an illustration.

######  Definition 2.18.

Fix
$W_{1},W_{2},W_{3}\in(0,\frac{\gamma^{2}}{2})\cup(\frac{\gamma^{2}}{2},\infty)$.
Let $I:=\\{i\in\\{1,2,3\\}:W_{i}<\frac{\gamma^{2}}{2}\\}$. Let
$\tilde{W}_{i}=W_{i}$ if $i\notin I$, and $\tilde{W}_{i}=\gamma^{2}-W_{i}$ if
$i\in I$. Sample $(S_{0},(S_{i})_{i\in I})$ from

$\textup{QT}(\tilde{W}_{1},\tilde{W}_{2},\tilde{W}_{3})\times\prod_{i\in
I}(1-\frac{2W_{i}}{\gamma^{2}})\mathcal{M}_{2}^{\textup{disk}}(W_{i}).$

For $i\in I$, concatenate $S_{i}$ with $S_{0}$ at the vertex of $S_{0}$ of
weight $\tilde{W}_{i}$. Let $\textup{QT}(W_{1},W_{2},W_{3})$ be the law of the
resulting quantum surface.

######  Remark 2.19.

We point out that when $W_{3}>\frac{\gamma^{2}}{2}$ with
$\beta_{3}=\gamma+\frac{2-W_{3}}{\gamma}$, by Definitions 2.15, 2.16 and 2.18,
the measure ${\mathcal{M}}^{\mathrm{disk}}_{2,\bullet}(W;\beta_{3})$ is some
multiple constant of the measure $\mathrm{QT}(W,W,W_{3})$. In this paper we
use both notations.

######  Definition 2.20.

For a quantum triangle with thin vertices as in Definition 2.18, we call
$S_{0}$ its _core_ , and we call each $S_{i}$ an _arm_ of weight $W_{i}$.

Since the thin quantum triangle is a concatenation of a thick quantum triangle
with one to three independent thin quantum disks, we embed the surface as
$(D,\phi,a_{1},a_{2},a_{3})$ where $D$ is not simply connected; see the
discussion after Definition 2.3. The vertices $a_{1},a_{2},a_{3}$ correspond
to the weight $W_{1},W_{2},W_{3}$ vertices respectively. To simplify the
notations, we shall call the boundary arc between the points with weights
$W_{1}$ and $W_{2}$ the _left_ boundary arc, the boundary arc between the
points with weights $W_{2}$ and $W_{3}$ the _bottom_ boundary arc, and the
points with weights $W_{3}$ and $W_{1}$ the _right_ boundary arc, as depicted
in Figure 1.

In the remaining of this section, we will work on the boundary length law of
quantum triangles. We begin with the integrability of boundary LQG measure as
obtained in [RZ20, RZ22]. To state the results we will need several functions.
The functions $\bar{R}$ and $\bar{H}$ are introduced for more general
parameters (see [RZ22, Page 6-8]) but for simplicity we only the ones which
will appear later. For $b>0$, recall the double-gamma function, the
meromorphic function $\Gamma_{b}(z)$ in $\mathbb{C}$ such that for
$\textup{Re}z>0$,

$\log\Gamma_{b}(z)=\int_{0}^{\infty}\frac{1}{t}\bigg{(}\frac{e^{-zt}-e^{\frac{(b^{2}+1)t}{2b}}}{(1-e^{-bt})(1-e^{-\frac{1}{b}t)}}-\frac{(b^{2}+1-2bz)^{2}}{2b^{2}}e^{-t}+\frac{2bz-b^{2}-1}{bt}\bigg{)}dt$
(2.14)

and it satisfies the shift equations

$\frac{\Gamma_{b}(z)}{\Gamma_{b}(z+b)}=\frac{1}{\sqrt{2\pi}}\Gamma(bz)b^{-bz+\frac{1}{2}},\
\
\frac{\Gamma_{b}(z)}{\Gamma_{b}(z+b^{-1})}=\frac{1}{\sqrt{2\pi}}\Gamma(b^{-1}z)b^{\frac{z}{b}-\frac{1}{2}}.$
(2.15)

For $\mu>0$, let

$\bar{R}(\beta,\mu,0):=\bar{R}(\beta,0,\mu)=\mu^{\frac{2(Q-\beta)}{\gamma}}\frac{(2\pi)^{\frac{2(Q-\beta)}{\gamma}-\frac{1}{2}}(\frac{2}{\gamma})^{\frac{\gamma(Q-\beta)}{2}-\frac{1}{2}}}{(Q-\beta)\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2(Q-\beta)}{\gamma}}}\frac{\Gamma_{\frac{\gamma}{2}}(\beta-\frac{\gamma}{2})}{\Gamma_{\frac{\gamma}{2}}(Q-\beta)}.$
(2.16)

Finally set $\bar{\beta}=\beta_{1}+\beta_{2}+\beta_{3}$ and

$\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})}:=\frac{(2\pi)^{\frac{2Q-\bar{\beta}+\gamma}{\gamma}(\frac{2}{\gamma})^{(\frac{\gamma}{2}-\frac{2}{\gamma})(Q-\frac{\bar{\beta}}{2})}-1}}{\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2Q-\bar{\beta}}{\gamma}}\Gamma(\frac{\bar{\beta}-2Q}{\gamma})}\frac{\Gamma_{\frac{\gamma}{2}}(\frac{\bar{\beta}}{2}-Q)\Gamma_{\frac{\gamma}{2}}(\frac{\bar{\beta}-2\beta_{2}}{2})\Gamma_{\frac{\gamma}{2}}(\frac{\bar{\beta}-2\beta_{1}}{2})\Gamma_{\frac{\gamma}{2}}(Q-\frac{\bar{\beta}-2\beta_{3}}{2})}{\Gamma_{\frac{\gamma}{2}}(Q)\Gamma_{\frac{\gamma}{2}}(Q-\beta_{1})\Gamma_{\frac{\gamma}{2}}(Q-\beta_{2})\Gamma_{\frac{\gamma}{2}}(\beta_{3})}.$
(2.17)

######  Proposition 2.21 (Theorem 1.1 of [RZ20]; also see Section 3.3.4 of
[RZ22]).

Fix $\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$ and set
$\bar{\beta}=\beta_{1}+\beta_{2}+\beta_{3}$. Let $h$ be sampled from
$P_{\mathbb{H}}$ and let $\phi(z)=h(z)-\beta_{1}\log|z|-\beta_{2}\log|1-z|$.
Then

$\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})}=\mathbb{E}\big{[}\nu_{\phi}([0,1])^{\frac{2Q-\bar{\beta}}{\gamma}}\big{]}$
(2.18)

if $\beta_{1},\beta_{2},\beta_{3}$ satisfies the constraints

$\beta_{1},\beta_{2}<Q,\ |\beta_{1}-\beta_{2}|<\beta_{3},\text{and}\
\bar{\beta}>\gamma.$ (2.19)

If (2.19) is not jointly satisfied, then the right hand side of (2.18) is
infinite.

The boundary length law of quantum disks can also be expressed in terms of
$\bar{R}$.

######  Proposition 2.22 (Propositions 3.3 and 3.6 of [AHS21]).

For $W<\gamma Q$, $\beta=\gamma+\frac{2-W}{\gamma}$, the left (or right)
boundary of a sample from $\mathcal{M}_{2}^{\textup{disk}}(W)$ has law

$\textup{{1}}_{\ell>0}\bar{R}(\beta;1,0)\ell^{-\frac{2}{\gamma^{2}}W}d\ell.$
(2.20)

When $W\geq\gamma Q$, for any subinterval $I$ of $(0,\infty)$, the event
$\\{\text{left boundary length}\in I\\}$ has infinite
$\mathcal{M}_{2}^{\textup{disk}}(W)$ measure.

Now we are ready to find the boundary length law for our quantum triangles.
For a sample from $\textup{QT}(W_{1},W_{2},W_{3})$, let $L_{12}$ be the
quantum length of the boundary arc between the $\beta_{1}$ and $\beta_{2}$
singularities.

######  Proposition 2.23.

Suppose $W_{1},W_{2},W_{3}>\frac{\gamma^{2}}{2}$ and let
$\beta_{i}=\gamma+\frac{2-W_{i}}{\gamma}$ for $i=1,2,3$. Set
$\bar{\beta}=\beta_{1}+\beta_{2}+\beta_{3}$. Suppose $(\beta_{i})$ satisfies
the bounds (2.19). Then for a sample from $\textup{QT}(W_{1},W_{2},W_{3})$,
$L_{12}$ has law

$\textup{{1}}_{\ell>0}\frac{2}{\gamma(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})}\ell^{\frac{\bar{\beta}-2Q}{\gamma}-1}d\ell.$
(2.21)

###### Proof.

By Definition 2.17, we can sample our quantum triangle by sampling $\phi$ from
$\textup{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}$
and outputting $(\mathbb{H},\phi,0,1,\infty)/\sim_{\gamma}$. Then one can
check that our $\phi$ has expression

$\phi(z)=h(z)+(\bar{\beta}-2Q)\log|z|_{+}-\beta_{1}\log|z|-\beta_{2}\log|z-1|+\mathbf{c}:=\phi_{0}(z)+\mathbf{c}$
(2.22)

where $(h,\mathbf{c})$ is sampled from $P_{\mathbb{H}}\times
e^{\frac{\bar{\beta}-2Q}{2}c}dc$. Now for $b>a>0$, we have

$\begin{split}\textup{QT}(W_{1},W_{2},W_{3})\big{[}\textbf{1}_{\nu_{\phi}([0,1])\in(a,b)}\big{]}&=\frac{1}{(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}\mathbb{E}_{P_{\mathbb{H}}}\bigg{[}\int_{0}^{\infty}\textbf{1}_{e^{\frac{\gamma}{2}c}\nu_{\phi_{0}}([0,1])\in(a,b)}e^{\frac{\bar{\beta}-2Q}{2}c}dc\bigg{]}\\\
&=\frac{2}{\gamma(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}\int_{a}^{b}\mathbb{E}_{P_{\mathbb{H}}}\big{[}\big{(}\nu_{\phi_{0}}([0,1])\big{)}^{{\frac{2Q-\bar{\beta}}{\gamma}}}\big{]}\ell^{\frac{\bar{\beta}-2Q}{\gamma}-1}d\ell\end{split}$
(2.23)

where we applied the substitution
$\ell=e^{\frac{\gamma}{2}c}\nu_{\phi_{0}}([0,1])$ and Fubini’s theorem. We
conclude the proof by noticing that our $\phi_{0}$ in (2.22) coincides with
the $\phi$ in Proposition 2.21 on the interval $[0,1]$ and applying (2.18). ∎

We can infer from (2.21) that

$\textup{QT}(W_{1},W_{2},W_{3})[e^{-\mu
L_{12}}]=\frac{2}{\gamma(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})}\Gamma(\frac{\bar{\beta}-2Q}{\gamma})\mu^{{\frac{2Q-\bar{\beta}}{\gamma}}}.$
(2.24)

Therefore we can further use the Laplace transform to compute boundary length
laws for thin quantum triangles.

######  Proposition 2.24.

Fix $W_{1},W_{2}\in(0,\frac{\gamma^{2}}{2})\cup(\frac{\gamma^{2}}{2},\infty)$
and $W_{3}>\frac{\gamma^{2}}{2}$. For $i=1,2,3$ again let
$\beta_{i}=\gamma+\frac{2-W_{i}}{\gamma}$, and $\tilde{\beta}_{i}$ be equal to
$\beta_{i}$ (resp. $2Q-\beta_{i}$) if $W_{i}>\frac{\gamma^{2}}{2}$ (resp.
$W_{i}<\frac{\gamma^{2}}{2}$). Suppose $(\tilde{\beta}_{i})$ satisfies the
bounds (2.19). Then for a sample from $\textup{QT}(W_{1},W_{2},W_{3})$,
$L_{12}$ has law

$\textup{{1}}_{\ell>0}\frac{2}{\gamma(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}{\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})}}\ell^{\frac{\bar{\beta}-2Q}{\gamma}-1}d\ell.$
(2.25)

###### Proof.

We first assume that $W_{1}>\frac{\gamma^{2}}{2}$ and
$W_{2}<\frac{\gamma^{2}}{2}$. Let $L_{2}$ be the left boundary length of a
weight $W_{2}$ disk, then by (2.20),

$\mathcal{M}_{2}^{\textup{disk}}(W_{2})[e^{-\mu
L_{2}}]=\bar{R}(\beta_{2};1,0)\Gamma(1-\frac{2W_{2}}{\gamma^{2}})\mu^{\frac{2W_{2}}{\gamma^{2}}-1}.$
(2.26)

By definition of $\textup{QT}(W_{1},W_{2},W_{3})$, if we independently sample
a triangle from $\textup{QT}(W_{1},\gamma^{2}-W_{2},W_{3})$ and let
$\tilde{L}_{12}$ be the corresponding edge length, then $L_{12}$ has the same
law as $L_{2}+\tilde{L}_{12}$. Therefore by combining (2.24) (where
$\beta_{2}$ is replaced by $2Q-\beta_{2}$) with (2.26) ,

$\begin{split}\textup{QT}(W_{1},W_{2},W_{3})&[e^{-\mu
L_{12}}]=\frac{2}{\gamma(Q-\beta_{1})(\beta_{2}-Q)(Q-\beta_{3})}\times\\\
&\bar{H}_{(0,1,0)}^{(\beta_{1},2Q-\beta_{2},\beta_{3})}(1-\frac{2W_{2}}{\gamma^{2}})\Gamma(\frac{\beta_{1}+\beta_{3}-\beta_{2}}{\gamma})\Gamma(\frac{2}{\gamma}(\beta_{2}-Q))\bar{R}(\beta_{2};1,0)\mu^{{\frac{2Q-\bar{\beta}}{\gamma}}}.\end{split}$
(2.27)

On the other hand, by [RZ22, Lemma 3.4], we have

$\bar{H}_{(0,1,0)}^{(\beta_{1},2Q-\beta_{2},\beta_{3})}=-\frac{\Gamma(\frac{2}{\gamma}(2Q-\beta_{2}-\frac{2}{\gamma}))\Gamma(\frac{\bar{\beta}-2Q}{\gamma})}{\Gamma(\frac{\beta_{1}+\beta_{3}-\beta_{2}}{\gamma})}\bar{R}(2Q-\beta_{2};1,0)\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})},$
(2.28)
$\bar{R}(\beta_{2};1,0)\bar{R}(2Q-\beta_{2};1,0)=\frac{1}{\Gamma(1-\frac{2(Q-\beta_{2})}{\gamma})\Gamma(1+\frac{2(Q-\beta_{2})}{\gamma})}$
(2.29)

Combining the equations (2.27), (2.28) and (2.29) implies

$\textup{QT}(W_{1},W_{2},W_{3})[e^{-\mu
L_{12}}]=-\frac{2}{\gamma(Q-\beta_{1})(\beta_{2}-Q)(Q-\beta_{3})}\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})}\Gamma(\frac{\bar{\beta}-2Q}{\gamma})$
(2.30)

which further implies (2.25). For the case when both $W_{1}$ and $W_{2}$ are
smaller than $\frac{\gamma^{2}}{2}$, we can start from independent samples of
$\textup{QT}(\gamma^{2}-W_{1},\gamma^{2}-W_{2},W_{3})$,
$\mathcal{M}_{2}^{\textup{disk}}(W_{1})$ and
$\mathcal{M}_{2}^{\textup{disk}}(W_{2})$. We omit the details. ∎

The above result gives the law of a quantum triangle boundary arc length. In
fact, for some range of parameters, we can identify the joint law of boundary
arc lengths and quantum area. Suppose $\sum\beta_{i}>2Q$,
$\beta_{1},\beta_{2},\beta_{3}<Q$, and $\mu_{1},\mu_{2},\mu_{3}>0$, then
[ARSZ22, Theorem 1.1] gives an explicit description of

$H^{(\beta_{1},\beta_{2},\beta_{3})}_{(\mu_{1},\mu_{2},\mu_{3})}:=\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}[\exp(-\mu_{\phi}(\mathbb{H})-\mu_{1}\nu_{\phi}(-\infty,0)-\mu_{2}\nu_{\phi}(0,1)-\mu_{3}\nu_{\phi}(1,\infty))],$
(2.31)

where
$\phi\sim\mathrm{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}$
is the Liouville field.

###  2.4 Quantum triangles with fixed boundary lengths

We start by proving that, the quantum triangles we defined a.s. has positive
finite length.

######  Lemma 2.25.

For any weights $W_{1},W_{2},W_{3}>0$, the $\textup{QT}(W_{1},W_{2},W_{3})$
measure of quantum triangles with edges having zero or infinite quantum length
is 0.

###### Proof.

We begin with the thick quantum triangles. Sample $\phi$ from
$\textup{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}$
with $\beta_{i}=\gamma+\frac{2-W_{i}}{\gamma}<Q$, so our quantum triangle is
$(\mathbb{H},\phi,0,1,\infty)/{\sim_{\gamma}}$. Using the expression (2.22)
for $\phi$, it suffices to check that under $P_{\mathbb{H}}$,
$\nu_{\phi_{0}}([0,1])$ is a.s. finite. Since $\beta_{i}<Q$, we can pick $p>0$
such that
$p<\frac{4}{\gamma^{2}}\wedge\frac{2}{\gamma}(Q-\beta_{1})\wedge\frac{2}{\gamma}(Q-\beta_{2})$.
By [RZ20, Theorem 1.1],
$\mathbb{E}_{P_{\mathbb{H}}}\big{[}\nu_{\phi_{0}}([0,1])^{p}\big{]}<\infty$,
which justifies our claim. The remaining case follows by noticing that thin
triangles are produced by concatenating independent samples of thick triangles
with thin quantum disks, while both of them have finite length almost surely.
∎

We are now ready to disintegrate $\textup{QT}(W_{1},W_{2},W_{3})$ over its
boundary length. Basically this is simply conditioning on edge length. Recall
that for any two-pointed disks, by [AHS20, Section 2.6], one can construct the
family of measures
$\\{\mathcal{M}_{2}^{\text{disk}}(W;\ell_{1}):\ell_{1}>0\\}$ and
$\\{\mathcal{M}_{2}^{\text{disk}}(W;\ell_{1},\ell_{2}):\ell_{1},\ell_{2}>0\\}$
for such that

$\mathcal{M}_{2}^{\text{disk}}(W)=\int_{0}^{\infty}\int_{0}^{\infty}\mathcal{M}_{2}^{\text{disk}}(W;\ell_{1},\ell_{2})d\ell_{1}d\ell_{2};\
\mathcal{M}_{2}^{\text{disk}}(W,\ell_{1})=\int_{0}^{\infty}\mathcal{M}_{2}^{\text{disk}}(W;\ell_{1},\ell_{2})d\ell_{2}.$
(2.32)

Each sample from $\mathcal{M}_{2}^{\text{disk}}(W;\ell_{1})$ has left (or
right) boundary length $\ell_{1}$, and each sample from
$\mathcal{M}_{2}^{\text{disk}}(W;\ell_{1},\ell_{2})$ has boundary lengths
$\ell_{1}$ and $\ell_{2}$. And the same disintegration can be applied for
$\mathcal{M}_{2,\cdot}(W;\alpha)$ over the length of unmarked boundary
[AHS21].

We formally state the definition below and again start with thick triangles.

######  Definition 2.26.

Suppose $W_{1},W_{2},W_{3}>\frac{\gamma^{2}}{2}$. Let
$\beta_{i}=\gamma+\frac{2-W_{i}}{\gamma}$ and
$\bar{\beta}=\beta_{1}+\beta_{2}+\beta_{3}$. Sample $h$ from $P_{\mathbb{H}}$
and set

$\tilde{h}(z)=h(z)+(\bar{\beta}-2Q)\log|z|_{+}-\beta_{1}\log|z|-\beta_{2}\log|z-1|.$

(i.e., The Liouville field
$\textup{LF}_{\mathbb{H}}^{(\beta_{1},0),(\beta_{2},1),(\beta_{3},\infty)}$
but without the constant $\mathbf{c}$.) Fix $\ell>0$. Let
$L_{12}=\nu_{\tilde{h}}([0,1])$ and we define the measure
$\textup{QT}(W_{1},W_{2},W_{3};\ell)$, the quantum triangles of weight
$W_{1},W_{2},W_{3}$ with _left boundary length_ $\ell$, to be the law of
$\tilde{h}+\frac{2}{\gamma}\log\frac{\ell}{L_{12}}$ under the reweighted
measure
$\frac{2}{\gamma(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}\frac{\ell^{\frac{1}{\gamma}(\bar{\beta}-2Q)-1}}{L_{12}^{\frac{1}{\gamma}(\bar{\beta}-2Q)}}P_{\mathbb{H}}(dh)$.

The above definition can be repeated for right or bottom boundary length.
(i.e., $L_{12}=\nu_{\tilde{h}}([0,1])$ replaced by
$L_{13}=\nu_{\tilde{h}}((-\infty,0])$ or
$L_{23}=\nu_{\tilde{h}}([1,+\infty))$.) The following lemma justifies our
disintegration.

######  Lemma 2.27.

In the setting of Definition 2.26, samples from
$\textup{QT}(W_{1},W_{2},W_{3};\ell)$ has left boundary length $\ell$, and we
have

$\textup{QT}(W_{1},W_{2},W_{3})=\int_{0}^{\infty}\textup{QT}(W_{1},W_{2},W_{3};\ell)d\ell.$
(2.33)

Furthermore, if $(\beta_{i})$ satisfies the Seiberg bounds (2.19), then

$\big{|}\textup{QT}(W_{1},W_{2},W_{3};\ell)\big{|}=\frac{2}{\gamma(Q-\beta_{1})(Q-\beta_{2})(Q-\beta_{3})}{\bar{H}_{(0,1,0)}^{(\beta_{1},\beta_{2},\beta_{3})}}\ell^{\frac{\bar{\beta}-2Q}{\gamma}-1}.$

###### Proof.

The proof is almost identical to that of [AHS21, Lemma 4.2] but we include it
here for completeness. The first claim is trivial as
$\nu_{\tilde{h}+\frac{2}{\gamma}\log\frac{\ell}{L_{12}}}([0,1])=\frac{\ell}{L_{12}}\nu_{\tilde{h}}([0,1])=\ell$.

Now for any nonnegative measurable function $F$ on $H^{-1}(\mathbb{H})$ we
have

$\int_{0}^{\infty}\int
F(\tilde{h}+\frac{2}{\gamma}\log\frac{\ell}{L_{12}})\frac{2}{\gamma}\frac{\ell^{\frac{1}{\gamma}(\bar{\beta}-2Q)-1}}{L_{12}^{\frac{1}{\gamma}(\bar{\beta}-2Q)}}P_{\mathbb{H}}(dh)d\ell=\int_{\mathbb{R}}\int
F(\tilde{h}+c)e^{(\frac{1}{2}\bar{\beta}-Q)c}P_{\mathbb{H}}(dh)dc$ (2.34)

using Fubini’s theorem and the change of variables
$c=\frac{2}{\gamma}\log\frac{\ell}{L_{12}}$. Therefore by definition, (2.33)
holds. The last statement follows directly from Proposition 2.23. ∎

Indeed if $W_{3}<\frac{\gamma^{2}}{2}$, the same disintegration applies by
starting from $\textup{QT}(W_{1},W_{2},\gamma^{2}-W_{3};\ell)$ and then
concatenating an independent weight $W_{3}$ disk (which does not affect the
left boundary length). If $W_{1}<\frac{\gamma^{2}}{2}$ and
$W_{2}>\frac{\gamma^{2}}{2}$, we can still define our disintegration over left
boundary length via

$\textup{QT}(W_{1},W_{2},W_{3};\ell)=(1-\frac{2W_{1}}{\gamma^{2}})\int_{0}^{\ell}\mathcal{M}_{2}^{\text{disk}}(W_{1};\ell-x)\times\textup{QT}(\gamma^{2}-W_{1},W_{2},W_{3};x)dx.$
(2.35)

Similarly, if $W_{1}<\frac{\gamma^{2}}{2}$ and $W_{2}<\frac{\gamma^{2}}{2}$,
we can also define

$\begin{split}&\textup{QT}(W_{1},W_{2},W_{3};\ell)=(1-\frac{2W_{1}}{\gamma^{2}})(1-\frac{2W_{2}}{\gamma^{2}})\\\
&\int_{0}^{\ell}\int_{0}^{\ell-x}\mathcal{M}_{2}^{\text{disk}}(W_{1};y)\times\textup{QT}(\gamma^{2}-W_{1},\gamma^{2}-W_{2},W_{3};x)\times\mathcal{M}_{2}^{\text{disk}}(W_{2};\ell-
x-y)dydx.\end{split}$ (2.36)

One can directly verify that (2.33) holds for our definition of
$\textup{QT}(W_{1},W_{2},W_{3};\ell)$ via (2.35) and (2.36), and each sample
from $\textup{QT}(W_{1},W_{2},W_{3};\ell)$ has left boundary length $\ell$.

We have defined disintegration (2.33) over a single boundary arc length. This
can naturally be extended to multiple edges, that is,

$\textup{QT}(W_{1},W_{2},W_{3})=\iiint_{\mathbb{R}_{+}^{3}}\mathrm{QT}(W_{1},W_{2},W_{3};\ell_{1},\ell_{2},\ell_{3})d\ell_{1}d\ell_{2}d\ell_{3}.$
(2.37)

See [AHS20, Section 2.6] for more details.

###  2.5 Vertices with weight $\frac{\gamma^{2}}{2}$

In this section we define quantum triangles where one or more vertices have
weight $\frac{\gamma^{2}}{2}$. Plugging in the relation
$\beta=Q+\frac{\gamma}{2}-\frac{W}{\gamma}$ gives $\beta=Q$, but the Liouville
field with boundary insertion $Q$ a.s. has infinite boundary length near the
insertion, so this does not give the correct definition. The correct
definition is obtained from the $\beta\uparrow Q$ limit of the previously
defined Liouville field, which we call the Liouville field with insertion of
size $\beta=Q^{-}$.

We first define an infinite measure $M^{Q^{-}}$ as follows. For $a>0$, let
$B_{t}$ be variance 2 Brownian motion run until the first time $\tau_{a}$ it
hits $a$, and independently let $B^{\prime}_{s}$ be variance 2 Brownian motion
conditioned on the event $\\{B^{\prime}_{s}\leq 0\text{ for all }s\geq 0\\}$.
Namely, $-\frac{1}{\sqrt{2}}B_{s}^{\prime}$ is a 3D Bessel process starting
from 0. Define $X_{t}=B_{t}$ for $t\leq\tau_{a}$ and
$X_{t}=a+B^{\prime}_{t-\tau_{a}}$ for $t>\tau_{a}$. Let $P_{a}$ be the law of
$X_{t}$. Slightly abusing notation, we sample $(X_{t},\mathbf{a})$ from
$P_{a}1_{a>0}da$ and let $M^{Q^{-}}$ be the marginal law of $X_{t}$ under this
infinite measure.

We will define the Liouville field with one or more insertions of size $Q^{-}$
via $M^{Q-}$. We will put insertions at the boundary points
$(+\infty,-\infty,1)$ of $\mathcal{S}$; we choose the third boundary point 1
rather than 0 to avoid interfering with the GFF normalization (mean zero on
$\\{0\\}\times[0,\pi]$). Recall that $H(\mathcal{S})$ is the closure of the
space of smooth functions on $\mathcal{S}$ of finite Dirichlet energy with
respect to the Dirichlet inner product. Let $H_{1}\subset H(\mathcal{S})$ be
the subspace of functions which are zero on $(-\infty,10]\times[0,\pi]$ and
constant on each segment $\\{t\\}\times[0,\pi]$ for $t\geq 10$. Let $H_{2}$ be
the subspace of functions which are zero on $[-10,\infty)\times[0,\pi]$ and
constant on each segment $\\{t\\}\times[0,\pi]$ for $t\leq-10$. Let $H_{3}$ be
the subspace of functions which are zero on $\\{z\in\mathcal{S}\>:\>|z-1|\geq
1\\}$ and constant on each semicircle $\\{z\in\mathcal{S}\>:\>|z-1|=e^{-t}\\}$
for $t\geq 0$. Let $H_{0}$ be the orthogonal complement of $H_{1}\oplus
H_{2}\oplus H_{3}$. Functions in $H_{0}$ are harmonic on
$(10,\infty)\times[0,\pi]$ (with Neumann boundary conditions on
$\\{10,\infty\\}\times[0,\pi]$) and have the same average value on
$\\{t\\}\times[0,\pi]$ for all $t\geq 10$. Functions in $H_{0}$ likewise have
similar behavior in $(-\infty,-10)\times[0,\pi]$ and
$\\{z\in\mathcal{S}:|z-1|<1\\}$.

Let $\mathcal{P}$ be the set of probability measures $\rho$ compactly
supported in $\\{z\in(-10,10)\times(0,\pi)\>:\>|z-1|>1\\}$ such that $\int
G_{\mathcal{S}}(z,w)\rho(dw)\rho(dz)<\infty$; such measures can be integrated
against a GFF. In particular $\mathcal{P}$ contains the uniform probability
measure on $\\{0\\}\times[0,\pi]$. Let $P_{\rho}$ be the law of the GFF $h$ on
$\mathcal{S}$ normalized so $(h,\rho)=0$. Using the decomposition
$H(\mathcal{S})=H_{0}\oplus H_{1}\oplus H_{2}\oplus H_{3}$ we can decompose a
GFF $h\sim P_{\rho}$ as

$h=g_{0}+g_{1}+g_{2}+g_{3}$ (2.38)

where the $g_{i}$ are independent and correspond to projections to $H_{i}$.

Let $\rho_{1}$, $\rho_{2}$ and $\rho_{3}$ be the uniform probability measures
on $\\{10\\}\times[0,\pi]$, $\\{-10\\}\times[0,\pi]$ and
$\\{z\in\mathcal{S}\>:\>|z|=1\\}$ respectively. For real
$\beta_{1},\beta_{2},\beta_{3}$ define the _non-probability_ measure

$P^{(\beta_{i})_{i}}_{\rho}(dh)=\varepsilon_{0}^{\frac{1}{4}((\beta_{1}-Q)^{2}+(\beta_{2}-Q)^{2})}e^{\frac{\beta_{1}-Q}{2}(h,\rho_{1})+\frac{\beta_{2}-Q}{2}(h,\rho_{2})+\frac{\beta_{3}}{2}(h,\rho_{3})}P_{\rho}(dh),\qquad\varepsilon_{0}:=e^{-10}.$

For $\beta\in\mathbb{R}$ let $M^{\beta}$ be the law of Brownian motion with
variance 2 and drift $-(Q-\beta)$; in particular $|M^{\beta}|=1$. We now
extend the definition of the Liouville field to allow insertions of size
$Q^{-}$.

######  Definition 2.28.

Suppose $\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}\cup\\{Q^{-}\\}$ and
$\rho\in\mathcal{P}$. Let $\hat{\beta}_{i}=Q$ if $\beta_{i}=Q^{-}$, and let
$\hat{\beta}_{i}=\beta_{i}$ otherwise. Let
$s=\frac{1}{2}(\sum\hat{\beta}_{i})-Q$. Sample
$(h,\mathbf{c},X_{t}^{1},X_{t}^{2},X_{t}^{3})\sim
P_{\rho}^{(\hat{\beta}_{i})_{i}}\times[e^{sc}\,dc]\times M^{\beta_{1}}\times
M^{\beta_{2}}\times M^{\beta_{3}}$. Decompose $h=g_{0}+g_{1}+g_{2}+g_{3}$ as
in (2.38). Let $\hat{g}_{1}$ be the function which is zero on
$(-\infty,10)\times[0,\pi]$ and equals $X_{t}^{1}$ on each segment
$\\{t+10\\}\times[0,\pi]$ for $t\geq 0$. Let $\hat{g}_{2}$ be the function
which is zero on $(-10,\infty)\times[0,\pi]$ and equals $X_{t}^{2}$ on each
segment $\\{-t-10\\}\times[0,\pi]$ for $t\geq 0$. Let $\hat{g}_{3}$ be the
function which is zero on $\\{z\in\mathcal{S}:|z|\geq 1\\}$ and equals
$X_{t}^{3}+Qt$ on each semicircle $\\{z\in\mathcal{S}\>:\>|z|=e^{-t}\\}$ for
$t\geq 0$. Let $\phi=g_{0}+\hat{g}_{1}+\hat{g}_{2}+\hat{g}_{3}+\mathbf{c}$. We
denote the law of $\phi$ by
$\mathrm{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},1)}$.

######  Lemma 2.29.

Definition 2.28 does not depend on the choice of $\rho$. Moreover, if
$\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$, then the definition agrees with
Definition 2.8.

###### Proof.

We first check that if $\rho$ is the uniform probability measure on
$\\{0\\}\times[0,\pi]$, then Definition 2.28 agrees with Definition 2.8. For
the special case $(\beta_{1},\beta_{2},\beta_{3})=(Q,Q,0)$, we have
$P_{\rho}^{(\beta_{i})_{i}}=P_{\mathcal{S}}$, and for $h\sim P_{\mathcal{S}}$
the field average processes described by $g_{1},g_{2},g_{3}$ from (2.38) each
have the law of variance 2 Brownian motion (see e.g. [DMS21, Section 4.1.6]),
so the claim is immediate. This gives the decomposition identifying
$\mathrm{LF}_{\mathcal{S}}^{(Q,+\infty),(Q,-\infty)}$ with
$P_{\mathcal{S}}\times dc\times M^{Q}\times M^{Q}\times M^{0}$. Now we explain
how to extend to the case $\beta_{1}\in\mathbb{R}$, $\beta_{2}=Q$ and
$\beta_{3}=0$. Parametrizing in $\mathcal{S}$ rather than $\mathbb{H}$, an
immediate consequence of Lemma 2.6 is

$\displaystyle\mathrm{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(Q,-\infty)}(d\phi)$
$\displaystyle=\lim_{\varepsilon\to
0}\varepsilon^{\frac{1}{4}(\beta_{1}-Q)^{2}}e^{\frac{1}{2}(\beta_{1}-Q)(\phi,\theta_{\varepsilon})}\mathrm{LF}_{\mathcal{S}}^{(Q,+\infty),(Q,-\infty)}(d\phi),$

where $\theta_{\varepsilon}$ is the uniform probability measure on
$\\{-\log\varepsilon\\}\times[0,\pi]$. Identifying
$\mathrm{LF}_{\mathcal{S}}^{(Q,+\infty),(Q,-\infty)}$ with
$P_{\mathcal{S}}\times dc\times M^{Q}\times M^{Q}\times M^{0}$, this limit can
be written as

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon^{\frac{1}{4}(\beta_{1}-Q)^{2}}e^{\frac{1}{2}(\beta_{1}-Q)((h,\rho_{1})+X^{1}_{-\log(\varepsilon/\varepsilon_{0})}+c)}P_{\mathcal{S}}(dh)dcM^{Q}(dX^{1})M^{Q}(dX^{2})M^{0}(dX^{3})$
$\displaystyle=\lim_{\varepsilon\to
0}(\varepsilon/\varepsilon_{0})^{\frac{1}{4}(\beta_{1}-Q)^{2}}e^{\frac{1}{2}(\beta_{1}-Q)X^{1}_{-\log(\varepsilon/\varepsilon_{0})}}P^{(\beta_{i})_{i}}_{\rho}(dh)e^{sc}dcM^{Q}(dX^{1})M^{Q}(dX^{2})M^{0}(dX^{3})$
$\displaystyle=P^{(\beta_{i})_{i}}_{\rho}(dh)e^{sc}dcM^{\beta_{1}}(dX^{1})M^{Q}(dX^{2})M^{0}(dX^{3}).$

To obtain the last equality above, by Girsanov’s theorem the law of
$X^{1}_{t}$ under the probability measure
$(\varepsilon/\varepsilon_{0})^{\frac{1}{4}(\beta_{1}-Q)^{2}}e^{\frac{1}{2}(\beta_{1}-Q)X^{1}_{-\log(\varepsilon/\varepsilon_{0})}}M^{Q}(dX^{1})$
is Brownian motion with variance $2$, with drift $-(Q-\beta_{1})$ until time
$-\log(\varepsilon/\varepsilon_{0})$ and zero drift afterwards; this converges
as $\varepsilon\to 0$ to $M^{\beta_{1}}$ in the topology of uniform
convergence on finite intervals. Thus
$\mathrm{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(Q,-\infty)}$ can be
identified with $P_{\rho}^{(\beta_{i})_{i}}\times e^{sc}dc\times
M^{\beta_{1}}\times M^{Q}\times M^{0}$ as desired. Here we discussed
$\beta_{1}\in\mathbb{R},\beta_{2}=Q,\beta_{3}=0$ to lighten notation, but the
same argument applies for $\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$. We
conclude that if $\rho$ is the uniform probability measure on
$\\{0\\}\times[0,\pi]$ then Definition 2.28 agrees with Definition 2.8.

Now let $\rho$ be arbitrary. It remains to verify that Definition 2.28 does
not depend on the choice of $\rho$. If $h\sim P_{\rho}$ then the law of $h$
viewed as a distribution modulo additive constant does not depend on $\rho$,
so by the translation invariance of $dc$, for $(h,\mathbf{c})$ sampled from
$P_{\rho}\times dc$ the law of $h+\mathbf{c}$ does not depend on $\rho$.
Consequently, if we sample $(h,\mathbf{c})$ from

$P_{\rho}^{(\beta_{i})_{i}}\times[e^{sc}dc]=e^{\frac{\beta_{1}-Q}{2}(h+c,\rho_{1})+\frac{\beta_{2}-Q}{2}(h+c,\rho_{2})+\frac{\beta_{3}}{2}(h+c,\rho_{3})}P_{\rho}(dh)dc,$

the law of $h+\mathbf{c}$ does not depend on $\rho$; since $\phi$ is a
function of $h+\mathbf{c}$ and randomness independent of $(h,\mathbf{c})$, the
claim follows. ∎

We will prove that the Liouville field with one or more $Q^{-}$ insertions
arises as a $\beta\uparrow Q$ limit. The key is the Brownian motion
description of $\frac{1}{Q-\beta}M^{\beta}$ and its convergence to $M^{Q^{-}}$
under a suitable topology.

######  Lemma 2.30.

Let $\beta<Q$. For $X_{t}\sim\frac{1}{Q-\beta}M^{\beta}$, the law of
$\mathbf{a}=\sup_{t}X_{t}$ is $1_{a>0}e^{-(Q-\beta)a}\,da$. Moreover,
conditioned on $\mathbf{a}$, the conditional law of $X_{t}$ is that of
variance 2 Brownian motion with upward drift $(Q-\beta)t$ run until it hits
$\mathbf{a}$, then variance 2 Brownian motion with downward drift
$-(Q-\beta)t$ started at $\mathbf{a}$ and conditioned to stay below
$\mathbf{a}$.

###### Proof.

The law of $\mathbf{a}$ follows from a standard Brownian motion computation,
and the conditional law of $X_{t}$ given $\mathbf{a}$ follows from the
Williams decomposition [Wil74]. ∎

######  Lemma 2.31.

For $A>0$ let $E^{\prime}_{A}$ be the event that a process $X_{t}$ satisfies
$\sup_{t}X_{t}<A$. Then we have the weak limit $\lim_{\beta\uparrow
Q}\frac{1}{Q-\beta}M^{\beta}|_{E^{\prime}_{A}}=M^{Q^{-}}|_{E^{\prime}_{A}}$,
where the topology on function space is uniform convergence on compact sets.

###### Proof.

Comparing the description of $\frac{1}{Q-\beta}M^{\beta}$ in Lemma 2.30 to the
definition of $M^{Q^{-}}$, the result follows. ∎

######  Proposition 2.32.

Let $\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}\cup\\{Q^{-}\\}$ and
$\rho\in\mathcal{P}$. For $\beta_{i}\in\mathbb{R}\backslash\\{Q\\}$ let
$(\beta_{i}^{n})_{n\geq 1}$ be a sequence with limit $\beta_{i}$. For
$\beta_{i}=Q$ let $(\beta_{i}^{n})_{n\geq 1}$ be a nonincreasing sequence with
limit $Q$. For $\beta_{i}=Q^{-}$ let $(\beta_{i}^{n})_{n\geq 1}$ be an
increasing sequence with $\lim_{n\to\infty}\beta_{i}^{n}=Q$.

Let $x_{1}=+\infty,x_{2}=-\infty,x_{3}=1$. Let
$\mathcal{I}\subset\\{(-\infty,1),(1,+\infty),\mathbb{R}\times\\{\pi\\}\\}$,
let $K>0$ and let $E_{K}=\\{\phi\>:\>|(\phi,\rho)|<K\text{ and
}\nu_{\phi}(I)\leq K\text{ for }I\in\mathcal{I}\\}$. Suppose that for any $i$
such that $\beta_{i}=Q^{-}$ (resp. $\beta_{i}\geq Q$) the point $x_{i}$ is an
endpoint of some (resp. no) interval in $\mathcal{I}$. Then, as a limit in the
space of finite measures,

$\lim_{n\to\infty}\left(\prod_{i\>:\>\beta_{i}=Q^{-}}\frac{1}{Q-\beta_{i}^{n}}\right)\mathrm{LF}_{\mathcal{S}}^{(\beta_{i}^{n},x_{i})_{i}}|_{E_{K}}=\mathrm{LF}_{\mathcal{S}}^{(\beta_{i},x_{i})_{i}}|_{E_{K}}.$

Here, we equip the space of distributions (on which
$\mathrm{LF}_{\mathcal{S}}^{(\beta_{i}^{n},x_{i})_{i}}$ is a measure) with the
weak-$*$ topology from testing against smooth compactly supported functions.

###### Proof.

For a field $\phi$ defined in Definition 2.28 and for $A>0$, let
$E^{\prime}_{K,A}$ be the event that $|(\phi,\rho)|<K$ and
$\sup_{t}X_{t}^{i}<A$ for all $i$. By Lemma 2.31 and the definition of
Liouville field from Definition 2.28, Proposition 2.32 holds when $E_{K}$ is
replaced by $E_{K,A}^{\prime}$.

By the above claim, since the conditional probabilities
$\mathrm{LF}_{\mathcal{S}}^{(\beta_{i}^{n},x_{i})_{i}}[E_{K}\mid
E^{\prime}_{K,A}]$ and
$\mathrm{LF}_{\mathcal{S}}^{(\beta_{i},x_{i})_{i}}[E_{K}\mid
E^{\prime}_{K,A}]$ are uniformly bounded from below uniformly for all $n$,
Proposition 2.32 holds when $E_{K}$ is replaced by $E_{K}\cap
E^{\prime}_{K,A}$. To bootstrap this to the desired statement, it suffices to
show

$\lim_{A\to\infty}\left(\prod_{i:\beta_{i}=Q^{-}}(Q-\beta_{i}^{n})^{-1}\right)\mathrm{LF}_{\mathcal{S}}^{(\beta_{i}^{n},x_{i})_{i}}[E_{K}\cap(E^{\prime}_{K,A})^{c}]=0\text{
uniformly in }n.$ (2.39)

To simplify notation we explain this for the case that
$\beta_{1},\beta_{2},\beta_{3}=Q^{-}$; the other cases are similarly shown.
Let $\rho$ be the uniform probability measure on $\\{0\\}\times[0,\pi]$. Let
$s^{n}=\frac{1}{2}\sum\beta_{i}^{n}-Q$. A sample $\phi$ from
$\left(\prod_{i:\beta_{i}=Q^{-}}(Q-\beta_{i}^{n})^{-1}\right)\mathrm{LF}_{\mathcal{S}}^{(\beta_{i}^{n},x_{i})_{i}}$
can be obtained by sampling

$(h,\mathbf{c},X_{t}^{1},X_{t}^{2},X_{t}^{3})\sim
P_{\rho}^{(\beta_{i}^{n})_{i}}\times[e^{s^{n}c}\,dc]\times(\frac{1}{Q-\beta_{1}^{n}}M^{\beta_{1}^{n}})\times(\frac{1}{Q-\beta_{2}^{n}}M^{\beta_{2}^{n}})\times(\frac{1}{Q-\beta_{3}^{n}}M^{\beta_{3}^{n}})$

and combining them to give $\phi$ as in Definition 2.28. Let
$\mathbf{a}_{i}=\sup_{t}X_{t}^{i}$ and let $F_{K,A}^{1}$ be the event that
$\mathbf{a}_{1}\geq\max(A,\mathbf{a}_{2},\mathbf{a}_{3})$ and
$|(\phi,\rho)|<K$. By Lemma 2.30, the law of
$(h,\mathbf{c},\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{a}_{3})$ restricted to
$F_{K,A}^{1}$ is

$P_{\rho}^{(\beta_{i}^{n})_{i}}\times[1_{|c|<K}e^{s^{n}c}dc]\times[1_{a_{1}>A}e^{-(Q-\beta_{1}^{n})a_{1}}\,da_{i}]\prod_{i=2}^{3}[1_{0<a_{i}<a_{1}}e^{-(Q-\beta_{i}^{n})a_{i}}\,da_{i}].$
(2.40)

Let $G$ be the average of $h$ on $\\{10\\}\times[0,\pi]$, so the maximum of
the field average on $\\{t\\}\times[0,\pi]$ for $t\geq 10$ is
$\mathbf{a}_{1}+G+\mathbf{c}$. When $\mathbf{a}_{1}+G+\mathbf{c}$ is large,
the LQG-length of any $I\in\mathcal{I}$ adjacent to $+\infty$ is likely large,
so $E_{K}$ likely does not occur. We quantify this via the existence of the
$-p$th GMC moment [RV10, Proposition 3.6] for any $p>0$, and Markov’s
inequality:

$\prod(Q-\beta_{i}^{n})^{-1}\mathrm{LF}_{\mathcal{S}}^{(\beta_{i}^{n},x_{i})_{i}}[E_{K}\cap
F_{K,A}^{1}]\lesssim\int_{-K}^{K}\int_{A}^{\infty}\mathbb{E}[(Ke^{-\frac{\gamma}{2}(a_{1}+G_{0}+c)})^{p}]e^{-(Q-\beta_{i}^{n})a_{1}}a_{1}^{2}\,da_{1}\,e^{s^{n}c}\,dc$

where the implicit constant depends on $p$ but not on $A$ or $n$, and the
expectation is taken with respect to the Gaussian $G_{0}$ defined as the
average of a GFF sampled from
$P_{\rho}^{(\beta_{i}^{n})_{i}}/|P_{\rho}^{(\beta_{i}^{n})_{i}}|$ on
$\\{10\\}\times[0,10]$. In the inequality, the term $a_{1}^{2}$ comes from the
integrals over $da_{2}$ and $da_{3}$ in (2.40). Since
$\int_{A}^{\infty}e^{-\frac{\gamma}{2}pa_{1}}a_{1}^{2}\,da_{1}\lesssim
A^{2}e^{-\frac{\gamma}{2}pA}$ and $G_{0}$ has mean and variance uniformly
bounded in $n$, the upper bound can be bounded above by a constant times
$K^{p+1}e^{-\frac{\gamma}{2}pK}e^{|s|K}A^{2}e^{-\frac{\gamma}{2}pA}$. Defining
$F_{K,A}^{i}$ for $i=2,3$, the above estimate also holds for these events, so
since $(E_{K,A}^{\prime})^{c}\subset\bigcup_{i}F_{K,A}^{i}$, we obtain

$\prod(Q-\beta_{i}^{n})^{-1}\mathrm{LF}_{\mathcal{S}}^{(\beta_{i}^{n},x_{i})_{i}}[E_{K}\cap(E_{K,A}^{\prime})^{c}]\lesssim
K^{p+1}e^{-\frac{\gamma}{2}pK}e^{|s|K}A^{2}e^{-\frac{\gamma}{2}pA}.$

This gives the desired uniform estimate (2.39). ∎

######  Definition 2.33.

Fix $W_{1},W_{2},W_{3}\geq\frac{\gamma^{2}}{2}$. For
$W_{i}>\frac{\gamma^{2}}{2}$ let $\beta_{i}=\gamma+\frac{2-W_{i}}{\gamma}<Q$,
and for $W_{i}=\frac{\gamma^{2}}{2}$ let $\beta_{i}=Q^{-}$. Sample $\phi$ from
$\prod_{i:\beta_{i}\neq
Q^{-}}(Q-\beta_{i})^{-1}\mathrm{LF}_{\mathcal{S}}^{(\beta_{1},+\infty),(\beta_{2},-\infty),(\beta_{3},1)}$.
Let $\textup{QT}(W_{1},W_{2},W_{3})$ be the law of
$(\mathcal{S},\phi,+\infty,-\infty,1)/{\sim_{\gamma}}$.
For general $W_{1},W_{2},W_{3}>0$ with one or more weights equal to
$\frac{\gamma^{2}}{2}$, define $\mathrm{QT}(W_{1},W_{2},W_{3})$ as in
Definition 2.18.

######  Remark 2.34.

A variant of the limiting statement in Proposition 2.32 for Liouville CFT on
the sphere was stated as a conjecture in [DKRV17, Remark 2.5]. We expect that
adapting our argument will lead to a proof.

##  3 Imaginary geometry and
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$

We briefly go over the GFF/SLE coupling in Imaginary Geometry [MS16a] in
Section 3.1. Then in Section 3.2 we state the _SLE resampling properties_
[MS16b, Section 4], which will frequently appear in our later proofs. Finally
we prove Theorem 1.5 in Section 3.3.

###  3.1 Background on $\mathrm{SLE}_{\kappa}(\underline{\rho})$ and imaginary
geometry

The SLEκ curves, as introduced in [Sch00], is a conformally invariant measure
on continuously growing compact hulls $K_{t}$ with the Loewner driving
function $W_{t}=\sqrt{\kappa}B_{t}$ (where $B_{t}$ is the standard Brownian
motion). When the background domain is the upper half plane, this can be
described by

$g_{t}(z)=z+\int_{0}^{t}\frac{2}{g_{s}(z)-W_{s}}ds,\ z\in\mathbb{H},$ (3.1)

and $g_{t}$ is the unique conformal transformation from $\mathbb{H}\backslash
K_{t}$ to $\mathbb{H}$ such that $\lim_{|z|\to\infty}|g_{t}(z)-z|=0$. SLEκ
curves also has a natural variant called SLE${}_{\kappa}(\underline{\rho})$,
which first appeared in [LSW03] and studied in [Dub05, MS16a]. Fix
$x^{k,L}<...<x^{1,L}\leq 0^{-}\leq 0^{+}\leq x^{1,R}<...<x^{\ell,R}$, which
are called force points, and set
$\underline{x}=(\underline{x}_{L},\underline{x}_{R})=(x^{1,L},...,x^{k,L};x^{1,R},...,x^{\ell,R})$.
For each for each force point $x^{i,q}$, $q\in\\{L,R\\}$ we assign a weight
$\rho^{i,q}\in\mathbb{R}$. Let $\underline{\rho}$ be the vector of weights.
The SLE${}_{\kappa}(\underline{\rho})$ process with force points
$\underline{x}$ is the measure on compact hulls $(K_{t})_{t\geq 0}$ growing
the same as ordinary SLEκ (i.e, satisfies (3.1)) except that the Loewner
driving function $(W_{t})_{t\geq 0}$ are now characterized by

$\begin{split}&W_{t}=\sqrt{\kappa}B_{t}+\sum_{q\in\\{L,R\\}}\sum_{i}\int_{0}^{t}\frac{\rho^{i,q}}{W_{s}-V_{s}^{i,q}}ds;\\\
&V_{t}^{i,q}=x^{i,q}+\int_{0}^{t}\frac{2}{V_{s}^{i,q}-W_{s}}ds,\
q\in\\{L,R\\}.\end{split}$ (3.2)

It has been shown in [MS16a] that SLE${}_{\kappa}(\underline{\rho})$ processes
a.s. exists, is unique and generates a continuous curve until the continuation
threshold, the first time $t$ such that $W_{t}=V_{t}^{j,q}$ with
$\sum_{i=1}^{j}\rho^{i,q}\leq-2$ for some $j$ and $q\in\\{L,R\\}$. Let
$f_{t}:=g_{t}-W_{t}$ be the centered Loewner flow.

Now we recall the notion of _the GFF flow lines_. Heuristically, given a GFF
$h$, $\eta(t)$ is a flow line of angle $\theta$ if

$\eta^{\prime}(t)=e^{i(\frac{h(\eta(t))}{\chi}+\theta)}\ \text{for}\ t>0,\
\text{where}\ \chi=\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2}.$ (3.3)

To be more precise, [MS16a, Theorem 1.1] introduces an exact coupling of a
Dirichlet GFF with an $\mathrm{SLE}_{\kappa}(\underline{\rho})$, which we
briefly recap as follows. Let $(K_{t})_{t\geq 0}$ be the hull at time $t$ of
the SLE${}_{\kappa}(\underline{\rho})$ process described by the Loewner flow
(3.1) with $(W_{t},V_{t}^{i,q})$ solving (3.2) with filtration
$\mathcal{F}_{t}$. Let $\mathfrak{h}_{t}^{0}$ be the harmonic function on
$\mathbb{H}$ with boundary values

$-\lambda(1+\sum_{i=0}^{j}\rho^{i,L})\ \ \text{on}\
[V_{t}^{j+1,L},V_{t}^{j,L})\ \ \text{and}\ \
\lambda(1+\sum_{i=0}^{j}\rho^{i,R})\ \text{on}\ \ [V_{t}^{j,R},V_{t}^{j+1,R})$

where $\lambda=\frac{\pi}{\sqrt{\kappa}}$, $\rho^{0,R}=\rho^{0,L}=0$,
$x^{0,L}=0^{-},x^{0,R}=0^{+},x^{k+1,L}=-\infty,x^{\ell+1,R}=+\infty$. Set
$\mathfrak{h}_{t}(z)=\mathfrak{h}_{t}^{0}(f_{t}(z))-\chi\arg
f_{t}^{\prime}(z)$. Let $\tilde{h}$ be a zero boundary GFF on $\mathbb{H}$ and
$h=\tilde{h}+\mathfrak{h}_{0}$. Then for any $\mathcal{F}_{t}$-stopping time
$\tau$ before the continuation threshold, $K_{\tau}$ is a local set for $h$
and the conditional law of $h|_{\mathbb{H}\backslash K_{\tau}}$ given
$\mathcal{F}_{\tau}$ is the same as the law of
$\mathfrak{h}_{\tau}+\tilde{h}\circ f_{\tau}$.

For $\kappa<4$, the SLE${}_{\kappa}(\underline{\rho})$ coupled with the GFF
$h$ as above is referred as the flow lines of $h$, and we say an
SLE${}_{\kappa}(\underline{\rho})$ curve is a flow line of angle $\theta$ if
it can be coupled with $h+\theta\chi$. Moreover, [MS16a, Theorem 1.2] shows
that these flow lines are a.s. determined by the GFF $h$, and we can
simultaneously consider flow lines starting from different boundary points.
Furthermore, by [MS16a, Theorem 1.5], the interaction of these flow lines
(i.e., crossing, merging, etc.) are completely determined by their angles. One
can also make sense of flow lines of GFF starting from interior points, and
see [MS17] for more details.

###  3.2 Coupling of two flow lines

One important consequence is that, as argued in [MS16a, Section 6], suppose
$\eta_{1}$ and $\eta_{2}$ are flow lines of $h$, then given $\eta_{1}$, the
conditional law of $\eta_{2}$ is the same as the law of the flow line (with
some angle) of the GFF in $\mathbb{H}\backslash\eta_{1}$ with the flow line
boundary conditions (one can go to [MS16a, Figure 1.10] for more explanation)
induced by $\eta_{1}$, and vice versa for the law of $\eta_{1}$ given
$\eta_{2}$. The _SLE resampling property_ states that these two conditional
laws actually uniquely characterize the joint law of $(\eta_{1},\eta_{2})$, at
least in some parameter ranges.

We summarize the imaginary geometry input we need for domains with three
marked points in Proposition 3.1 below. Suppose $\eta_{1}$ is a simple curve
in $\mathbb{H}$ from $0$ to $\infty$ which does not hit 1. We want to make
sense of the curve
$\eta_{2}\sim\mathrm{SLE}_{\kappa}(\rho_{-},\rho_{0};\rho_{+})$ in
$\mathbb{H}\backslash\eta_{1}$ in the region to the right of $\eta_{1}$ from 1
to $\infty$. The definition is clear if $\eta_{1}\cap(0,\infty)=\emptyset$.
Otherwise, let $p$ be the rightmost point of $\eta_{1}\cap[0,1)$ and $q$ the
leftmost point of $\eta_{1}\cap(1,+\infty)$ (with $q=\infty$ if $\eta_{1}$ is
disjoint from $(1,\infty)$). Let $D$ be the connected component of
$\mathbb{H}\backslash\eta_{1}$ with $1$ on its boundary, and sample an
$\mathrm{SLE}_{\kappa}(\rho_{-},\rho_{0};\rho_{+})$ curve in $D$ from $1$ to
$q$ with left force point at $p$. If $q=\infty$ then $\eta_{2}$ is this curve.
Otherwise, in each connected component of $\mathbb{H}\backslash\eta_{1}$ to
the right of $D$ we sample an independent
$\mathrm{SLE}(\rho_{0}+\rho_{-};\rho_{+})$ and let $\eta_{2}$ be the
concatenation of all the sampled curves. Similarly, if $\eta_{2}$ is a curve
in $\mathbb{H}$ from $1$ to $\infty$ which does not hit $0$, we can define
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ in
$\mathbb{H}\backslash\eta_{2}$ in the region to the left of $\eta_{2}$ from 0
to $\infty$.

######  Proposition 3.1.

Let $\theta_{1},\theta_{2},x_{1},x_{2},x_{3}\in\mathbb{R}$ such that
$\theta_{1}>\theta_{2}$ and

$x_{1}<\lambda-\theta_{1}\chi;\ x_{3}>-\lambda-\theta_{2}\chi;\
-\lambda-\theta_{1}\chi<x_{2}<\lambda-\theta_{2}\chi.$ (3.4)

The following two laws on pairs of curves $(\eta_{1},\eta_{2})$ agree:

  * •

Sample $\eta_{1}$ in $\mathbb{H}$ from $0$ to $\infty$ as
$\mathrm{SLE}_{\kappa}(-\frac{x_{1}+\theta_{1}\chi}{\lambda}-1;\frac{x_{2}+\theta_{1}\chi}{\lambda}-1,\frac{x_{3}-x_{2}}{\lambda})$.
In $\mathbb{H}\backslash\eta_{1}$ in the region to the right of $\eta_{1}$,
sample $\eta_{2}$ from $1$ to $\infty$ as
$\mathrm{SLE}_{\kappa}(-\frac{x_{2}+\theta_{2}\chi}{\lambda}-1,\frac{x_{2}+\theta_{1}\chi}{\lambda}-1;\frac{x_{3}+\theta_{2}\chi}{\lambda}-1)$.

  * •

Sample $\eta_{2}$ in $\mathbb{H}$ from $1$ to $\infty$ as
$\mathrm{SLE}_{\kappa}(-\frac{x_{2}+\theta_{2}\chi}{\lambda}-1,\frac{x_{2}-x_{1}}{\lambda};\frac{x_{3}+\theta_{2}\chi}{\lambda}-1)$.
In $\mathbb{H}\backslash\eta_{2}$ in the region to the left of $\eta_{2}$,
sample $\eta_{1}$ from $0$ to $\infty$ as
$\mathrm{SLE}_{\kappa}(-\frac{x_{1}+\theta_{1}\chi}{\lambda}-1;\frac{x_{2}+\theta_{1}\chi}{\lambda}-1,\frac{-\theta_{2}\chi-
x_{2}}{\lambda}-1)$.

Furthermore, for $(\theta_{1}-\theta_{2})\chi\geq\frac{\sqrt{\kappa}\pi}{2}$,
this law on $(\eta_{1},\eta_{2})$ is characterized by the following:

  * •

Almost surely $\eta_{1}\cap[1,\infty)=\emptyset$ and
$\eta_{2}\cap(-\infty,0]=\emptyset$. Moreover, the conditional law of
$\eta_{1}$ given $\eta_{2}$ is
$\mathrm{SLE}_{\kappa}(-\frac{x_{1}+\theta_{1}\chi}{\lambda}-1;\frac{x_{2}+\theta_{1}\chi}{\lambda}-1,\frac{-\theta_{2}\chi-
x_{2}}{\lambda}-1)$, and the conditional law of $\eta_{2}$ given $\eta_{1}$ is
$\mathrm{SLE}_{\kappa}(-\frac{x_{2}+\theta_{2}\chi}{\lambda}-1,\frac{x_{2}+\theta_{1}\chi}{\lambda}-1;\frac{x_{3}+\theta_{2}\chi}{\lambda}-1)$.

See Figure 4 for an illustration of the setting. The first statement is clear
from the flow line conditioning in [MS16a, Section 6]. The second statement is
the resampling property of flow lines in [MS16b]. We remark that the original
statement [MS16b, Theorem 4.1] is for curves with same starting and ending
points; the proof is based on a Markov chain mixing argument and the first
step is to apply the SLE duality argument to separate the initial and terminal
points of $\eta_{1}$ and $\eta_{2}$. Therefore the same argument readily
applies (and is simpler) in the case described in Proposition 3.1.

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Figure 4: Left: Suppose $h$ is a Dirichlet GFF with piecewise constant
boundary condition
$x_{1}\textbf{1}_{(-\infty,0)}(x)+x_{2}\textbf{1}_{[0,1)}(x)+x_{3}\textbf{1}_{[1,+\infty)}(x)$.
Let $\eta_{1}$ (resp. $\eta_{2}$) be the flow line of $h$ starting from 0
(resp. 1) of angle $\theta_{1}$ (resp. $\theta_{2}$). Then the marginal law of
$\eta_{1}$ is
SLE${}_{\kappa}(-\frac{x_{1}+\theta_{1}\chi}{\lambda}-1;\frac{x_{2}+\theta_{1}\chi}{\lambda}-1,\frac{x_{3}-x_{2}}{\lambda})$
with force points at $(0^{-};0^{+},1)$, and the marginal law of $\eta_{2}$ is
SLE${}_{\kappa}(-\frac{x_{2}+\theta_{2}\chi}{\lambda}-1,\frac{x_{2}-x_{1}}{\lambda};\frac{x_{3}+\theta_{2}\chi}{\lambda}-1)$
force points at $(1^{-},0;1^{+})$. Suppose $\theta_{1}\geq\theta_{2}$. By
[MS16a, Section 6], one can also read the conditional law of $\eta_{1}$ given
$\eta_{2}$, which is
SLE${}_{\kappa}(-\frac{x_{1}+\theta_{1}\chi}{\lambda}-1;\frac{x_{2}+\theta_{1}\chi}{\lambda}-1,\frac{-\theta_{2}\chi-
x_{2}}{\lambda}-1)$ in the left component of $\mathbb{H}\backslash\eta_{2}$,
and similarly conditional law of $\eta_{2}$ given $\eta_{1}$ is
SLE${}_{\kappa}(-\frac{x_{2}+\theta_{2}\chi}{\lambda}-1,\frac{x_{2}+\theta_{1}\chi}{\lambda}-1;\frac{x_{3}+\theta_{2}\chi}{\lambda}-1)$
in the right component of $\mathbb{H}\backslash\eta_{1}$. If
$\theta_{1}<\theta_{2}$, then given $\eta_{2}$, the segment of $\eta_{1}$
before crossing $\eta_{2}$ is the same as non-crossing case and $\eta_{1}$ can
be continued after crossing. Finally, by the resampling property, the
conditional laws of $\eta_{1}|\eta_{2}$ and $\eta_{2}|\eta_{1}$ uniquely
characterize the joint distribution of $(\eta_{1},\eta_{2})$, as constructed
using Imaginary Geometry. Right: One can similarly consider the flow lines
from 0 and read off the marginal and conditional laws as in Lemma 3.3.

###  3.3 Reversibility of SLE${}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$

In this section, as an application of the Imaginary Geometry flow lines and
the curve resampling properties, we extend the result on reversibility of
SLE${}_{\kappa}(0;\rho_{+},\rho_{1})$ in [Zha22] to
SLE${}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ curves.

To begin with, let us recall the notion of _SLE weighted by conformal
derivative_. Given $\rho_{-},\rho_{+}>-2$, $\rho_{1}>-2-\rho_{+}$ (which
implies that the continuation threshold is never hit) and
$\alpha\in\mathbb{R}$, we define the measure
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)$ on
curves $\eta$ from 0 to $\infty$ on $\mathbb{H}$ as follows. Let $D_{\eta}$ be
the component of $\mathbb{H}\backslash\eta$ containing $1$, and $\psi_{\eta}$
the unique conformal map from $D_{\eta}$ to $\mathbb{H}$ fixing 1 and sending
the first (resp. last) point on $\partial D_{\eta}$ hit by $\eta$ to 0 (resp.
$\infty$). Then our
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)$ on
$\mathbb{H}$ is defined by

$\frac{d\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)}{d{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})}(\eta)=|\psi_{\eta}(1)^{\prime}|^{\alpha}$
(3.5)

where the force points of
${\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$ is $0^{-},0^{+},1$. This
definition can be extended to other domains via conformal transforms, while by
symmetry, we can also define the version
$d\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-},\rho_{-1};\rho_{+};\alpha)$ with
$1$ replaced by $-1$ and force points $-1,0^{-},0^{+}$ similarly. Also let
$\mathcal{R}(\eta)$ be the time reversal of $\eta$. With these notations, we
state the result in [Zha22] as follows.

######  Theorem 3.2.

Suppose $\eta$ an $\mathrm{SLE}_{\kappa}(0;\rho_{+},\rho_{1})$ curve in
$\mathbb{H}$ from $0$ to $\infty$ with force located at $0^{+}$ and 1, and
$\rho_{+}>-2,\rho_{+}+\rho_{1}>-2$. Let $\mathcal{L}$ be the law of the time
reversal $\mathcal{R}(\eta)$ under the conformal mapping
$z\mapsto-\frac{1}{z}$. Then $\mathcal{L}$ is a constant multiple of the
measure
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{+}+\rho_{1},-\rho_{1};0;\frac{\rho_{1}(4-\kappa)}{2\kappa})$.

We note that the theorem above is implicitly shown in Theorem 1.1 and Section
3.2 of [Zha22] via the construction of the reversed curve. The statement is
for general $\mathrm{SLE}_{\kappa}(\underline{\rho})$ curves with all force
points lying on the same side of 0, while in this paper we only work on the
$0^{+},1$ force point case for simplicity. To prove Theorem 1.5, we begin with
the following variant of Proposition 3.1. Again suppose we want to sample
$\eta_{2}\sim\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)$
to the right of $\eta_{1}$ in $\mathbb{H}\backslash\eta_{1}$ when $\eta_{1}$
is hitting $(0,\infty)$, let $p,q$ be the left and right most point on
$\mathbb{R}$ on the boundary of the connected component of $D_{\eta_{1}}$. In
each component of $\mathbb{H}\backslash\eta_{1}$ whose boundary contains a
segment of $(0,p)$ (resp. $(q,\infty)$), we sample an independent
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+})$ (resp.
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+}+\rho_{1})$), and in $D_{\eta_{1}}$ we
sample an
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{-};\rho_{+},\rho_{1};\alpha)$ curve
from $p$ to $q$. Then $\eta_{2}$ is the concatenation of these curves.

######  Lemma 3.3.

Let $x_{1},x_{2},x_{3},\alpha,\theta_{1},\theta_{2}\in\mathbb{R}$ such that

$\theta_{1}>\theta_{2};\ x_{1}<\lambda-\theta_{1}\chi;\
x_{2},x_{3}>-\lambda-\theta_{2}\chi.$

The following two laws on pairs of curves $(\eta_{1},\eta_{2})$ agree:

  * •

Sample $\eta_{1}$ in $\mathbb{H}$ from $0$ to $\infty$ as
$\widetilde{\mathrm{SLE}}_{\kappa}(-\frac{x_{1}+\theta_{1}\chi}{\lambda}-1;\frac{x_{2}+\theta_{1}\chi}{\lambda}-1,\frac{x_{3}-x_{2}}{\lambda};\alpha)$.
In $\mathbb{H}\backslash\eta_{1}$ in the region to the right of $\eta_{1}$,
sample $\eta_{2}$ from $0$ to $\infty$ as
$\widetilde{\mathrm{SLE}}_{\kappa}(-\frac{(\theta_{1}-\theta_{2})\chi}{\lambda}-2;\frac{x_{2}+\theta_{2}\chi}{\lambda}-1,\frac{x_{3}-x_{2}}{\lambda};\alpha)$.

  * •

Sample $\eta_{2}$ in $\mathbb{H}$ from $0$ to $\infty$ as
$\widetilde{\mathrm{SLE}}_{\kappa}(-\frac{x_{1}+\theta_{2}\chi}{\lambda}-1;\frac{x_{2}+\theta_{2}\chi}{\lambda}-1,\frac{x_{3}-x_{2}}{\lambda};\alpha)$.
In $\mathbb{H}\backslash\eta_{2}$ in the region to the left of $\eta_{2}$,
sample $\eta_{1}$ from $0$ to $\infty$ as
$\mathrm{SLE}_{\kappa}(-\frac{x_{1}+\theta_{1}\chi}{\lambda}-1;\frac{(\theta_{1}-\theta_{2})\chi}{\lambda}-2)$.

###### Proof.

For $\alpha=0$ case, the result is straightforward from the flow line
conditioning argument in [MS16a, Section 6] as drawn in Figure 4. If
$\alpha\neq 0$, let $\mathcal{P}$ be the measure on curves
$(\hat{\eta}_{1},\hat{\eta}_{2})$ as in the statement with $\alpha=0$. If we
let $\mathcal{L}$ be the joint law of $(\eta_{1},\eta_{2})$ constructed from
the second way (i.e. start with $\eta_{2}$ then sample $\eta_{1}$)

$\frac{d\mathcal{L}}{d\mathcal{P}}(\eta_{1},\eta_{2})=|\psi_{\eta_{2}}^{\prime}(1)|^{\alpha}.$
(3.6)

Meanwhile, if we first sample $\eta_{1}$ and then $\eta_{2}$ conditioned on
$\eta_{1}$ as in the statement and let $\tilde{\mathcal{L}}$ be the joint law
of $(\eta_{1},\eta_{2})$, then

$\frac{d\tilde{\mathcal{L}}}{d\mathcal{P}}({\eta}_{1},{\eta}_{2})=|\psi_{{\eta}_{1}}^{\prime}(1)|^{\alpha}|\psi_{{\eta}_{2}|{\eta}_{1}}^{\prime}(1)|^{\alpha}$
(3.7)

where $\psi_{{\eta}_{2}|{\eta}_{1}}$ is the conformal map from the component
of $\mathbb{H}\backslash\psi_{{\eta}_{1}}({\eta}_{2})$ containing 1 to
$\mathbb{H}$ fixing 1 and sending the first (resp. last) point hit by
$\psi_{{\eta}_{1}}({\eta}_{2})$ to 0 (resp. $\infty$). Then we observe that
$\psi_{{\eta}_{2}}=\psi_{{\eta}_{2}|{\eta}_{1}}\circ\psi_{{\eta}_{1}}$ and
therefore the two Radon-Nikodym derivatives (3.6) and (3.7) are the same. ∎

###### Proof of Theorem 1.5.

We start with the case $\rho_{-}\leq 0$. We sample a curve $\eta_{1}$ from
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{+}+\rho_{1},-\rho_{1};0;\frac{\rho_{1}(4-\kappa)}{2\kappa})$
and a curve $\eta_{2}\sim\mathrm{SLE}_{\kappa}(\rho_{-};-\rho_{-}-2)$ with
force points at $0^{-};0^{+}$ on the right component of
$\mathbb{H}\backslash\eta_{1}$. Then by Theorem 3.2 we know that the marginal
law of $\mathcal{R}(\eta_{1})$ is now
$\mathrm{SLE}_{\kappa}(0;\rho^{+},\rho_{1})$; furthermore, by [MS16b, Theorem
1.1], the conditional law of $\mathcal{R}(\eta_{2})$ given
$\mathcal{R}(\eta_{1})$ is $\mathrm{SLE}_{\kappa}(-\rho_{-}-2;\rho_{-})$.
Therefore by Lemma 3.3, the conditional law of $\mathcal{R}(\eta_{1})$ given
$\mathcal{R}(\eta_{2})$ is precisely
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$. See Figure 5.

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Figure 5: Left: The curves $(\eta_{1},\eta_{2})$, whose the law has Radon-
Nikodym derivative
$|\psi_{\eta_{1}}^{\prime}(-1)|^{\frac{\rho_{1}(4-\kappa)}{2\kappa}}$ with
respect to the corresponding flow lines of the GFF with the depicted boundary
values. By Lemma 3.3 the conditional law of $\eta_{1}$ given $\eta_{2}$ is
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{+}+\rho_{1},-\rho_{1};\rho_{-};\frac{\rho_{1}(4-\kappa)}{2\kappa})$.
Right: An Imaginary Geometry coupling of $\mathcal{R}(\eta_{1})$ and
$\mathcal{R}(\eta_{2})$ where the marginal law of $\mathcal{R}(\eta_{1})$ is
$\mathrm{SLE}_{\kappa}(0;\rho^{+},\rho_{1})$ by Theorem 3.2 and the law of
$\mathcal{R}(\eta_{2})$ given $\mathcal{R}(\eta_{1})$ is
$\mathrm{SLE}_{\kappa}(-\rho_{-}-2;\rho_{-})$. Another application of Lemma
3.3 gives that the law of $\mathcal{R}(\eta_{1})$ given $\eta_{2}$ is
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$.

Now suppose $\rho^{-}\in[0,2]$. We first sample a curve $\eta_{2}$ on
$\mathbb{H}$ from 0 to $\infty$ from
$\widetilde{\mathrm{SLE}}_{\kappa}(2+\rho_{+}+\rho_{1},-\rho_{1};\rho_{-}-2;\frac{\rho_{1}(4-\kappa)}{2\kappa})$
and then $\eta_{1}$ from
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{+}+\rho_{1},-\rho_{1};0;\frac{\rho_{1}(4-\kappa)}{2\kappa})$
on the left component of $\mathbb{H}\backslash\eta_{2}$. Then using the same
conformal map composing argument, we observe that the Radon-Nikodym derivative
of the law of $(\eta_{1},\eta_{2})$ with respect to the flow lines of the GFF
with the corresponding boundary values in Figure 6 is
$|\psi_{\eta_{1}}^{\prime}(-1)|^{\frac{\rho_{1}(4-\kappa)}{2\kappa}}$, and the
marginal law of $\eta_{1}$ is
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{+}+\rho_{1},-\rho_{1};\rho_{-};\frac{\rho_{1}(4-\kappa)}{2\kappa})$.
By Theorem 3.2, we know that the conditional law of $\mathcal{R}(\eta_{1})$
given $\mathcal{R}(\eta_{2})$ is $\mathrm{SLE}_{\kappa}(0;\rho_{+},\rho_{1})$,
while by what we have just proved, since $\rho_{-}-2\leq 0$, the marginal law
of $\mathcal{R}(\eta_{2})$ is
$\mathrm{SLE}_{\kappa}(\rho_{-}-2;2+\rho^{+},\rho_{1})$. Therefore using the
Imaginary Geometry coupling we observe that the marginal law of
$\mathcal{R}(\eta_{1})$ is
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$, which concludes the proof
for $\rho_{-}\in[0,2]$ case. Also see Figure 6 for an illustration.

Finally we notice that the above argument (i.e., the coupling in Figure 6) can
be iterated, giving the reversibility for $\rho_{-}\in[2,4],[4,6]$, …, etc..
This finishes the proof of Theorem 1.5.

|  |
---|---|---
Figure 6: Left: The curves $(\eta_{1},\eta_{2})$, whose the law has Radon-
Nikodym derivative
$|\psi_{\eta_{1}}^{\prime}(-1)|^{\frac{\rho_{1}(4-\kappa)}{2\kappa}}$ with
respect to the corresponding flow lines of the GFF with the depicted boundary
values. One can show that the marginal law of $\eta_{1}$ is
$\widetilde{\mathrm{SLE}}_{\kappa}(\rho_{+}+\rho_{1},-\rho_{1};\rho_{-};\frac{\rho_{1}(4-\kappa)}{2\kappa})$.
Right: An Imaginary Geometry coupling of $\mathcal{R}(\eta_{1})$ and
$\mathcal{R}(\eta_{2})$ where the marginal law of $\mathcal{R}(\eta_{2})$ is
$\mathrm{SLE}_{\kappa}(\rho_{-}-2;2+\rho^{+},\rho_{1})$ and the conditional
law of $\mathcal{R}(\eta_{1})$ given $\mathcal{R}(\eta_{2})$ is
$\mathrm{SLE}_{\kappa}(0;\rho_{+},\rho_{1})$. Then we see that the marginal
law of $\mathcal{R}(\eta_{1})$ is
$\mathrm{SLE}_{\kappa}(\rho_{-};\rho_{+},\rho_{1})$.

∎

##  4 Conformal welding of $\mathrm{QT}(W,W,2)$ and two thin quantum disks

The aim of this section is to prove the following result. See Figure 7 for an
illustration.

######  Theorem 4.1.

Fix $W>\frac{\gamma^{2}}{2}$ and $U\in(0,\frac{\gamma^{2}}{2})$. Take a
triangle from $\mathrm{QT}(W+U,W+U,2+2U)$ embedded as
$(\mathbb{H},\phi,0,\infty,1)$ with 1 being the weight $2+2U$ point. Then
there exists some constant $c$ depending only on $W$ and $U$, and some
probability measure $\mathsf{m}(W;U)$ of pairs of curves $(\eta_{1},\eta_{2})$
where $\eta_{1}$ runs from 1 to 0 and $\eta_{2}$ runs from 1 to $\infty$ such
that the following welding equation holds:

$\mathrm{QT}(W+U,W+U,2+2U)\otimes\mathsf{m}(W;U)=c\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{Weld}(\mathrm{QT}(W,W,2;\ell,\ell^{\prime}),{\mathcal{M}}^{\mathrm{disk}}_{2}(U;\ell),{\mathcal{M}}^{\mathrm{disk}}_{2}(U;\ell^{\prime})d\ell
d\ell^{\prime}$ (4.1)

where $\mathrm{QT}(W,W,2;\ell,\ell^{\prime})$ is disintegration over the
length of the two boundary arcs containing the weight 2 vertex and
$\mathrm{Weld}$ stands for identifying the edges of lengths $\ell$,
$\ell^{\prime}$.

Figure 7: Setup of Theorem 4.1. We claim that welding a thick disk from
${\mathcal{M}}^{\mathrm{disk}}_{2,\bullet}(W)$ (which is the same as
$\mathrm{QT}(W,W,2)$) with two weight $U$ thin disks along the boundary arc
containing the third marked point produces a three-pointed disk with law
$\mathrm{QT}(W+U,W+U,2+2U)$ embedded as $(\mathbb{H},\phi,0,\infty,1)$.

This section is organized as follows. In Section 4.1 we recall the notion of
conformal welding and the result from [AHS21, Proposition 4.5], which states
the welding of a two-pointed disk with a three-pointed disk. Then using a
limiting procedure over this result, in Section 4.2 we give the proof of
Theorem 4.1.

###  4.1 Conformal welding of two-pointed and three-pointed disks

We first recall the the _conformal welding_ of quantum surfaces. Let $n\geq 1$
and $\mathcal{M}^{1},...,\mathcal{M}^{n}$ be measures on quantum surfaces. Fix
some boundary arcs $\tilde{e}_{1},e_{1},...,\tilde{e}_{n},e_{n}$ such that
$e_{i}$ and $\tilde{e}_{i}$ are different boundary arcs on samples from
$\mathcal{M}_{i}$. Suppose we have the disintegration

$\mathcal{M}^{i}=\int_{0}^{\infty}\int_{0}^{\infty}\mathcal{M}^{i}(\ell_{i-1},\ell_{i})d\ell_{i}d\ell_{i-1}\
\ \text{for}\ \ i=1,...,n$

over the quantum lengths of $e_{i}$ and $\tilde{e}_{i}$. Given a tuple of
_independent_ surfaces from
$\mathcal{M}^{1}(\ell_{0},\ell_{1})\times\mathcal{M}^{2}(\ell_{1};\ell_{2})\times\cdots\mathcal{M}^{n}(\ell_{n-1},\ell_{n})$,
suppose that they can a.s. be _conformally welded_ along the pairs of arcs
$(e_{1},\tilde{e}_{i+1})$ for $i=1,...,n-1$, yielding a large surface
decorated with interfaces from the gluing. We write

$\text{Weld}(\mathcal{M}^{1}(\ell_{0},\ell_{1}),\mathcal{M}^{2}(\ell_{1},\ell_{2}),...,\mathcal{M}^{n}(\ell_{n-1},\ell_{n}))$

for the law of the resulting curve-decorated surface. On the other hand,
suppose we have a quantum surface sampled from some measure $\mathcal{M}$ and
embedded on domain $D$ and we also sample an _independent_ family of curves on
$D$ from some measure $\mathcal{P}$ with conformal invariance property. Then
we write $\mathcal{M}\otimes\mathcal{P}$ for the law of this curve-decorated
surface.

We emphasize that for all the quantum surfaces discussed in this paper,
including the (two and three pointed) quantum disks and quantum triangles, the
conformal welding as above is well-defined. This is because near a point $x$
with weight $W\geq\frac{\gamma^{2}}{2}$, the field is locally absolutely
continuous to that of a weight $W$ quantum wedge near its finite-volume
endpoint, while near a point $x$ with weight $W<\frac{\gamma^{2}}{2}$ the
surface is a Possionian chain of weight $\gamma^{2}-W$ disks so local absolute
continuity with respect to the weight $W$ quantum wedge still holds. Therefore
from the conformal welding of quantum wedges [DMS21, Theorem 1.2], our
conformal weldings for quantum disks and triangles are well-defined. See e.g.
[She16], [DMS21, Section 3.5] or [GHS19, Section 4.1] for more background on
conformal welding.

We state the conformal welding of two-pointed quantum disks as below. Recall
the notion of the measure $\mathcal{P}^{\text{disk}}(W_{1},...,W_{n})$ in
[AHS20, Definition 2.25] on tuple of curves $(\eta_{1},...,\eta_{n-1})$ in a
domain $(D,x,y)$, which is the same as
$\textup{SLE}_{\kappa}(W_{1}-2;W_{2}-2)$ from $x$ to $y$ for $n=2$ and defined
recursively for $n\geq 3$ by first sampling $\eta_{n-1}$ from
$\textup{SLE}_{\kappa}(W_{1}+...+W_{n-1}-2;W_{n}-2)$ then
$(\eta_{1},...,\eta_{n-2})$ from
$\mathcal{P}^{\text{disk}}(W_{1},...,W_{n-1})$ on each connected component
$(D^{\prime},x^{\prime},y^{\prime})$ on the left of $D\backslash\eta_{n-1}$
where $x^{\prime}$ and $y^{\prime}$ are the first and the last point hit by
$\eta_{n-1}$.

######  Theorem 4.2 (Theorem 2.2 of [AHS20]).

Fix $W_{1},...,W_{n}>0$ and $W=W_{1}+...+W_{n}$. Then there exists a constant
$c=c_{W_{1},...,W_{n}}\in(0,\infty)$ such that for all $\ell,r>0$, the
identity

$\begin{split}&\mathcal{M}_{2}^{\textup{disk}}(W;\ell,r)\otimes\mathcal{P}^{\textup{disk}}(W_{1},...,W_{n})\\\
&=c\iiint_{0}^{\infty}\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W_{1};\ell,\ell_{1}),\mathcal{M}_{2}^{\textup{disk}}(W_{2};\ell_{1},\ell_{2}),...,\mathcal{M}_{2}^{\textup{disk}}(W_{n};\ell_{n-1},r))d\ell_{1}...d\ell_{n-1}\end{split}$
(4.2)

holds as measures on the space of curve-decorated quantum surfaces.

Next we present the welding of two-pointed quantum disk with three-pointed
quantum disks as in [AHS21, Proposition 4.5], which adds a marked point to the
boundary arc in Theorem 4.2 above. Recall the notion of SLE weighted by
conformal radius in Section 3.3.

######  Proposition 4.3.

Suppose $W_{1},W_{2}>0$, $W_{1}+W_{2}\neq\frac{\gamma^{2}}{2}$ and
$W_{2}\neq\frac{\gamma^{2}}{2}$. Then there exists a constant
$c_{W_{1},W_{2}}\in(0,\infty)$ such that for all $\beta\in\mathbb{R}$ and
$\ell>0$,

$\begin{split}\mathcal{M}_{2,\bullet}^{\textup{disk}}(W_{1}+W_{2};\beta;\ell)&\otimes\widetilde{\mathrm{SLE}}_{\kappa}(W_{1}-2;W_{2}-2,0;1-\Delta_{\beta})\\\
&=c_{W_{1},W_{2}}\int_{0}^{\infty}\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W_{1};\ell,x),\mathcal{M}_{2,\bullet}^{\textup{disk}}(W_{2};\beta;x))dx.\end{split}$
(4.3)

where again $\Delta_{\beta}$ is determined by (2.8).

Note that if $W_{1}+W_{2}<\frac{\gamma^{2}}{2}$, the interface above is
understood as a chain of $\mathrm{SLE}_{\kappa}(W_{1}-2;W_{2}-2)$ curves
except that the segment of curve on the disk containing the marked point is
replaced by
$\widetilde{\mathrm{SLE}}_{\kappa}(W_{1}-2;W_{2}-2,0;1-\Delta_{\beta})$. If
$\alpha=\gamma$ then since $\Delta_{\gamma}=1$, the interface is simply
$\mathrm{SLE}_{\kappa}(W_{1}-2;W_{2}-2)$ without any reweighting.

###### Proof.

When $W_{1}\geq\frac{\gamma^{2}}{2}$, the statement is precisely the same as
[AHS21, Proposition 4.5]. Now suppose $W_{1}<\frac{\gamma^{2}}{2}$. We start
with a sample from
$\mathcal{M}_{2,\bullet}^{\textup{disk}}(2+W_{1}+W_{2};\beta)$ embedded as
$(\mathbb{H},\phi,0,\infty,1)$ where the $\beta$-insertion is located at 1.
Sample an independent curve $\eta_{2}$ from
$\widetilde{\mathrm{SLE}}_{\kappa}(W_{1},W_{2}-2;1-\Delta_{\beta})$, and given
$\eta_{2}$, independently sample a curve $\eta_{1}$ from
SLE${}_{\kappa}(0;W_{1}-2)$ on the left component of
$\mathbb{H}\backslash\eta_{2}$. Let $\tilde{\mathcal{P}}(2,W_{1},W_{2})$ be
the joint law of $(\eta_{1},\eta_{2})$. Then by Theorem 4.2, we obtain that
for some $c=c_{W_{1},W_{2}}\in(0,\infty)$,

$\mathcal{M}_{2,\bullet}^{\textup{disk}}(2+W_{1}+W_{2};\beta)\otimes\tilde{\mathcal{P}}(2,W_{1},W_{2})=c\iint_{[0,\infty)^{2}}\text{Weld}(\mathcal{M}_{2}^{\textup{disk}}(2;\ell),\mathcal{M}_{2}^{\textup{disk}}(W_{1};\ell,x),\mathcal{M}_{2,\bullet}^{\textup{disk}}(W_{2};\beta;x))dxd\ell.$
(4.4)

On the other hand, using the same trick as in the proof of Theorem 1.5, the
marginal law of $\eta_{1}$ under $\tilde{\mathcal{P}}$ is
$\widetilde{\mathrm{SLE}}_{\kappa}(0;W_{1}+W_{2}-2,0;1-\Delta_{\beta})$, and
by the existing argument for $W_{1}\geq\frac{\gamma^{2}}{2}$, given the
interface and its quantum length $\ell$, the quantum surface to the right of
$\eta_{1}$ has law
$\mathcal{M}_{2,\bullet}^{\textup{disk}}(W_{1}+W_{2};\beta;\ell)$. The law of
$\eta_{2}$ given $\eta_{1}$ is
$\widetilde{\mathrm{SLE}}_{\kappa}(W_{1}-2;W_{2}-2,0;1-\Delta_{\beta})$ on the
right component of $\mathbb{H}\backslash\eta_{1}$, and therefore
disintegrating (4.4) over $\ell$ and $\eta_{1}$ yields the proposition. ∎

Recall from Remark 2.19, if $W_{3}>\frac{\gamma^{2}}{2}$ and
$\beta_{3}=\gamma+\frac{2-W_{3}}{\gamma}$, then the measure
$\mathcal{M}_{2,\bullet}^{\textup{disk}}(W;\beta_{3};\ell)$ is some multiple
constant of our quantum triangle QT$(W,W,W_{3};\ell)$. Therefore we can
rewrite (4.3) as

$\begin{split}\text{QT}(W_{1}+W_{2},W_{1}+W_{2},W_{3};\ell)&\otimes\widetilde{\mathrm{SLE}}_{\kappa}(W_{1}-2;W_{2}-2,0;1-\Delta_{\beta_{3}})=\\\
&c_{W_{1},W_{2}}\int_{0}^{\infty}\textup{Weld}(\mathcal{M}_{2}^{\textup{disk}}(W_{1};\ell,x),\text{QT}(W_{2},W_{2},W_{3};x))dx.\end{split}$
(4.5)

We emphasize that (4.5) continues to hold for $W_{3}<\frac{\gamma^{2}}{2}$ by
the thick-thin duality. This is because concatenating weight $W_{3}$ quantum
disks to both sides of (4.5) (with $W_{3}$ replaced by $\gamma^{2}-W_{3}$)
does not affect the equation, while from (2.8), the corresponding
$\Delta_{\beta}$’s are the same for $W_{3}$ and $\gamma^{2}-W_{3}$ and
therefore the interfaces are the same.

###  4.2 Proof of Theorem 4.1

The idea of proving Theorem 4.1 is as follows. First assume
$W\in(\frac{\gamma^{2}}{2},2)$ and $U\in(0,\frac{\gamma^{2}}{2})$. We take
$(W_{1},W_{2})$ to be $(W,U)$ in Proposition 4.3 and let
$\beta\downarrow\beta_{0}:=\gamma-\frac{2U}{\gamma}$. In this limiting
procedure, we will show that the SLE excursion containing the point 1 shrinks
into a single point, yielding the desired welding picture. Finally if $W>2$,
we can split the weight $W$ disk into a weight $W-2$ quantum disk and a weight
$2$ quantum disk and apply Proposition 4.3.

---
Figure 8: Illustration of the proof of Theorem 4.1. Top: A three-pointed disk
from ${\mathcal{M}}^{\mathrm{disk}}_{2,\bullet}(W+U;\beta;\ell_{0})$ embedded
as $(\mathbb{H},\phi,0,\infty,1)$ decorated by an independent
$\widetilde{\mathrm{SLE}}_{\kappa}(W-2;U-2,0;1-\Delta_{\beta})$ curve on the
top. This splits the surface into a weight $W$ thick disk and a three-pointed