# Distances on the $\mathrm{CLE}_{4}$, critical Liouville quantum gravity and
$3/2$-stable maps

Emmanuel Kammerer CMAP, École polytechnique, Institut Polytechnique de Paris,
91120 Palaiseau, France<EMAIL_ADDRESS>
###### Abstract

The purpose of this article is threefold. First, we show that when one
explores a conformal loop ensemble of parameter $\kappa=4$
($\mathrm{CLE}_{4}$) on an independent $2$-Liouville quantum gravity ($2$-LQG)
disk, the surfaces which are cut out are independent quantum disks. To achieve
this, we rely on approximations of the explorations of a $\mathrm{CLE}_{4}$:
we first approximate the $\mathrm{SLE}_{4}^{\langle\mu\rangle}$ explorations
for $\mu\in\mathbb{R}$ using explorations of the $\mathrm{CLE}_{\kappa}$ as
$\kappa\uparrow 4$ and then we approximate the uniform exploration by letting
$\mu\to\infty$. Second, we describe the relation between the so-called natural
quantum distance and the conformally invariant distance to the boundary
introduced by Werner and Wu. Third, we establish the scaling limit of the
distances from the boundary to the large faces of $3/2$-stable maps and relate
the limit to the $\mathrm{CLE}_{4}$-decorated $2$-LQG.

Figure 1: Left: illustration of the $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$
exploration of the $\mathrm{CLE}_{4}$ ensemble. Right: illustration of the
uniform exploration of the $\mathrm{CLE}_{4}$ ensemble. The black curve $\eta$
is the trunk of the exploration. The unexplored region is in green, the
domains encircled by loops are in orange and the cut out domains are in blue.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main results
    3. 1.3 Outline
  2. 2 Preliminaries on the uniform exploration of $\mathrm{CLE}_{4}$ and LQG
    1. 2.1 The uniform exploration of the $\mathrm{CLE}_{4}$
    2. 2.2 Critical Liouville quantum gravity and quantum disks
    3. 2.3 The locally largest component and the quantum natural distance
  3. 3 The convergence $\kappa\uparrow 4$
    1. 3.1 Carathéodory topology
    2. 3.2 Background on $\mathrm{CLE}_{\kappa}$ explorations
    3. 3.3 Carathéodory convergence of the chordal exploration
    4. 3.4  The $\sqrt{\kappa}$-quantum disks and $\mathrm{CLE}_{\kappa}$
    5. 3.5 Joint convergence of the $\mathrm{CLE}_{\kappa}$ and the $\sqrt{\kappa}$-quantum disks as $\kappa\uparrow 4$
  4. 4 The $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ converges to the uniform exploration as $\mu\to\infty$
    1. 4.1 From the Carathéodory convergence to the Markov property for quantum disks
    2. 4.2 Convergence of the branch towards the origin
    3. 4.3 Joint convergence of the exploration branches
  5. 5 Conformally invariant distance and quantum natural distance
  6. 6 Scaling limit of the distances to the boundary in $3/2$-stable maps
    1. 6.1 Background on $3/2$-stable maps and the peeling exploration
    2. 6.2 Metric growth for infinite $3/2$-stable maps with increasing perimeter
    3. 6.3 Large $3/2$-stable maps and $\mathrm{CLE}_{4}$-decorated quantum disks
  7. 7 Discussion
    1. 7.1 Application to $O(2)$-decorated maps
    2. 7.2 The conformally invariant distance is not the Lamperti transform of the quantum distance

##  1 Introduction

###  1.1 Background

The conformal loop ensemble $\mathrm{CLE}_{4}$ is a random collection of
disjoint simple loops included in some simply connected domain. It was
introduced and studied in [She09] as part of the one parameter family of
simple conformal loop ensembles $\mathrm{CLE}_{\kappa}$ for
$\kappa\in(8/3,8)$, whose loops are no longer simple when $\kappa>4$. In
[She09], Sheffield also defined a family of branching explorations
parametrized by some “drift” $\mu$ of the $\mathrm{CLE}_{4}$ starting from the
boundary and discovering the loops, called the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration. According to [MSW17]
this exploration can be described using a random simple continuous curve
called the trunk of the exploration which hits the loops it discovers (see
Figure 1). In [WW13], Werner and Wu introduced a uniform exploration of the
$\mathrm{CLE}_{4}$ which can be seen as the limit of the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration as $\mu\to\infty$.
Informally, it consists in discovering the loop next to a random point chosen
according to the harmonic measure on the boundary of the explored region and
then updating this random point. The time parametrization of the uniform
exploration in [WW13] is interpreted as a distance to the boundary. In an
unpublished work [SWW], Sheffield, Watson and Wu argue that this distance from
the boundary to the loops of the $\mathrm{CLE}_{4}$, which will be denoted by
$\mathrm{d}_{\mathrm{WW}}$, can be extended to a conformally invariant
distance between all the loops of the $\mathrm{CLE}_{4}$, including the
boundary of the domain. However, we will not use this result in this work.

More recently, the $\mathrm{CLE}_{\kappa}$ was shown to interact nicely with
an independent Liouville quantum gravity surface of parameter $\sqrt{\kappa}$
on the same simply connected domain ($\sqrt{\kappa}$-LQG). An LQG disk
parametrized by a simply connnected domain can be defined as a well-chosen
version of a Gaussian free-field on this domain which provides a way to
measure quantum lengths of some curves and quantum area of sub-domains. In
[MSW22], the law of the quantum boundary length of the unexplored region along
the exploration of the $\mathrm{CLE}_{\kappa}$ for $\kappa<4$ parametrized by
the quantum length of the trunk is shown to behave as some branching Markov
process called a growth-fragmentation process, which was introduced by Bertoin
in [Ber17]. For the case $\kappa=4$, building on the rich interplay between
$\mathrm{CLE}$’s, LQG’s, and Brownian motion, and particularly on [DMS21],
Aru, Holden, Powell and Sun identified in [AHPS23] a coupling between a
uniform $\mathrm{CLE}_{4}$ exploration, an independent $2$-LQG disk and a
Brownian excursion in the upper half-plane. Thanks to [AdS22], this coupling
actually enables to describe the quantum boundary length of the explored
region along the uniform exploration, parametrized by a so-called quantum
natural distance to the boundary, as a growth-fragmentation process.

Independently, Boltzmann $3/2$-stable maps were first studied in [BCM18] in
the infinite volume setting. In this introduction, we will only talk about
finite $3/2$-stable maps. Recall that a planar map $\mathfrak{m}$ is a planar
graph embedded in the sphere seen up to orientation preserving homeomorphism,
with a distinguished oriented edge $\vec{e}_{r}$. The face on the right of the
root edge is called the root face and is written $f_{r}$. We focus on
bipartite planar maps, which only have faces with even degrees. For all
$\ell\in\mathbb{N}$, let us denote by $\mathcal{M}^{(\ell)}$ the set of
bipartite planar maps of perimeter $2\ell$, i.e. such that the degree of the
root face is $2\ell$. Let ${\bf q}=(q_{k})_{k\geq 0}$ be a non-zero sequence
of non-negative real numbers. The Boltzmann weight of a map
$\mathfrak{m}\in\mathcal{M}^{(\ell)}$ is defined by

$w_{\bf
q}(\mathfrak{m})=\prod_{f\in\mathrm{Faces}(\mathfrak{m})\setminus\left\\{f_{r}\right\\}}q_{\deg(f)/2}.$

When the partition function
$W^{(\ell)}\coloneqq\sum_{\mathfrak{m}\in\mathcal{M}^{(\ell)}}w_{\bf
q}(\mathfrak{m})$ is finite, we say that the weight sequence $\bf q$ is
admissible. We further say that $\bf q$ is critical non-generic of type $2$
when

$W^{(\ell)}\mathop{\sim}\limits_{\ell\to\infty}\frac{p_{\bf q}}{2}c_{\bf
q}^{\ell+1}\ell^{-2}$ (1.1)

for some constants $c_{\bf q},p_{\bf q}>0$. The Boltzmann probability measure
on $\mathcal{M}^{(\ell)}$ is then defined by
$\mathbb{P}^{(\ell)}(\mathfrak{m})=\frac{w_{\bf q}(\mathfrak{m})}{W^{(\ell)}}$
for all $\mathfrak{m}\in\mathcal{M}^{(\ell)}$. The associated mathematical
expectation is written $\mathbb{E}^{(\ell)}$. A random variable of law
$\mathbb{P}^{(\ell)}$ is called a $3/2$-stable map of perimeter $2\ell$. In
what follows, ${\bf q}$ will be assumed critical non-generic of type $2$.

The distances that we will study on these maps are the graph distance on the
dual map, obtained by exchanging the vertices with the faces, written
$\mathrm{d}^{\dagger}_{\mathrm{gr}}$, and the first passage percolation
distance, $\mathrm{d}^{\dagger}_{\mathrm{fpp}}$, which is obtained by putting
i.i.d. parameter $1$ exponential random lengths on each edge of the dual map.

We studied the geometry of $3/2$-stable maps at the macroscopic scale in
[Kam23] and proved that large $3/2$-stable planar maps do not satisfy scaling
limits in the usual sense of Gromov-Hausdorff or Gromov-Prokhorov. The main
technique in [BCM18] and [Kam23] to study these distances is the peeling
exploration first introduced in [Bud16]. It consists in a step by step
Markovian exploration of the map where at each step, we “peel” an edge on the
boundary of the unexplored region and discover what is behind it. In this
setting, we can also observe the evolution the perimeter of the unexplored
region (which is the number of edges on its boundary).

###  1.2 Main results

Let us already state our main results more precisely even if the exact
definitions will only arise in the next sections.

#### Exploration of $\mathrm{CLE}_{4}$ and LQG.

We choose to write the results for a particular branch of the exploration of
the $\mathrm{CLE}_{4}$ on the unit disk $\mathbb{D}$ equipped with an
independent $2$-LQG disk, namely the branch of the locally largest component:
at each time the remaining to explore region splits in two regions, we choose
to explore the region which has the largest quantum boundary length. We
parametrize the uniform exploration by the so-called quantum natural distance
to the boundary and the $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration
by the quantum length of the trunk. These parametrizations are characterized
as follows: if we denote by $(Y_{t})_{t\geq 0}$ (resp. $(Y^{\mu}_{t})_{t\geq
0}$) the càdlàg process of the quantum boundary length of the locally largest
component along the uniform exploration (resp.
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration), then we have for all
$t\geq 0$ the limit in probability

$\lim_{\varepsilon\to 0}\varepsilon\\#\\{s\in[0,t],\ \lvert\Delta
Y_{s}\rvert\in[\varepsilon,2\varepsilon]\\}=\lim_{\varepsilon\to
0}\varepsilon\\#\\{s\in[0,t],\ \lvert\Delta
Y^{\mu}_{s}\rvert\in[\varepsilon,2\varepsilon]\\}=t,$

where for any càdlàg process $(X_{t})_{t\geq 0}$ we write $\Delta
X_{t}\coloneqq X_{t}-X_{t-}$ for all $t\geq 0$. The positive jumps of
$(Y_{t})_{t\geq 0}$ or $(Y^{\mu}_{t})_{t\geq 0}$ occur when the exploration
draws a loop while the negative jumps happen when a part of the unexplored
region is cut out by the exploration.

Let us recall the positive self-similar Markov process studied in [AdS22],
which corresponds to a particular case in a family of such processes
introduced in [BBCK18] for $\theta=1$. Let $\Lambda_{1}$ be the image by
$x\mapsto\log x$ of the measure ${1/(\pi(x(1-x))^{2})}{\bf
1}_{x\in(1/2,1)\cup(1,\infty)}dx.$ We denote by $\xi$ the Lévy process with no
Brownian part, with drift $-2/\pi$ and Lévy measure $\Lambda_{1}$. If
$\alpha\in\mathbb{R}$, we define the Lamperti time-substitution

$\tau(t)=\inf\left\\{r\geq 0\ ;\ \int_{0}^{r}e^{-\alpha\xi(s)}ds\geq
t\right\\}.$ (1.2)

For all $x>0$, we write $P_{x}^{(\alpha)}$ the distribution of the time-
changed process $X^{(\alpha)}(t)=xe^{\xi(\tau({tx^{\alpha}}))}$ with the
convention that $X^{(\alpha)}(t)=\partial$ for $t\geq\zeta\coloneqq
x^{-\alpha}\int_{0}^{\infty}e^{-\alpha\xi(s)}ds$. Then $X^{(\alpha)}$ is a
self-similar Markov process of index $\alpha$, in the sense that for every
$x>0$, the law of $(xX^{(\alpha)}(x^{\alpha}t))_{t\geq 0}$ under
$P_{1}^{(\alpha)}$ is $P_{x}^{(\alpha)}$. By the definition of $\xi$, we can
see that either $X^{(\alpha)}$ is absorbed a.s. at the cemetery point
$\partial$ after a finite time, or it converges a.s. exponentially fast to
zero. In the case $\alpha=-1$, we have $\zeta<+\infty$.

Our first main result is the analogue of Theorem 1.2 of [MSW22] in the case
$\kappa=4$ for the $\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ exploration and
for the uniform exploration.

######  Theorem 1.1.

Let $\mu\in\mathbb{R}$. The processes $(Y_{t})_{t\geq 0}$ and
$(Y^{\mu}_{t})_{t\geq 0}$ have the same law as $(X^{(-1)}(\pi t))_{t\geq 0}$.
Moreover, conditionally on $(Y_{s})_{0\leq s\leq t}$ (resp.
$(Y^{\mu}_{s})_{0\leq s\leq t}$), if we denote by $(x_{i})_{i\geq 1}$ the
sizes of the jumps of $(Y_{s})_{0\leq s\leq t}$ (resp. $(Y^{\mu}_{s})_{0\leq
s\leq t}$) up to time $t$ ranked in the non-increasing order of absolute
value, then the corresponding domains which are encircled by loops, associated
with the positive jumps, or which are cut out, associated with the negative
jumps, and the unexplored region are independent $2$-quantum disks of boundary
lengths $|x_{i}|$’s and $Y_{t}$ (resp. $Y^{\mu}_{t}$).

Note that the fact that $(Y_{t})_{t\geq 0}$ has the same law as $(X^{(-1)}(\pi
t))_{t\geq 0}$ can be seen as a direct consequence of Theorem 5.5 of [AHPS23]
and Theorem 1.1 of [AdS22]. Besides, by focusing on the positive jumps, from
the above theorem one recovers Theorem 1.1 of the recent article [AG23a].

#### Quantum natural distance and the conformally invariant distance.

For all $u\geq 0$, we denote by $C_{u}$ the unexplored region at time $u$ when
following the locally largest component in the uniform exploration of [WW13]
parametrized by the distance $\mathrm{d}_{\mathrm{WW}}$. We also denote by
$\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},C_{u})$ the quantum natural
distance from $C_{u}$ to the boundary $\partial\mathbb{D}$. Let
$(\sigma(t))_{t\geq 0}$ be the right-continuous inverse of
$u\mapsto\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},C_{u})$. The next result,
a consequence of Theorem 1.1, gives some insight on $(\sigma(t))_{t\geq 0}$.

######  Theorem 1.2.

The process $(\sigma(t))_{t\geq 0}$ is continuous and increasing. Moreover,
the continuous increasing process $S$ characterized by

$\forall t\geq 0,\qquad\sigma(t)=S\left(\int_{0}^{t}\frac{1}{Y_{s}}ds\right)$

has stationary increments.

However, we do not believe that $S$ has independent increments, see Subsection
7.2 for some heuristics. The above result motivates the introduction of
another distance to the boundary: for every loop $\mathcal{L}$ which is
discovered by the exploration of the locally largest component, we set
$\mathrm{d}_{\mathrm{Lamperti}}(\partial\mathbb{D},\mathcal{L})\coloneqq\int_{0}^{\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},\mathcal{L})}{ds}/{Y_{s}}$.
Then we have
$\mathrm{d}_{\mathrm{WW}}(\partial\mathbb{D},\mathcal{L})=S(\mathrm{d}_{\mathrm{Lamperti}}(\partial\mathbb{D},\mathcal{L}))$.

####  $3/2$-stable maps and $\mathrm{CLE}_{4}$-decorated LQG.

Denote by $(P_{*}(n))_{n\geq 0}$ the perimeter process during the peeling
exploration of the locally largest component (in terms of perimeter) under the
law $\mathbb{P}^{(\ell)}$. Then it is known from Proposition 6.6 of [BBCK18]
(which comes from Theorem 1 of [BCM18]) that under $\mathbb{P}^{(\ell)}$ the
rescaled process $(P_{*}(\ell t)/\ell)_{t\geq 0}$ converges towards
$(X^{(-1)}(\pi p_{\bf q}t))_{t\geq 0}$ where $p_{\bf q}$ is introduced in
(1.1), so that by Theorem 1.1 the quantum length of the trunk of the
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ exploration and the quantum natural
distance corespond to the scaling limit of the time in the peeling
exploration. The next results extend this relation between $3/2$-stable maps
and $\mathrm{CLE}_{4}$-decorated LQG by establishing the scaling limit of the
distances to the boundary.

By using some particular peeling algorithm, we can explore the map in a way
that follows the growth of the balls centred at the root face $f_{r}$ or
equivalently such that the distances from the edges on the boundary of the
unexplored region to the root face are rouhly the same. For example, for
$\mathrm{d}^{\dagger}_{\mathrm{gr}}$, the idea is to explore the map layer by
layer: at first the edges at distance $0$ from the root face, then at distance
$1$, etc. For $\mathrm{d}^{\dagger}_{\mathrm{fpp}}$, we use the uniform
peeling exploration which takes at each step an edge uniformly at random in
the boundary. See Subsection 6.1 for the definition of the peeling
explorations. Let $\partial\overline{\mathfrak{e}}_{n}$ be the boundary of the
explored region at time $n\geq 0$ using the appropriate peeling exploration
for each distance. Then we prove that the radius of the balls during the
exploration evolves in the following way:

$\displaystyle\text{under
}\mathbb{P}^{(\ell)},\qquad(\mathrm{d}^{\dagger}_{\mathrm{fpp}}(f_{r},\partial\overline{\mathfrak{e}}_{\lfloor\ell
t\rfloor}))_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{\mathrm{(d)}}\left(\int_{0}^{t}\frac{1}{2Y(p_{\bf
q}s)}ds\right)_{t\geq 0};$ (1.3) $\displaystyle\text{under
}\mathbb{P}^{(\ell)},\qquad(\mathrm{d}^{\dagger}_{\mathrm{gr}}(f_{r},\partial\overline{\mathfrak{e}}_{\lfloor\ell
t\rfloor})/(\log\ell))_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{\mathrm{(d)}}\left(\int_{0}^{t}\frac{p_{\bf
q}}{2Y(p_{\bf q}s)}ds\right)_{t\geq 0};$ (1.4)

for the $J_{1}$ Skorokhod topology.

All our statements only focused so far on the locally largest component. But
by the Markov property of Theorem 1.1 for the $\mathrm{CLE}_{4}$ exploration,
and by the spatial Markov property of the peeling exploration, they can be
extended to the convergence of the whole exploration tree. This enables us to
state the scaling limit of the degree, the time of exploration and the
distance to the boundary of large faces of $3/2$-stable maps towards the
quantum boundary length, the quantum natural distance (or quantum length of
the trunk depending on the exploration) and its Lamperti transform of a
$\mathrm{CLE}_{4}$-decorated LQG disk.

######  Theorem 1.3.

For all $\ell\geq 1$, let $\mathfrak{M}^{(\ell)}$ of law
$\mathbb{P}^{(\ell)}$. Let $(f_{i}^{(\ell)})_{i\geq 1}$ be the collection of
faces of $\mathfrak{M}^{(\ell)}$ ranked in the non-increasing order of degree.
Let $(n(f_{i}^{(\ell)}))_{i\geq 1}$ be the time at which they are discovered
by the peeling by layers exploration. Let $(\mathcal{L}_{i})_{i\geq 1}$ be the
collection of (outermost) loops of the $\mathrm{CLE}_{4}$ ranked in the non-
increasing order of quantum boundary length. Let us denote by
$\nu^{2}_{h^{2}}(\mathcal{L})$ the quantum boundary length of a loop
$\mathcal{L}$, by $\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},\mathcal{L})$
the quantum natural distance to $\mathcal{L}$ and by
$\mathrm{d}_{\mathrm{Lamperti}}(\partial\mathbb{D},\mathcal{L})$ its Lamperti
transform defined by

$\mathrm{d}_{\mathrm{Lamperti}}(\partial\mathbb{D},\mathcal{L})\coloneqq\int_{0}^{\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},\mathcal{L})}\frac{dt}{Y_{\mathcal{L}}(t)},$

where $(Y_{\mathcal{L}}(t))_{0\leq
t\leq\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},\mathcal{L})}$ is the quantum
boundary length of the unexplored region surrounding $\mathcal{L}$
parametrized by the quantum natural distance. Then, for the product topology,

$\left(\frac{\deg(f_{i}^{(\ell)})}{\ell},\frac{n(f_{i}^{(\ell)})}{p_{\bf
q}\ell},\frac{2\mathrm{d}^{\dagger}_{\mathrm{gr}}(f_{r},f_{i}^{(\ell)})}{\log\ell}\right)_{i\geq
1}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left(\nu^{2}_{h^{2}}(\mathcal{L}_{i}),\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},\mathcal{L}_{i}),\mathrm{d}_{\mathrm{Lamperti}}(\partial\mathbb{D},\mathcal{L}_{i})\right)_{i\geq
1}.$

The same theorem can be stated for the
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ exploration, replacing the quantum
natural distance by the quantum length of the trunk. For the fpp distance, the
factor $\log\ell$ is replaced by a factor $1/p_{\bf q}$ and the peeling by
layers exploration is replaced by the uniform peeling exploration.

###  1.3 Outline

In Section 2 we recall the definition of the uniform exploration of the
$\mathrm{CLE}_{4}$ from [WW13] and the definitions of $\gamma$-LQG disks for
$\gamma\in(\sqrt{2},2]$ from [AHPS23]. The definition of the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration is postponed to the
next section.

Then, in Section 3, after introducing the remaining definitions, we prove
Theorem 1.1 for the $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration by
approximating the exploration with some well-chosen
$\mathrm{SLE}_{\kappa}^{\beta}(\kappa-6)$ exploration, using the results of
[Leh21] and by approximating the $2$-LQG disks with $\sqrt{\kappa}$-LQG disks
relying on [AHPS23].

Next, in Section 4, so as to obtain the statement of Theorem 1.1 for the
uniform exploration, we approximate the uniform exploration with the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration as $\mu\to\infty$. This
section relies partly on the same ideas as in Section 3.

On the basis of Theorem 1.1, using the Poissonian structure of the uniform
exploration in [WW13], we prove Theorem 1.2 in Section 5.

In Section 6, we first establish the metric growth in the infinite volume
setting using results of [BCM18]. By absolute continuity, we prove the
convergences (1.3) and (1.4). We then describe the limit of the whole
exploration tree and deduce Theorem 1.3. Our approximation results are
informally summarized by the following diagram:

$\mathrm{CLE}_{\kappa}$
exploration$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$Uniform
exploration$3/2$-stable mapon a $\sqrt{\kappa}$-LQG diskon a $2$-LQG diskon a
$2$-LQG diskof perimeter $\ell$$\kappa\uparrow 4$Sec.
3$\mu\to\infty$Sec.4$\ell\to\infty$Sec.6

Finally in Section 7, we describe the applications to $O(2)$-loop decorated
planar maps and we argue why the conformally invariant distance from [WW13] is
not the Lamperti transform of the quantum natural distance.

##  2 Preliminaries on the uniform exploration of $\mathrm{CLE}_{4}$ and LQG

In this section, we first define the uniform exploration of the
$\mathrm{CLE}_{4}$ and then we given some background on Liouville quantum
gravity anf quantum disks.

###  2.1 The uniform exploration of the $\mathrm{CLE}_{4}$

We recall here the uniform exploration of the $\mathrm{CLE}_{4}$ introduced in
[WW13]. Let $E$ be the space of positive excursions defined by

$E=\\{e\in\mathcal{C}(\mathbb{R}_{+},\mathbb{R}_{+});\ e(0)=0\text{ and
}\tau(e)\coloneqq\sup\\{s>0;\ e(s)>0\\}\in(0,\infty)\\}.$

Let $\lambda$ be the Itô excursion measure on $E$, which is characterized by
the fact that for any $x>0$, the measure of the subset $E_{x}$ of excursions
of height at least $x$ is finite and if one renormalizes $\lambda$ so that it
is a probabillity measure on $E_{x}$, then after reaching $x$ the excursion
has the same law as a Brownian motion which is stopped when attaining zero. We
denote by $\mathbb{H}$ the upper half-plane. If $e\in E$ is an excursion of
duration $\tau(e)$, we define the driving function $w$ on $[0,\tau(e)]$ by

$\forall t\in[0,\tau(e)],\qquad w_{t}=2e(t)-\int_{0}^{t}\frac{ds}{e(s)}.$

It is shown in [SW12] that for $\lambda$-almost all $e\in E$, the conformal
maps $g_{t}$ characterized by the ODE

$g_{0}(z)=z\qquad\text{and}\qquad\forall s\geq
t,\enskip\partial_{s}g_{s}(z)=\frac{2}{g_{s}(z)-w_{s}}$

define a simple loop $\gamma=\gamma(e)$ in the upper half plane starting at
zero, in the sense that the domain of definition of $g_{t}$ can be written
$\mathbb{H}\setminus\gamma((0,t])$. We define the measure $\mu^{0}$ as the
pushforward measure $\gamma_{*}\lambda$. For all $x\in\mathbb{R}$, we denote
by $\mu^{x}$ the measure on loops rooted at $x$ obtained by translating
$\mu^{0}$.

Let $z\in\mathbb{H}$. Let us first define the uniform exploration targeted at
$z$. Let $(\hat{\gamma}_{u})_{u\geq 0}$ be a PPP of intensity
$M=\int_{\mathbb{R}}dx\mu^{x}$. For every loop $\hat{\gamma}_{u}$, we denote
by $\hat{f}_{u}$ the conformal transformation from the connected component of
$\mathbb{H}\setminus\hat{\gamma}_{u}$ containing $z$ onto $\mathbb{H}$ which
fixes the point $z$ and such that $\hat{f}_{u}^{\prime}(z)\in\mathbb{R}_{+}$.
We define $\hat{F}^{z}_{u}=\hat{F}_{u}\coloneqq\circ_{0\leq v<u}\hat{f}_{v}$
where the composition is done in the order of appearance of the maps
$\hat{f}_{u}$. This indeed defines a conformal map by p.19 of [WW13]. We also
denote by $\tau^{z}$ the first time at which the loop $\hat{\gamma}_{u}$
surrounds the point $z$. We define
$(\widetilde{\gamma}^{z}_{u})_{u\leq\tau^{z}}\coloneqq(\hat{F}_{u}^{-1}(\hat{\gamma}_{u}))_{u\leq\tau^{z}}$.

Lemma 8 from [WW13] shows that it is possible to couple two processes
$(\widetilde{\gamma}^{z}_{u})_{u\leq\tau^{z}}$ and
$(\widetilde{\gamma}^{z^{\prime}}_{u})_{u\leq\tau^{z^{\prime}}}$ so that they
coincide until the time at which they disconnect $z$ from $z^{\prime}$. If we
perform this coupling for a dense countable subset of $\mathbb{H}$ then the
union of the loops surrounding points of this subset of $\mathbb{H}$ is the
$\mathrm{CLE}_{4}$ ensemble. If we keep on drawing loops with
$(\hat{F}_{u}^{-1}(\hat{\gamma}_{u}))_{u\geq 0}$ after the time $\tau^{z}$, we
then build the nested $\mathrm{CLE}_{4}$ ensemble.

For future use, we come back to the unit disk $\mathbb{D}$ via a conformal
transformation $\Phi:\mathbb{H}\rightarrow\mathbb{D}$ and we define the domain
$D_{u}\subseteq\mathbb{D}$ of the unexplored points (during the uniform
exploration of the non-nested $\mathrm{CLE}_{4}$) at time $u\geq 0$ by
$D_{u}=\Phi(\left\\{z\in\mathbb{H};\ \tau^{z}>u\right\\})$. We also define for
all $u\geq 0$ and $z\in\mathbb{D}$ the domain
$\widetilde{D}^{z}_{u}\subseteq\mathbb{D}$ consisting in the unexplored points
at time $u$ in the uniform exploration of the nested $\mathrm{CLE}_{4}$
targeted at $z$ by
$\widetilde{D}^{z}_{u}=\Phi((F_{u}^{\Phi^{-1}(z)})^{-1}(\mathbb{H}))$. The
choice of $\Phi$ does not change the law of the exploration by Proposition 9
of [WW13].

###  2.2 Critical Liouville quantum gravity and quantum disks

We recall here the definitions of the free boundary Gaussian free field (GFF)
and of the $\gamma$-quantum disks for $\gamma\in(\sqrt{2},2]$ following
[AHPS23]. See also [She07] for more details on GFF and [BP] for more about GFF
and Liouville quantum gravity. We follow exactly the definitions in Subsection
4.1 of [AHPS23]. Let $D\subset\mathbb{C}$ be a simply connected domain with
harmonically non-trivial boundary. We denote by $H(D)$ the Hilbert space
closure of the space of real functions $f$ of class $\mathcal{C}^{\infty}$ on
$D$ modulo constants which have finite Dirichlet energy (i.e.
$\int_{D}\lvert\nabla f\rvert^{2}<\infty$), equipped with the Dirichlet inner
product defined for any pair $f,g$ of such functions by

$(f|g)_{\nabla}=\frac{1}{2\pi}\int_{D}\nabla f(z)\cdot\nabla g(z)d^{2}z,$

where $d^{2}z$ is the Lebesgue measure on $\mathbb{C}$. Let $(f_{n})_{n\geq
0}$ be a Hilbert basis of $H(D)$. Let $(X_{n})_{n\geq 0}$ be i.i.d. random
variables of law $\mathcal{N}(0,1)$. The free boundary GFF on $D$ is defined
as the sum

$h=\sum_{n\geq 0}X_{n}f_{n},$

which converges in the space of distributions modulo constants. Even though
the resulting random variable $h$ lives in the space of distributions modulo
constants, one can fix that constant by assigning some particular value to $h$
on an arbitrary test function. The resulting random distribution is actually
in the Sobolev space $H^{-1}_{\mathrm{loc}}(D)$ which is the space of
distributions such that their restrictions to any relatively compact open
domain $U\subset D$ are in the Sobolev space $H^{-1}(U)$. We endow the space
$H^{-1}_{\mathrm{loc}}(D)$ with the corresponding limit topology which is
metrizable and separable, such that a sequence converges in
$H^{-1}_{\mathrm{loc}}(D)$ if and only if the restrictions to any relatively
compact open domain $U$ converge in $H^{-1}(U)$.

Next, we recall that a $\gamma$-LQG surface for $\gamma\in(\sqrt{2},2]$ can be
seen as an equivalence class of domains $D\subset\mathbb{C}$ equipped with a
variant of the GFF $h$ (a free-boundary GFF plus a continuous random function
on $\overline{D}$) under conformal maps. To that instance of the GFF, we
associate an area and a boundary length measures $\mu^{\gamma}_{h}$ and
$\nu^{\gamma}_{h}$. We define the area and boundary length measures
$\mu^{\gamma}_{h}$ and $\nu^{\gamma}_{h}$ by the limits in probability: for
$\gamma=2$, for all bounded open set $A\subseteq D$ we set

$\mu^{2}_{h}(A)=\lim_{\delta\to
0}\int_{A}\left(-h_{\delta}+\log\left(\frac{1}{\delta}\right)\right)e^{2h_{\delta}(z)}\delta
d^{2}z$

and for all bounded open set $B\subseteq\partial D$

$\nu^{2}_{h}(B)=\lim_{\delta\to
0}\int_{B}\left(-\frac{h_{\delta}}{2}+\log\left(\frac{1}{\delta}\right)\right)e^{h_{\delta}(z)}\delta
dz,$

while for $\gamma\in(\sqrt{2},2)$, for all bounded open set $A\subseteq D$, we
set

$\mu_{h}^{\gamma}(A)=\lim_{\delta\to
0}(8(2-\gamma))^{-1}\int_{A}\exp\left(\gamma
h_{\delta}(z)\right)\delta^{\gamma^{2}/2}d^{2}z$

and for all bounded open set $B\subseteq\partial D$

$\nu_{h}^{\gamma}(B)=\lim_{\delta\to
0}(8(2-\gamma))^{-1}\int_{B}\exp\left(\frac{\gamma}{2}h_{\delta}(z)\right)\delta^{\gamma^{2}/4}dz,$

where $d^{2}z$ is the Lebesgue measure on $\mathbb{C}$, $dz$ is the Lebesgue
measure on $\mathbb{R}$ when $D$ is the upper half-plane $\mathbb{H}$ and
$h_{\delta}(z)$ is the circular mean of $h$ on the circle of center $z$ and
radius $\delta$.

We say that two pairs $(D,h)$ and $(\widetilde{D},\widetilde{h})$, where
$D,\widetilde{D}$ are domains and $h,\widetilde{h}$ are distributions, are
equivalent (as $\gamma$-quantum surfaces) if there exists a conformal map
$\varphi:D\rightarrow\widetilde{D}$ such that

$h=\widetilde{h}\circ\varphi+Q_{\gamma}\log|\varphi^{\prime}|\qquad\text{where}\qquad
Q_{\gamma}=2/\gamma+\gamma/2.$ (2.1)

If $h,\widetilde{h}$ are absolutely continuous with respect to a GFF plus a
continuous function, then
$\mu^{\gamma}_{\widetilde{h}}=\varphi_{*}\mu^{\gamma}_{h}$ and
$\nu^{\gamma}_{\widetilde{h}}=\varphi_{*}\nu^{\gamma}_{h}$.

Informally, a $\gamma$-quantum disk (also called $\gamma$-LQG disk)
corresponds to the domain encircled by a $\mathrm{CLE}_{\gamma^{2}}$ loop with
the restriction of an infinite volume GFF. A quantum disk has a finite
boundary length and a finite volume. More precisely, our unit-boundary length
$\gamma$-quantum disks are defined as in Definition 4.1 for
$\gamma=\sqrt{\kappa}\in(\sqrt{2},2)$ in [AHPS23] and Definition 4.3 for
$\gamma=2$ from [AHPS23]. Let us summarize their definition although we will
not rely on it since we will use directly the results of [AHPS23]. The
definition is easier to state when we parametrize the disk by
$\mathcal{S}\coloneqq\mathbb{R}\times(0,\pi)\subset\mathbb{C}$, the infinite
strip. Let $h_{1}^{\gamma}$ be a function on $\mathcal{S}$ defined for all
$s\in\mathbb{R}$ and $y\in(0,\pi)$ by
$h_{1}^{\gamma}(s+iy)=\mathcal{B}^{\gamma}_{s}$, where $\mathcal{B}^{\gamma}$
is defined by:

  1. (i)

For $\gamma\in(\sqrt{2},2)$, the function $(\mathcal{B}^{\gamma}_{s})_{s\geq
0}$ has the same law as $B_{2s}-Q_{\gamma}s$ conditioned to stay negative for
all $s\geq 0$, where $(B_{s})_{s\geq 0}$ is a standard Brownian motion, while
$(\mathcal{B}^{2}_{s})_{s\geq 0}$ is $(-\sqrt{2})$ times a Bessel process of
dimension $3$ (which corresponds in some sense to $(B_{2s})_{s\geq 0}$
conditioned to stay negative by a famous result of Williams).

  2. (ii)

The process $(\mathcal{B}^{\gamma}_{-s})_{s\geq 0}$ is independent of
$(\mathcal{B}^{\gamma}_{s})_{s\geq 0}$ and has the same law as
$(\mathcal{B}^{\gamma}_{s})_{s\geq 0}$.

Let $h_{2}$ be the orthogonal projection of a free boundary GFF on
$\mathcal{S}$ on the closure in $H(\mathcal{S})$ of the functions of mean zero
along the vertical lines of $\mathcal{S}$. Set
$h^{\gamma}_{s}=h_{1}^{\gamma}+h_{2}$ and let $\hat{h}^{\gamma}$ be a random
distribution which has the law of
$h^{\gamma}_{s}-(2/\gamma)\log\nu^{\gamma}_{h^{\gamma}_{s}}(\partial\mathcal{S})$
reweighted by
$\nu^{\gamma}_{h^{\gamma}_{s}}(\partial\mathcal{S})^{(4/\gamma^{2})-1}$. Then
the $\gamma$-quantum disk can be defined as the equivalence class of
$(\mathcal{S},\hat{h}^{\gamma})$. We denote by $(\mathbb{D},h^{\gamma})$ the
parametrization by the unit disk $\mathbb{D}$ of the unit boundary length
$\gamma$-quantum disk.

One can define a marked quantum disk by sampling a marked boundary point
according to the quantum boundary length measure, but we will not do this in
this work. For $b>0$, we define the $b$-boundary length $\gamma$-quantum disk
to be the quantum surface parametrized by
$(\mathbb{D},h^{\gamma}+(2/\gamma)\log b)$. Note that
$\nu^{\gamma}_{h^{\gamma}+(2/\gamma)\log b}=b\nu^{\gamma}_{h^{\gamma}}$ and
$\mu^{\gamma}_{h^{\gamma}+(2/\gamma)\log b}=b^{2}\mu^{\gamma}_{h^{\gamma}}$.

###  2.3 The locally largest component and the quantum natural distance

Let $(\mathbb{D},h^{2})$ be a parametrization of a $2$-quantum disk by
$\mathbb{D}$ and consider an independent uniform exploration of a
$\mathrm{CLE}_{4}$ on $\mathbb{D}$. We want to focus on the quantum boundary
length of the locally largest component of the exploration in terms of quantum
boundary length. Informally, at each discovery of a loop, it goes on exploring
the exterior of the loop and at each splitting event, it explores the domain
which has the largest quantum boundary length. Let us first define formally
this locally largest component. Let $(z_{j})_{j\geq 1}$ be an enumeration of
$\mathbb{D}\cap\mathbb{Q}^{2}$. For all $j\geq 1$, for all
$u\in[0,\tau^{z_{j}})$, let $D_{u}^{z_{j}}$ be the connected component of
$D_{u}$ containing $z_{j}$. Using the restriction of $h^{2}$ to
$D_{u}^{z_{j}}$, one can define the quantum boundary length

$\mathcal{Y}^{z_{j}}_{u}\coloneqq\nu^{2}_{h^{2}_{|D_{u}^{z_{j}}}}(\partial
D_{u}^{z_{j}})$

of $D_{u}^{z_{j}}$ since it is encircled by $\mathrm{SLE}_{4}$-type curves.

Then, by the results of [AHPS23], the process
$(\mathcal{Y}^{z_{j}}_{u})_{0\leq u<\tau^{z_{j}}}$ is a càdlàg process. Each
positive jump corresponds to the discovery of a loop of the $\mathrm{CLE}_{4}$
and the size of the jump is exactly the quantum boundary length of the loop
(computed using the restriction of $h^{2}$ to the domain encircled by the
loop) while each negative jump at some time $u$ corresponds to the splitting
of $D_{u-}^{z_{j}}$ into two components $D_{u}^{z_{j}}$ and
$D_{u-}^{z_{j}}\setminus D_{u}^{z_{j}}$ and the size of the negative jump
corresponds to the quantum boundary length of $D_{u-}^{z_{j}}\setminus
D_{u}^{z_{j}}$.

The locally largest component $(C_{u})_{u\geq 0}$ is defined as follows. Let

$\tau_{1}=\tau^{z_{1}}\wedge\inf\\{u\geq 0;\
\mathcal{Y}^{z_{1}}_{u}<\mathcal{Y}^{z_{1}}_{u-}/2\\}$

be the first time $u$ at which $z_{1}$ is encircled by a loop or
$\mathcal{Y}^{z_{1}}_{u}<\mathcal{Y}^{z_{1}}_{u-}/2$. For all $u<\tau_{1}$, we
set $C_{u}=D^{z_{1}}_{u}$. We then define by induction a sequence
$(\tau_{k})_{k\geq 1}$ and a subsequence $(z_{j_{k}})_{k\geq 1}$ by setting
for all $k\geq 1$:

  * •

If $\tau_{k}=\tau^{z_{j_{k}}}$, then $\tau_{k}$ is the time at which a loop
$\mathcal{L}$ encircles $z_{j_{k}}$. Let $j_{k+1}$ be the first integer
$j>j_{k}$ for which $z_{j}\in D_{\tau_{k}-}^{z_{j_{k}}}$ and such that $z_{j}$
is not encircled by $\mathcal{L}$.

  * •

If $\tau_{k}=\inf\\{u\geq 0;\
\mathcal{Y}^{z_{j_{k}}}_{u}<\mathcal{Y}^{z_{j_{k}}}_{u-}/2\\}$, then
$\tau_{k}$ is the first splitting time at which $z_{j_{k}}$ lies in the
component with smallest quantum boundary length. Let $j_{k+1}$ be the first
integer $j>j_{k}$ for which $z_{j}\in D_{\tau_{k}-}^{z_{j_{k}}}\setminus
D_{\tau_{k}}^{z_{j_{k}}}$.

In both cases we set

$\tau_{k+1}=\tau^{z_{j_{k+1}}}\wedge\inf\\{u\geq 0;\
\mathcal{Y}^{z_{j_{k+1}}}_{u}<\mathcal{Y}^{z_{j_{k+1}}}_{u-}/2\\}\qquad\text{and}\qquad\forall
u\in[\tau_{k},\tau_{k+1}],\ C_{u}=D^{z_{j_{k+1}}}_{u}.$

Let $z\in\bigcap_{k\geq 0}C_{\tau_{k}}$ (it exists since the intersection can
be written as a decreasing intersection of compact sets). Notice that $z$
cannot be part of a loop otherwise it would not be in the $C_{\tau_{k}}$ for
all $k\geq 0$. Then for all $k\geq 0$, for all $u\leq\tau_{k}$, we have
$C_{u}=\widetilde{D}_{u}^{z}$ and as a result $\tau_{k}\to\infty$ as
$k\to\infty$. Thus, $C_{u}$ is well-defined for all $u\geq 0$. Notice that by
Lemma 8 of [WW13], the process $(C_{u})_{u\geq 0}$ does not depend on the
choice of the enumeration $(z_{j})_{j\geq 1}$. For all $u\geq 0$, we set

$\mathcal{Y}(u)\coloneqq\nu^{2}_{h^{2}_{|C_{u}}}(\partial C_{u}).$

Following Equation (5.2) of [AHPS23] and Remark 2.3 of [MSW22], we define the
quantum natural distance from $\partial\mathbb{D}$ to $C_{u}$ as the limit in
probability

$\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},C_{u})=\lim_{\varepsilon\to
0}\varepsilon\\#\\{s\in[0,u],\
\lvert\Delta\mathcal{Y}_{s}\rvert\in[\varepsilon,2\varepsilon]\\}.$ (2.2)

It is clear that $(\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},C_{u}))_{u\geq
0}$ is non-decreasing. Moreover, it is adapted with respect to the natural
filtration of $\mathcal{Y}$. We denote by $\sigma$ the right-continuous
inverse of $(\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},C_{u}))_{u\geq 0}$ and
for all $t\geq 0$ we set $Y_{t}=\mathcal{Y}_{\sigma(t)}$.

##  3 The convergence $\kappa\uparrow 4$

In this section, we prove Theorem 1.1 for the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration. To this end, we
approximate the $\mathrm{SLE}_{\kappa}^{\langle\mu\rangle}$ explorations of
the $\mathrm{CLE}_{4}$ decorated $2$-quantum disk by explorations of a
$\mathrm{CLE}_{\kappa}$-decorated $\sqrt{\kappa}$-quantum disk as
$\kappa\uparrow 4$, relying on the convergence results of [Leh21] and [AHPS23]
so as to transfer the analogous property of [MSW22]. But first, we recall the
definition on the Carathéodory topology and the
$\mathrm{SLE}_{\kappa}^{\beta}(\kappa-6)$ and
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ explorations.

###  3.1 Carathéodory topology

We first provide some useful results on Carathéodory topology. Let
$z\in\mathbb{C}$. For all $n\geq 0$, let $U_{n}\subset\mathbb{C}$ be a simply
connected domain. Let $U\subset\mathbb{C}$ be a simply connected domain
containing $z$. When $z\in U_{n}$, let $f_{n}$ (resp. $f$) be the unique
conformal map from $\mathbb{D}$ to $U_{n}$ (resp. $U$) such that $f_{n}(0)=z$
(resp. $f(0)=z$) and $f^{\prime}_{n}(0)>0$ (resp. $f^{\prime}(0)>0$). We
recall that $U_{n}$ converges to $U$ as $n\to\infty$ in the Carathéodory
topology if $z\in U_{n}$ for $n$ large enough and $f_{n}$ converges to $f$
uniformly on compact subsets of $\mathbb{D}$. Note that the convergence does
not depend on the choice of $z\in U$ (see e.g. Lemma 2.2 of [BRY19]). The
Carathéodory topology on the space of simply connected open domains of
$\mathbb{C}$ containing $z$ is clearly metrizable. Still, with this
definition, notice that a sequence $U_{n}$ which converges to $U$ in the sense
of Carathéodory may also converge to another simply connected open domain $V$
such that $V\cap U=\emptyset$.

An equivalent definition of Carathéodory convergence (see for instance
Definition 2.1 of [BRY19]), in fact the original one, states that $U_{n}$
converges to $U$ for the Carathéodory topology if and only if for every
compact $K\subset U$, we have $K\subset U_{n}$ for all $n$ large enough and
for every connected open set $V\ni z$, if $V\subset U_{n}$ for infinitely many
$n$ then $V\subset U$. From this equivalent definition, it is not hard to see
that the space of simply connected open domains of $\mathbb{C}$ containing $z$
is separable.

From this equivalent definition, one can see that for all compact $K\subset U$
the restriction of $f_{n}^{-1}$ to $K$ is well defined for $n$ large enough.
Actually, as a consequence of Arzelà-Ascoli’s theorem, the convergence in the
sense of Carathéodory of $U_{n}$ to $U$ implies the uniform convergence of
$f_{n}^{-1}$ towards $f^{-1}$ on every compact subset of $U$. See Proposition
2.1 of [BRY19] for more details.

###  3.2 Background on $\mathrm{CLE}_{\kappa}$ explorations

We recall here the explorations of $\mathrm{CLE}_{\kappa}$ for
$\kappa\in(8/3,4]$ defined in [She09]. We will not rely on the precise
definition of the explorations and we refer to [She09] for details. If
$\beta\in[-1,1]$, we define the skew Bessel process $(X_{u})_{u\geq 0}$ of
dimension $\delta>0$ as the process such that $(|X_{u}|)_{u\geq 0}$ is a
Bessel process of dimension $\delta$ and to each excursion of
$(|X_{u}|)_{u\geq 0}$ we assign an independent sign: the corresponding
excursion of $(X_{u})_{u\geq 0}$ is positive with probability
$\frac{1+\beta}{2}$ and negative with probability $\frac{1-\beta}{2}$. When
$\delta\neq 1$, we associate a process $(Y_{u})_{u\geq 0}$ to the skew Bessel
process $X$ which is characterized by Proposition 3.8 of [She09]. In
particular, it satisfies

$\frac{dY_{u}}{du}=\frac{1}{X_{u}}\qquad\text{on}\qquad\left\\{u\geq 0;\
X_{u}\neq 0\right\\}.$

In the case $\delta=1$, the process $(Y_{u})_{u\geq 0}$ is only defined when
$\beta=0$ (and can be defined as the limit in law when $\delta\to 1$). Let
$\mu\in\mathbb{R}$. One can obtain processes which satisfy the same properties
as $(Y_{u})_{u\geq 0}$ in Proposition 3.8 of [She09] for $\delta=1$ by
replacing $(Y_{u})_{u\geq 0}$ by $(Y_{u}-\mu\ell^{0}_{u})_{u\geq 0}$, where
$(\ell^{0}_{u})_{u\geq 0}$ is the local time at zero of $(X_{u})_{u\geq 0}$.
Let $\delta=3-8/\kappa$. If $\delta\neq 1$, then let $\beta\in[-1,1]$,
otherwise let $\mu\in\mathbb{R}$. Let $(X_{u})_{u\geq 0}$ and $(Y_{u})_{u\geq
0}$ be the two processes defined above associated to $\delta,\beta$ when
$\delta\neq 1$ and to $\delta,\mu$ when $\delta=1$. Let $(O_{u})_{u\geq 0}$
and $(W_{u})_{u\geq 0}$ be the processes given by $O_{0}=W_{0}=0$ and for all
$u\geq 0$,

$O_{u}=-\frac{2}{\sqrt{\kappa}}Y_{u}\qquad\text{and}\qquad
W_{u}=O_{u}+\sqrt{\kappa}X_{u}.$

Then the chordal $\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ for
$\kappa\in(8/3,4)$ and the chordal
$\mathrm{SLE}^{\left\langle\mu\right\rangle}_{4}(-2)$ from $0$ to $\infty$ in
$\mathbb{H}$ are defined as the growing family of closed sets $(K_{u})_{u\geq
0}$ determined by the ODE

$g_{0}(z)=z\qquad\text{and}\qquad\partial_{u}g_{u}(z)=\frac{2}{g_{u}(z)-W_{u}},$

in the sense that the domain of definition of $g_{u}$ is $\mathbb{H}\setminus
K_{u}$. In fact, it was shown in [MSW17] (in Theorem 7.4 for
$\kappa\in(8/3,4)$ and in Proposition 5.3 for $\kappa=4$) that
$\mathbb{H}\setminus K_{u}$ is the unbounded connected component of
$\mathbb{H}\setminus\gamma([0,u])$ for some continuous path $\gamma$. On each
excursion of $(X_{u})_{u\geq 0}$, the path $\gamma$ draws a simple loop. We
also recall from Proposition 3.10 of [She09] that $(K_{u})_{u\geq 0}$
satisfies a “renewal property” stating that conditionally on $W_{u}$ up to a
stopping time $T$ for which $O_{T}=W_{T}$ amost surely, the process
$(W_{T+u}-W_{T})_{u\geq 0}$ has the same law as $(W_{u})_{u\geq 0}$, and a
“conformal Markov property” stating that conditionally on $W_{u}$ up to some
stoppping time $T$ for which a.s. $O_{T}\neq W_{T}$, the growing family given
by the closure of $g_{T}(K_{T+u})$ up to the first time $u\geq 0$ such that
$W_{T+u}=O_{T+u}$ has the same law (up to a time-change) as a chordal
$\mathrm{SLE}_{\kappa}$ in $\mathbb{H}$ from $W_{T}$ to $O_{T}$. A chordal
$\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\left\langle\mu\right\rangle}_{4}(-2)$ when $\kappa=4$) from
$a$ to $b$ which are on the boundary of some simply connected domain
$D\subset\mathbb{C}$ is just the image of the above
$\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\left\langle\mu\right\rangle}_{4}(-2)$ when $\kappa=4$) by a
conformal map sending $0$ to $a$ and $\infty$ to $b$. One can define the trunk
$\eta$ of $\gamma$, a chordal $\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\left\langle\mu\right\rangle}_{4}(-2)$ when $\kappa=4$) as the
path $\gamma$ where we removed the loops. In Theorem 7.4 of [MSW17], the trunk
is shown to be an $\mathrm{SLE}_{\kappa^{\prime}}$-type curve for
$\kappa^{\prime}=16/\kappa$. The path $\gamma$ can be decomposed into the
trunk $\eta$ and the loops it traces along each excursion of $X$, see [MSW17]
for details.

One can also define a radial version of the
$\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ for $\kappa=4$) from $-i$ to $0$ in
the unit disk $\mathbb{D}$, which is characterized by Proposition 3.13 of
[She09] using the chordal $\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ for $\kappa=4$). As explained in
[WW13, p. 11] one can define this radial version relying only on chordal
$\mathrm{SLE}$’s as follows. Consider $\gamma_{1}$ a chordal
$\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ for $\kappa=4$) from $-i$ to $i$,
up to the first time $u_{1}$ at which it finishes to draw a loop intersecting
the circle of radius $1/2$ centred at $-i$. Then, let $F_{1}$ be the conformal
transformation from $D_{u_{1}}^{0}$ to $\mathbb{D}$ that keeps $0$ fixed and
maps $\gamma_{1}(u_{1})$ to $-i$ and we repeat the same procedure again:
growing $\gamma_{2}$ another independent chordal
$\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ for $\kappa=4$) from $-i$ to $i$
until it draws a loop intersecting the circle of radius $1/2$ centred at $-i$,
looking at its preimage by $F_{1}$ etc. The radial
$\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ for $\kappa=4$) targeted at
$z\in\mathbb{D}$ can be defined in a similar way.

By Proposition 3.14 of [She09], one can couple two radial
$\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\left\langle\mu\right\rangle}_{4}(-2)$ when $\kappa=4$),
$\gamma_{1}$ starting from $-i$ and targeted at $z_{1}\in\mathbb{D}$ and
$\gamma_{2}$ from $-i$ to some $z_{2}\in\mathbb{D}$ so that they coincide up
to some time-change until the time that $z_{1}$ and $z_{2}$ fail to lie in the
same connected component of $\mathbb{H}\setminus\gamma_{1}([0,u])$. This
property is called the target-invariance property. By performing this coupling
with targets in a countable dense subset of $\mathbb{D}$, one can thus build
the branching $\mathrm{SLE}^{\beta}_{\kappa}(\kappa-6)$ (or
$\mathrm{SLE}^{\left\langle\mu\right\rangle}_{4}(-2)$ when $\kappa=4$)
starting at $-i$ in $\mathbb{D}$, hence exploring all the loops of a
$\mathrm{CLE}_{\kappa}$. It is also called the exploration tree. The radial
and chordal $\mathrm{SLE}$’s can be seen as “branches” of this exploration
tree.

###  3.3 Carathéodory convergence of the chordal exploration

We now describe an approximation result of the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration in the sense of
Carathéodory based on the results of [Leh21]. Let $\mu\in\mathbb{R}$. If
$\kappa\in(8/3,4)$, let $(\gamma^{\kappa}(u))_{u\geq 0}$ be a chordal
$\mathrm{SLE}^{\mu((4/\kappa)-1)}_{\kappa}(\kappa-6)$ in $\mathbb{D}$ from
$-i$ to $i$. For all $u\geq 0$, if $X(u)=0$, then let $\upsilon^{\kappa}(u)=u$
and otherwise let $\upsilon^{\kappa}(u)$ be the time of the beginning of the
excursion $X$ which is being drawn at time $u$. For all $u\geq 0$, for all
$z\in\mathbb{D}\cap\mathbb{Q}^{2}$, let $D^{z,\kappa}_{u}$ be the connected
component of $\mathbb{D}\setminus\gamma^{\kappa}([0,\upsilon^{\kappa}(u)])$
containing $z$. Let $(\gamma^{4}(u))_{u\geq 0}$ be a chordal
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ in $\mathbb{D}$ from $-i$ to $i$.
Let $\upsilon^{4}(u)$ for $u\geq 0$ be defined as before using the
corresponding driving function. For all $u\geq 0$, for all $z\in\mathbb{D}$,
let $D^{z,4}_{u}$ be the connected component of
$\mathbb{D}\setminus\gamma^{4}([0,\upsilon^{4}(u)])$ containing $z$. Note that
we choose to restrict the definition of the $D^{z,\kappa}_{u}$’s to the times
$\upsilon^{\kappa}(u)$ since the relation with quantum disks that we will
recall in the next subsection is only valid for these times. Then a
consequence of the results of [Leh21] is the following proposition.

######  Proposition 3.1.

We have the convergence in law

$\left(D^{z,\kappa}_{u}\right)_{u\geq
0,z\in\mathbb{Q}^{2}\cap\mathbb{D}}\enskip\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}^{(\mathrm{d})}\enskip(D^{z,4}_{u})_{u\geq
0,z\in\mathbb{Q}^{2}\cap\mathbb{D}}$ (3.1)

in terms of finite-dimensional distributions for the Carathéodory topology.

###### Proof.

We fix $\mu\in\mathbb{R}$. For all $\kappa\in(8/3,4]$, we denote by
$(W^{\kappa}_{u})_{u\geq 0}$, $(O^{\kappa}_{u})_{u\geq 0}$,
$(g^{\kappa}_{u})_{u\geq 0}$ and $(K^{\kappa}_{u})_{u\geq 0}$ the processes
with which the $\mathrm{SLE}^{\mu((4/\kappa)-1)}_{\kappa}(\kappa-6)$ for
$\kappa<4$ and $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ for $\kappa=4$ is
defined as described above. Let $\kappa_{n}$ be an increasing sequence tending
to $4$. By Proposition 3.10 of [Leh21] we have the convergence of the driving
functions:

$(W^{\kappa_{n}},O^{\kappa_{n}})\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{d})}(W^{4},O^{4})$
(3.2)

for the topology of uniform convergence on compact sets. By Skorokhod’s
representation theorem, we may suppose that the above convergence is almost
sure. In particular, by Proposition 4.47 of [Law05], for all $u_{0}>0$, for
all $\varepsilon>0$, we get a.s. the convergence

$g^{\kappa_{n}}\mathop{\longrightarrow}\limits_{n\to\infty}g^{4}$ (3.3)

uniformly on the set $[0,u_{0}]\times\left\\{z\in\mathbb{H};\
d(z,K^{4}_{u_{0}})\geq\varepsilon\right\\}$. In particular, we obtain the
Carathéodory convergence of the unexplored region: for all $u\geq 0$,

$\mathbb{H}\setminus
K^{\kappa_{n}}_{u}\mathop{\longrightarrow}\limits_{n\to\infty}\mathbb{H}\setminus
K^{4}_{u}\qquad\text{and}\qquad\mathbb{H}\setminus
K^{\kappa_{n}}_{\upsilon^{\kappa_{n}}(u)}\mathop{\longrightarrow}\limits_{n\to\infty}\mathbb{H}\setminus
K^{4}_{\upsilon^{4}(u)}.$ (3.4)

We then look into the domains encircled by loops. Let $\varepsilon>0$. For all
$i\geq 1$, for all $\kappa\in(8/3,4]$, let $e_{i,\varepsilon}^{\kappa}$ be the
$i$-th excursion of $W^{\kappa}-O^{\kappa}$ of height larger than
$\varepsilon$ and let $S_{i,\varepsilon}^{\kappa}<T_{i,\varepsilon}^{\kappa}$
be its starting time and end time. First notice that by (3.2) we get readily
the uniform convergence of the excursions $e_{i,\varepsilon}^{\kappa_{n}}$’s
to the $e_{i,\varepsilon}^{4}$’s and the convergence of their endpoints
$S_{i,\varepsilon}^{\kappa_{n}},T_{i,\varepsilon}^{\kappa_{n}}$ to
$S^{4}_{i,\varepsilon},T_{i,\varepsilon}^{4}$ as $n\to\infty$.

Let $u\in(S^{4}_{i,\varepsilon},T^{4}_{i,\varepsilon})$ and $\delta>0$.
Applying Proposition 4.47 of [Law05] to
$(W^{\kappa_{n}}_{(S^{\kappa_{n}}_{i,\varepsilon}+u)\wedge
T^{\kappa_{n}}_{i,\varepsilon}})_{u\geq 0}$ which converges uniformly to
$(W^{4}_{(S^{4}_{i,\varepsilon}+u)\wedge T^{4}_{i,\varepsilon}})_{u\geq 0}$ as
$n\to\infty$, we know that for all $n$ large enough,

$\left((v,z)\mapsto
g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}+v}((g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}})^{-1}(z))\right)\mathop{\longrightarrow}\limits_{n\to\infty}\left((v,z)\mapsto
g^{4}_{S^{4}_{i,\varepsilon}+v}((g^{4}_{S^{4}_{i,\varepsilon}})^{-1}(z))\right)$
(3.5)

uniformly on $[0,u]\times\left\\{z\in\mathbb{H};\
d(z,g^{4}_{S^{4}_{i,\varepsilon}}(K^{4}_{u}))\geq\delta\right\\}$. In
particular, for $n$ large enough,

$\left\\{z\in\mathbb{H};\
d(z,g^{4}_{S^{4}_{i,\varepsilon}}(K^{4}_{u}))\geq\delta\right\\}\subset\mathbb{H}\setminus
g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}(K^{\kappa_{n}}_{u}).$

In other words, for all $n$ large enough, the curve
$(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([S^{\kappa_{n}}_{i,\varepsilon},u])$
is included in the $\delta$-neighbourhood of
$(g^{4}_{S^{4}_{i,\varepsilon}}\circ\gamma^{4})([S^{4}_{i,\varepsilon},u])$.
Besides, let $U$ be the open ball centred at
$g^{4}_{S^{4}_{i,\varepsilon}}(\gamma^{4}(u))$ of radius $\delta$. Then, by
choosing $\delta$ small enough, $U$ is an open connected subset of
$\mathbb{H}$ such that $U\cap(\mathbb{H}\setminus
g^{4}_{S^{4}_{i,\varepsilon}}(K^{4}_{u}))\neq\emptyset$. By the Carathéodory
convergence of $\mathbb{H}\setminus
g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}(K^{\kappa_{n}}_{u})$ towards
$\mathbb{H}\setminus g^{4}_{S^{4}_{i,\varepsilon}}(K^{4}_{u})$ (which stems
from (3.5)), we cannot have $U\subset\mathbb{H}\setminus
g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}(K^{\kappa_{n}}_{u})$
infinitely often, otherwise we would get that
$\gamma^{4}(u)\in\mathbb{H}\setminus K^{4}_{u}$, which is not the case. Thus,
for all $n$ large enough, $U\cap
g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}(K^{\kappa_{n}}_{u})\neq\emptyset$.
By choosing $\delta$ small enough, by connectedness of
$(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([S^{\kappa_{n}}_{i,\varepsilon},u])$,
we obtain that for $n$ large enough,
$(g^{4}_{S^{4}_{i,\varepsilon}}\circ\gamma^{4})([S^{4}_{i,\varepsilon},u])$ is
in the $\delta$-neighbourhood of
$(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([S^{\kappa_{n}}_{i,\varepsilon},u])$.
We have thus proven that for all
$u\in(S^{4}_{i,\varepsilon},T^{4}_{i,\varepsilon})$,

$(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([S^{\kappa_{n}}_{i,\varepsilon},u])\mathop{\longrightarrow}\limits_{n\to\infty}(g^{4}_{S^{4}_{i,\varepsilon}}\circ\gamma^{4})([S^{4}_{i,\varepsilon},u])$
(3.6)

with respect to the Hausdorff distance.

For all $\kappa\in(8/3,4]$, let $R^{\kappa}_{i,\varepsilon}$ be the $i$-th
time at which $W^{\kappa}-O^{\kappa}$ first reaches $\varepsilon$ after
hitting zero. Then $R^{\kappa_{n}}_{i,\varepsilon}$ converges to
$R^{4}_{i,\varepsilon}$ as $n\to\infty$. Moreover, for all $n\geq 0$, by the
conformal Markov property the curve
$g^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}}_{|[R^{\kappa_{n}}_{i,\varepsilon},T^{\kappa_{n}}_{i,\varepsilon}]}$
has the same law as a chordal $\mathrm{SLE}(\kappa_{n})$ in $\mathbb{H}$ from
$g^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}}(\gamma^{\kappa_{n}}(R^{\kappa_{n}}_{i,\varepsilon}))=W^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}}$
to
$g^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}}(\gamma^{\kappa_{n}}(T^{\kappa_{n}}_{i,\varepsilon}))=O^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}}$.
Furthermore, by taking Möbius transformations sending one of these two
endpoints to $\infty$, and by applying Theorem 1.10 of [KS17] (which states
that the chordal $\mathrm{SLE}(\kappa_{n})$ from $0$ to $\infty$ converges
uniformly on compact sets to the $\mathrm{SLE}(4)$ as $n\to\infty$), we obtain
the convergence

$(g^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([R^{\kappa_{n}}_{i,\varepsilon},T^{\kappa_{n}}_{i,\varepsilon}])\mathop{\longrightarrow}\limits_{n\to\infty}(g^{4}_{R^{4}_{i,\varepsilon}}\circ\gamma^{4})([R^{4}_{i,\varepsilon},T^{4}_{i,\varepsilon}])$
(3.7)

for the Hausdorff distance. Besides, by $(\ref{eq cv g excursion})$ and by the
convergence of $R^{\kappa_{n}}_{i,\varepsilon}$ to $R^{4}_{i,\varepsilon}$, we
know that

$g^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}}\circ(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}})^{-1}\mathop{\longrightarrow}\limits_{n\to\infty}g^{4}_{R^{4}_{i,\varepsilon}}\circ(g^{4}_{S^{4}_{i,\varepsilon}})^{-1}$

uniformly on compact subsets of
$\mathbb{H}\setminus(g^{4}_{S^{4}_{i,\varepsilon}}\circ\gamma^{4})([S^{4}_{i,\varepsilon},R^{4}_{i,\varepsilon}])$.
Since the limiting function is a conformal map, it is easy to see that
$g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ(g^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}})^{-1}$
converges pointwise to
$g^{4}_{S^{4}_{i,\varepsilon}}\circ(g^{4}_{R^{4}_{i,\varepsilon}})^{-1}$. By
Montel’s theorem, we deduce that

$g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ(g^{\kappa_{n}}_{R^{\kappa_{n}}_{i,\varepsilon}})^{-1}\mathop{\longrightarrow}\limits_{n\to\infty}g^{4}_{S^{4}_{i,\varepsilon}}\circ(g^{4}_{R^{4}_{i,\varepsilon}})^{-1}$

uniformly on compact subsets of $\mathbb{H}$. Combining this with (3.7), we
get that for all $\delta>0$ small enough,

$(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([R^{\kappa_{n}}_{i,\varepsilon},T^{\kappa_{n}}_{i,\varepsilon}])\cap
V_{\delta}\mathop{\longrightarrow}\limits_{n\to\infty}(g^{4}_{S^{4}_{i,\varepsilon}}\circ\gamma^{4})([R^{4}_{i,\varepsilon},T^{4}_{i,\varepsilon}])\cap
V_{\delta}$ (3.8)

for the Hausdorff distance, where

$V_{\delta}\coloneqq\left\\{z\in\mathbb{H};\
d(z,g^{4}_{S^{4}_{i,\varepsilon}}(\gamma^{4}([S^{4}_{\varepsilon,i},R^{4}_{i,\varepsilon}]))\cup\mathbb{R})>\delta\right\\}.$

We are now in position to prove the Carathéodory convergence of the domain
encircled by the loop. For all $n\in\mathbb{N}$, let
$\mathcal{L}_{n}=g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}(\gamma^{\kappa_{n}}([S^{\kappa_{n}}_{\varepsilon,i},T^{\kappa_{n}}_{\varepsilon,i}]))$
be the loop drawn by $u\mapsto
W^{\kappa_{n}}_{(S^{\kappa_{n}}_{i,\varepsilon}+u)\wedge
T^{\kappa_{n}}_{i,\varepsilon}}$ and let
$\mathcal{L}=g^{4}_{S^{4}_{i,\varepsilon}}(\gamma^{4}([S^{4}_{\varepsilon,i},T^{4}_{\varepsilon,i}]))$
be the loop drawn by $u\mapsto W^{4}_{(S^{4}_{i,\varepsilon}+u)\wedge
T^{4}_{i,\varepsilon}}$. By the convergence of $g^{\kappa_{n}}$ and
$S_{i,\varepsilon}^{\kappa_{n}}$ as $n\to\infty$, to prove the Carathéodory
convergence of the open domain encircled by the loop
$\gamma^{\kappa_{n}}([S^{\kappa_{n}}_{\varepsilon,i},T^{\kappa_{n}}_{\varepsilon,i}])$
towards the domain encircled by
$\gamma^{4}([S^{4}_{\varepsilon,i},T^{4}_{\varepsilon,i}])$, it suffices to
prove the Carathéodory convergence of the domain $\mathcal{B}_{n}$ encircled
by $\mathcal{L}_{n}$ towards the domain $\mathcal{B}$ encircled by
$\mathcal{L}$.

Figure 2: Illustration of the proof of the Carathéodory convergence of the
domain encircled by
$\mathcal{L}_{n}=g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}(\gamma^{\kappa_{n}}([S^{\kappa_{n}}_{\varepsilon,i},T^{\kappa_{n}}_{\varepsilon,i}]))$,
in green dashes, towards the domain encircled by
$\mathcal{L}=g^{4}_{S^{4}_{i,\varepsilon}}(\gamma^{4}([S^{4}_{\varepsilon,i},T^{4}_{\varepsilon,i}]))$,
in black. The complement of $V_{\delta}$ is in orange. A compact $\mathcal{K}$
encircled by $\mathcal{L}$ is drawn in blue.

Let $\mathcal{K}$ be a compact subset of the open domain $\mathcal{B}$. Then
we may take $\delta>0$ small enough such that $\mathcal{K}\subset V_{\delta}$,
so that by (3.8), we obtain that
$\mathcal{K}\cap(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([R^{\kappa_{n}}_{i,\varepsilon},T^{\kappa_{n}}_{i,\varepsilon}])=\emptyset$
for $n$ large enough. Moreover, by (3.6), we have that for $u>0$ small enough,
for all $n$ large enough, we have
$(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([S^{\kappa_{n}}_{i,\varepsilon},R^{\kappa_{n}}_{i,\varepsilon}+u])\subset
V_{\delta}$. Therefore, by the convergence of $R^{\kappa_{n}}_{i,\varepsilon}$
to $R^{4}_{i,\varepsilon}$ we deduce that for all $n$ large enough,
$\mathcal{K}\cap\mathcal{L}_{n}=\emptyset$. One can actually see that
$\mathcal{K}$ is encircled by $\mathcal{L}_{n}$ for $n$ large enough using
that the endpoints of the path
$(g^{\kappa_{n}}_{S^{\kappa_{n}}_{i,\varepsilon}}\circ\gamma^{\kappa_{n}})([R^{\kappa_{n}}_{i,\varepsilon},T^{\kappa_{n}}_{i,\varepsilon}])$
are in $\mathbb{H}\setminus V_{\delta}$ and applying (3.8) once more (see
Figure 2).

Let $z\in\mathcal{B}$. To end the proof of the Carathéodory convergence of
$\mathcal{B}_{n}$ towards $\mathcal{B}$, it suffices to show that if $U$ is a
connected open set containing $z$ such that $U\subset\mathcal{B}_{n}$
infinitely often, then $U\subset\mathcal{B}$. Since $U$ and $\mathcal{B}$ are
open it is enough to see that $U\subset\overline{\mathcal{B}}$. Assume by
contradiction that there exists $w\in U\setminus\overline{\mathcal{B}}$. Take
$\delta>0$ small enough so that $B(w,\delta)\subset
U\setminus\overline{\mathcal{B}}$. In particular, $w\in V_{\delta}$. As a
result, due to the Hausdorff convergences (3.6) and (3.8), the point $w$ is
not encircled by $\mathcal{L}_{n}$ for $n$ large enough, hence a
contradiction. We have thus proven that

$\mathcal{B}_{n}\mathop{\longrightarrow}\limits_{n\to\infty}\mathcal{B}$

in the sense of Carathéodory. This entails the convergence of the domains
encircled by loops
$\gamma^{\kappa_{n}}([S^{\kappa_{n}}_{\varepsilon,i},T^{\kappa_{n}}_{\varepsilon,i}])$
for $i\geq 1$ towards the domain encircled by
$\gamma^{4}([S^{4}_{\varepsilon,i},T^{4}_{\varepsilon,i}])$. By letting
$\varepsilon\to 0$, we get the Carathéodory convergence of all the domains
encircled by loops.

Finally, we focus on the domains which are cut out by the exploration. Let
$u\geq 0$. Let $x\in\mathbb{R}$. For all $n$, let $\zeta^{n}_{x}$ be the first
time that $x$ is swallowed by
$(g^{\kappa_{n}}_{u}({K}^{\kappa_{n}}_{v+u}))_{v\geq 0}$ and let $\zeta_{x}$
be the first time that $x$ is swallowed by $(g^{4}_{u}(K^{4}_{u+v}))_{v\geq
0}$. For all $n\geq 1$, let $C^{n}_{x}$ be the interior of
$g^{\kappa_{n}}_{u}({K}^{\kappa_{n}}_{u+\zeta^{n}_{x}})\setminus
g^{\kappa_{n}}_{u}({K}^{\kappa_{n}}_{u+\zeta^{n}_{x}-})$ and let $C_{x}$ be
the interior of $g^{4}_{u}({K}^{4}_{u+\zeta_{x}})\setminus
g^{4}_{u}({K}^{4}_{u+\zeta_{x}-})$. By the proof of Theorem 1.2 of [Leh21] and
by another application of Skorokhod’s representation theorem, we know that
$\zeta_{x}$ is the almost sure limit as $n\to\infty$ of $\zeta^{n}_{x}$. To be
more precise, the proof of Theorem 1.2 of [Leh21] gives the convergence of
$\zeta^{n}_{1}$, while the convergence of $\zeta^{n}_{x}$ follows by scale-
invariance and reflexion-invariance. As a consequence, by considering chordal
$\mathrm{SLE}_{\kappa_{n}}^{\mu((4/\kappa_{n})-1)}(\kappa_{n}-6)$’s and an
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ targeted at $x$ and coupled with
the chordal
$\mathrm{SLE}_{\kappa_{n}}^{\mu((4/\kappa_{n})-1)}(\kappa_{n}-6)$’s and
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ targeted at $\infty$ until $x$ is
separated from $\infty$ and by applying the convergence (3.3), we obtain the
a.s. convergence

$C^{n}_{x}\mathop{\longrightarrow}\limits_{n\to\infty}C_{x}$

as $n\to\infty$ in the sense of Carathéodory. By (3.4), we deduce the
convergence of the cut out domain

$(g_{u}^{\kappa_{n}})^{-1}(C^{n}_{x})\mathop{\longrightarrow}\limits_{n\to\infty}(g_{u}^{4})^{-1}(C_{x})$

in the sense of Carathéodory. To conclude we just need to check that any cut
out domain can be written in the form $(g_{u}^{4})^{-1}(C_{x})$ for some
$u\geq 0$ and $x\in\mathbb{R}$. This is straightforward by taking $u\geq 0$
such that the frontier of the image by $g_{u}^{4}$ of the cut out domain has a
non-trivial intersection with $\mathbb{R}$.

∎

###  3.4  The $\sqrt{\kappa}$-quantum disks and $\mathrm{CLE}_{\kappa}$

Here, we describe the results of [MSW22] that we want to transfer to
$\kappa=4$. Recall Subsection 2.2. Let $\kappa\in(8/3,4]$. Let
$(\mathbb{D},h^{\sqrt{\kappa}})$ be a parametrization of a unit boundary
$\sqrt{\kappa}$-quantum disk. By Lemma 4.5 of [AHPS23], we know that

$(h^{\sqrt{\kappa}},\mu_{h^{\sqrt{\kappa}}}^{\sqrt{\kappa}},\nu_{h^{\sqrt{\kappa}}}^{\sqrt{\kappa}})\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}(h^{2},\mu^{2}_{h^{2}},\nu^{2}_{h^{2}}),$ (3.9)

where the convergence of the first coordinate holds in the space
$H^{-1}_{\mathrm{loc}}(\mathbb{D})$ while the convergences of the other
coordinates hold with respect to the weak topology of measures. In what
follows, when we speak of the convergence of some quantum disks, it always
means the convergence of the GFF and of the quantum area and boundary length
measures associated to the parametrization by the unit disk as in the above
equation.

When we run an independent chordal $\mathrm{SLE}_{\kappa}^{\beta}(\kappa-6)$
(or $\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ for $\kappa=4$) $\gamma$ on a
$\sqrt{\kappa}$-quantum disk $(\mathbb{D},h^{\sqrt{\kappa}})$, the trunk of
$\gamma$ can be parametrized by its quantum length since it is an
$\mathrm{SLE}_{\kappa^{\prime}}$-type process by Theorem 7.4 of [MSW17] and
Remark 2.3 of [MSW22]. Notice that this quantum length is defined up to some
multiplicative constant. Besides, we can define the quantum boundary length of
$\mathrm{SLE}_{\kappa}$ type curves, such as the loops drawn by $\gamma$ or
the boundary of the domains cut out by $\gamma$, via a quantum boundary length
measure induced by $h^{\sqrt{\kappa}}$, using the quantum zipper properties
first discovered in [She16]. The quantum boundary length measure on such a
domain $D$ is equal to $\nu^{\sqrt{\kappa}}_{h^{\sqrt{\kappa}}_{|D}}$. We will
see in the next proposition that they correspond exactly to the boundary
length measures of some quantum disks.

We then state a result of [MSW22] for $\kappa\in(8/3,4)$, which gives the law
of the quantum boundary length of the unexplored region and a Markov property
in terms of quantum disks. Let $L_{0}$ (resp. $R_{0}$) be the quantum boundary
length along the clockwise (resp. counterclockwise) segment from $-i$ to $i$
in $\partial\mathbb{D}$. Let $\beta\in(-1,1)$. Let $\gamma$ be a chordal
$\mathrm{SLE}_{\kappa}^{\beta}(\kappa-6)$ from $-i$ to $i$ taken independently
from the quantum disk $(\mathbb{D},h^{\sqrt{\kappa}})$. We parametrize its
trunk $\eta$ by its quantum length. Let $T^{\kappa,\beta}$ be the total
quantum length of the trunk. For each $t$ (corresponding here to the quantum
length of the trunk), let $E_{t}$ be the event that the trunk has not yet
attained $i$ before having quantum length $t$, i.e.
$E_{t}=\\{T^{\kappa,\beta}>t\\}$. For all $t<T^{\kappa,\beta}$, we set
$L^{\kappa,\beta}_{t}$ and $R^{\kappa,\beta}_{t}$ to be the left and right
quantum boundary lengths of the unexplored region $\mathcal{D}^{\kappa}_{t}$
($\mathcal{D}^{\kappa}_{t}$ is the connected component of
$\mathbb{D}\setminus\gamma([0,\tau_{t}])$ containing $i$ in its closure, where
$\tau_{t}$ is the time for $\gamma$ at which the quantum length of the trunk
$\eta$ reaches $t$). More precisely, $L^{\kappa,\beta}_{t}$ is the quantum
boundary length of the arc of $\partial\mathcal{D}^{\kappa}_{t}$ going from
the tip of the trunk to $i$ in the clockwise direction and
$R^{\kappa,\beta}_{t}$ is the quantum boundary length of the arc of
$\partial\mathcal{D}^{\kappa}_{t}$ going from the tip of the trunk to $i$ in
the anti-clockwise direction. Note that for all $t<T^{\kappa,\beta}$, we have
$L^{\kappa,\beta}_{t}>0$ and $R^{\kappa,\beta}_{t}>0$. For all $t\geq
T^{\kappa,\beta}$ we set $L^{\kappa,\beta}_{t}=R^{\kappa,\beta}_{t}=0$. If we
write $S^{\kappa,\beta}_{t}=L^{\kappa,\beta}_{t}+R^{\kappa,\beta}_{t}$ for all
$t\geq 0$, then $S^{\kappa,\beta}_{t}$ is the boundary length of
$\mathcal{D}^{\kappa}_{t}$ for all $t<T^{\kappa,\beta}$. Let $a=1+4/\kappa$.
Let $\widetilde{L}^{\beta,a}$ (resp. $\widetilde{R}^{\beta,a}$) be the
$(a-1)$-stable Lévy process of Lévy measure
$\Lambda_{L}^{\beta,a}\coloneqq\frac{1-\beta}{2}\cos(a\pi)(dx/x^{a}){\bf
1}_{x>0}+\frac{1}{2}(dx/|x|^{a}){\bf 1}_{x<0}$ (resp.
$\Lambda_{R}^{\beta,a}\coloneqq\frac{1+\beta}{2}\cos(a\pi)(dx/x^{a}){\bf
1}_{x>0}+\frac{1}{2}(dx/|x|^{a}){\bf 1}_{x<0}$). The two processes
$\widetilde{L}^{\beta,a}$ and $\widetilde{R}^{\beta,a}$ are taken independent.
Let $\widetilde{S}^{a}=\widetilde{L}^{\beta,a}+\widetilde{R}^{\beta,a}$.

######  Proposition 3.2.

(Proposition 5.1 of [MSW22]) Up to some linear time-change, the process
$(L^{\kappa,\beta}_{t},R^{\kappa,\beta}_{t})_{t\geq 0}$ is a Doob
$h$-transform of the process
$(\widetilde{L}^{\beta,a},\widetilde{R}^{\beta,a})$, in the sense that there
exists a constant $c>0$ such that for all $t>0$, for all bounded measurable
function $F:\mathbb{D}([0,t],\mathbb{R})^{2}\to\mathbb{R}$, for all $x,y>0$,

$\displaystyle\mathbb{E}$
$\displaystyle\left(\left.F\left((L^{\kappa,\beta}_{s},R^{\kappa,\beta}_{s})_{0\leq
s\leq t}\right){\bf
1}_{E_{t}}\right|(L^{\kappa,\beta}_{0},R^{\kappa,\beta}_{0})=(x,y)\right)$
$\displaystyle=\mathbb{E}_{x,y}\left(\left(\frac{x+y}{\widetilde{S}^{a}_{ct}}\right)^{a}F\left((\widetilde{L}^{\beta,a}_{cs},\widetilde{R}^{\beta,a}_{cs})_{0\leq
s\leq t}\right){\bf 1}_{\forall s\in[0,t],\
\widetilde{L}^{\beta,a}_{cs},\widetilde{R}^{\beta,a}_{cs}>0}\right),$

where the Lévy process $\widetilde{L}^{\beta,a}$ (resp.
$\widetilde{R}^{\beta,a}$) starts at $x$ (resp. $y$). Moreover, conditionally
on the quantum boundary lengths
$(L^{\kappa,\beta}_{s},R^{\kappa,\beta}_{s})_{0\leq s\leq t}$, the quantum
surfaces which are cut out to the left, cut out to the right, inside the left
loops or inside the right loops are independent $\sqrt{\kappa}$-quantum disks
of quantum boundary lengths corresponding to the sizes of the positive and
negative jumps of $L^{\kappa,\beta}$ and $R^{\kappa,\beta}$, while
$(\mathcal{D}^{\kappa}_{t},h^{\sqrt{\kappa}}_{|\mathcal{D}_{t}})$ is an
independent $\sqrt{\kappa}$-quantum disk of quantum boundary length
$S^{\kappa,\beta}_{t}$.

When we parametrize the trunk by its quantum length, the parametrization is
chosen up to a multiplicative constant, so that for our purpose we choose to
take $c=1$ in the above proposition. This proposition will be a key ingredient
to obtain analogous statements for $\kappa=4$.

###  3.5 Joint convergence of the $\mathrm{CLE}_{\kappa}$ and the
$\sqrt{\kappa}$-quantum disks as $\kappa\uparrow 4$

In this subsection, we combine the convergences (3.1) and (3.9) to extend
Proposition 3.2 to the case $\kappa=4$ for the chordal
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$. Then the statement of Theorem 1.1
for the $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration will be a
direct corollary.

Let $(\mathbb{D},h^{2})$ be a unit boundary length $2$-quantum disk. Let
$L_{0}^{4}$ (resp. $R_{0}^{4}$) be the quantum boundary length along the
clockwise (resp. counterclockwise) segment from $-i$ to $i$ in
$\partial\mathbb{D}$. Let $\mu\in\mathbb{R}$. Let $\gamma$ be an independent
chordal $\mathrm{SLE}_{4}^{\mu}(-2)$ on $\mathbb{D}$ from $-i$ to $i$. We
again parametrize the trunk by its quantum length and we denote by $T^{4,\mu}$
its total quantum length. For each $t\geq 0$ (corresponding here to the
quantum length of the trunk), let $E_{t}$ be the event that the trunk has not
yet attained $i$ before having quantum length $t$, i.e.
$E_{t}=\\{T^{4,\mu}>t\\}$. For all $t<T^{4,\mu}$, we set $L_{t}^{4,\mu}$ and
$R_{t}^{4,\mu}$ to be the left and right quantum boundary lengths of the
remaining to be explored region $\mathcal{D}_{t}^{4}$ ($\mathcal{D}^{4}_{t}$
is the connected component of $\mathbb{D}\setminus\gamma([0,\tau_{t}])$
containing $i$ in its closure, where $\tau_{t}$ is the time for $\gamma$ at
which the quantum length of the trunk reaches $t$). Note that for all
$t<T^{4,\mu}$, we have $L_{t}^{4,\mu}>0$ and $R_{t}^{4,\mu}>0$. For all $t\geq
T^{4,\mu}$, we set $L_{t}^{4,\mu}=R_{t}^{4,\mu}=0$. If we write
$S^{4,\mu}_{t}=L^{4,\mu}_{t}+R^{4,\mu}_{t}$ for all $t\geq 0$, then
$S^{4,\mu}_{t}$ is the quantum boundary length of $\mathcal{D}^{4}_{t}$ when
$t<T^{4,\mu}$. Let $\widetilde{L}^{2}$ and $\widetilde{R}^{2}$ be two
independent symmetric Cauchy processes of Lévy measure
$\frac{1}{2}(dx/x^{2}){\bf 1}_{x\neq 0}$. Let
$\widetilde{S}^{2}=\widetilde{L}^{2}+\widetilde{R}^{2}$.

######  Proposition 3.3.

The process $(S^{4,\mu}_{t})_{t\geq 0}$ satisfies the following absolute
continuity relation: there exists a constant $c>0$ such that for all $t>0$,
for all bounded measurable function
$F:\mathbb{D}([0,t],\mathbb{R})^{2}\to\mathbb{R}$, for all $x,y>0$,

$\mathbb{E}\left(\left.F\left((S^{4,\mu}_{s})_{0\leq s\leq t}\right){\bf
1}_{E_{t}}\right|(L^{4}_{0},R^{4}_{0})=(x,y)\right)=\mathbb{E}_{x,y}\left(\left(\frac{x+y}{\widetilde{S}^{2}_{ct}}\right)^{2}F\left((\widetilde{S}^{2}_{cs})_{0\leq
s\leq t}\right){\bf 1}_{\forall s\in[0,t],\
\widetilde{L}^{2}_{cs},\widetilde{R}^{2}_{cs}>0}\right),$ (3.10)

where the Cauchy process $\widetilde{L}^{2}$ (resp. $\widetilde{R}^{2}$)
starts at $x$ (resp. $y$). Moreover, conditionally on $(S^{4,\mu}_{s})_{0\leq
s\leq t}$, the quantum surfaces which are cut out or inside discovered loops
before time $t$ are independent quantum disks of quantum boundary length
corresponding to the sizes of the jumps of $(S^{4,\mu}_{s})_{0\leq s\leq t}$,
while $(\mathcal{D}^{4}_{t},h^{2}_{|\mathcal{D}^{4}_{t}})$ is an independent
quantum disk of quantum boundary length $S^{4,\mu}_{t}$.

Here we also choose the multiplicative constant for the quantum length of the
trunk so that $c=1$ to be consistent with our choice of $c$ for Proposition
3.2. Note that the quantum length of the trunk of the chordal
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ can also be defined using the
quantum boundary length measure since it is an $\mathrm{SLE}_{4}$-type curve,
so that there should be a choice for $c$ such that the quantum length of the
trunk and the quantum boundary lengths of the cut out and encircled domains
are defined using the same multiplicative constant. However, our approach does
not enable us to identify this constant.

######  Remark 3.4.

The above proposition is not exatly the proper counterpart of Proposition 3.2
insofar as we do not describe the left and right quantum boundary lengths of
the unexplored region, but only the whole quantum boundary length. To get the
evolution of the left and right quantum boundary lengths, one has to identify
the quantum surfaces which are cut out or encircled by a loop on the right
side and on the left side of the trunk. For the cut out domains one can see
from Proposition 3.1 that the domains which are cut out on the right (resp.
left) side of the trunk are limits of domains which are cut out on the right
(resp. left) side of the trunk. For the loops, one has to look at the
direction the trunk takes after hitting the loops, using the proof of
Proposition 3.1 once more. We do not go further in that direction since it is
not the main purpose of this work.

Using Proposition 3.3, one can deduce Theorem 1.1 for the
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ exploration:

###### Proof of Theorem 1.1 for $(Y^{\mu}_{t})_{t\geq 0}$ using Proposition
3.3.

Using the $\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ exploration tree, one can
define the branch which follows the locally largest component in terms of
quantum boundary length. It can be defined through an approximation as in
Subsection 5.3 of [MSW22]: let $\varepsilon>0$, for each time of the form
$\varepsilon n$ for $n\in\mathbb{N}$, one updates the target point for the
chordal exploration and chooses the point such that the quantum boundary
lengths on both sides of the unexplored region are equal. When $\varepsilon\to
0$, this exploration approximates the branch of the locally largest component.
Denote by $(Y^{\mu}_{t})_{t\geq 0}$ the quantum boundary length of the locally
largest component parametrized by the quantum length of the trunk (which is
given by a concatenation of parts of trunks of chordal
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$’s). Then, using exactly the same
reasoning as in Subsection 5.3 of [MSW22], we deduce from Proposition 3.3 the
analogous result for the branch of the locally largest component, which is no
more than the statement of Theorem 1.1 for $(Y^{\mu}_{t})_{t\geq 0}$. ∎

All this subsection is then devoted to the proof of Proposition 3.3. Let
$(\mathbb{D},h^{\sqrt{\kappa}})$ be a parametrization of a unit-boundary
$\sqrt{\kappa}$-quantum disk. We run an independent chordal
$\mathrm{SLE}^{\mu((4/\kappa)-1)}_{\kappa}(\kappa-6)$ in $\mathbb{D}$ from
$-i$ to $i$. We will prove the above result by letting $\kappa\uparrow 4$. The
first step towards the above proposition is the convergence in law of the
processes involved in Proposition 3.2. Let
$(\overline{L}^{4}_{0},\overline{R}^{4}_{0})=(L^{4}_{0},R^{4}_{0})$ (see the
beginning of Subsection 3.5). Next, conditionnally on
$(\overline{L}^{4}_{0},\overline{R}^{4}_{0})=(x,y)$, define a process
$(\overline{L}_{t}^{4},\overline{R}_{t}^{4})_{t\geq 0}$ via the same Doob
$h$-transform as in Proposition 3.2 for $\kappa=4$, $a=1+4/\kappa=2$ and
$\beta=0$, replacing $E_{t}$ by
$\\{\overline{L}_{t}^{4},\overline{R}_{t}^{4}>0\\}$ and setting
$\overline{L}_{t}^{4}=\overline{R}_{t}^{4}=0$ on the complement.

######  Lemma 3.5.

We have the convergence

$(L_{t}^{\kappa,\mu((4/\kappa)-1)},R_{t}^{\kappa,\mu((4/\kappa)-1)})_{t\geq
0}\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}^{(\mathrm{d})}(\overline{L}_{t}^{4},\overline{R}_{t}^{4})_{t\geq 0}$ (3.11)

in the sense of Skorokhod.

###### Proof.

The convergence of
$(L_{0}^{\kappa,\mu((4/\kappa)-1},R^{\kappa,\mu((4/\kappa)-1}_{0})$ stems from
the convergence of the quantum boundary length measure in (3.9). Besides, one
can see easily see that for all continuous bounded function
$\varphi:\mathbb{R}\rightarrow\mathbb{R}$ vanishing on a neighbourhood of
zero, we have

$\int_{\mathbb{R}}\varphi(x)\Lambda_{L}^{\mu((4/\kappa)-1),(4/\kappa)+1}(dx)\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}\int_{\mathbb{R}}\varphi(x)\frac{dx}{2x^{2}}$

and

$\int_{\mathbb{R}}\varphi(x)\Lambda_{R}^{\mu((4/\kappa)-1),(4/\kappa)+1}(dx)\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}\int_{\mathbb{R}}\varphi(x)\frac{dx}{2x^{2}}.$

It is classical (see e.g. [JS87] VII.3.6) that this implies the convergence of
Lévy processes

$\left(\widetilde{L}^{\mu((4/\kappa)-1),(4/\kappa)+1}_{t}\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}^{(\mathrm{d})}\left(\widetilde{L}^{2}_{t}\right)_{t\geq
0}\enskip\text{and}\enskip\left(\widetilde{R}^{\mu((4/\kappa)-1),(4/\kappa)+1}_{t}\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}^{(\mathrm{d})}\left(\widetilde{R}^{2}_{t}\right)_{t\geq 0},$

for the $J_{1}$ Skorokhod topology. We then reason by absolute continuity. Let
$t,\varepsilon>0$. By Proposition 3.2, if $F$ is a bounded continuous function
on $\mathbb{D}([0,t],\mathbb{R})^{2}$ vanishing on $\\{(f,g);\ \exists
s\in[0,t],f(s)<\varepsilon\text{ or }g(s)<\varepsilon\\}$, we have

$\displaystyle\mathbb{E}$
$\displaystyle\left(\left.F\left((L^{\kappa,\beta}_{s},R^{\kappa,\beta}_{s})_{0\leq
s\leq t}\right){\bf
1}_{E_{t}}\right|(L^{\kappa,\beta}_{0},R^{\kappa,\beta}_{0})=(x,y)\right)$
$\displaystyle=\mathbb{E}_{x,y}\left(\left(\frac{x+y}{\widetilde{S}^{a}_{ct}}\right)^{a}F\left((\widetilde{L}_{cs}^{\beta,a},\widetilde{R}^{\beta,a}_{cs})_{0\leq
s\leq t}\right){\bf 1}_{\forall s\in[0,t],\
\widetilde{L}^{\beta,a}_{cs},\widetilde{R}^{\beta,a}_{cs}>0}\right),$

where we have set $\beta=\mu((4/\kappa)-1)$ and $a=(4/\kappa)+1$. One
concludes by dominated convergence. ∎

Another ingredient is the fact that the convergences (3.1) and (3.9) imply the
convergence of the underlying fields and quantum area measure of the cut out,
encircled and unexplored surfaces:

######  Lemma 3.6.

If for all $z\in\mathbb{D}\cap\mathbb{Q}^{2}$, the function $f^{z,\kappa}_{u}$
is the unique conformal map from $\mathbb{D}$ to $D^{z,\kappa}_{u}$ which
sends $0$ to $z$ such that $(f^{z,\kappa}_{u})^{\prime}(0)>0$, then jointly
with (3.1) and (3.9),

$\\!\\!\\!\begin{pmatrix}f^{z,\kappa}_{u}\\\ h^{\sqrt{\kappa}}\circ
f^{z,\kappa}_{u}+Q_{\sqrt{\kappa}}\log|(f^{z,\kappa}_{u})^{\prime}|\\\
\mu^{\sqrt{\kappa}}_{h^{\sqrt{\kappa}}}\circ f^{z,\kappa}_{u}\\\
\end{pmatrix}_{u\geq
0,z\in\mathbb{Q}^{2}\cap\mathbb{D}}\\!\\!\\!\\!\\!\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}^{(\mathrm{d})}\qquad\begin{pmatrix}f^{z,4}_{u}\\\ h^{2}\circ
f^{z,4}_{u}+Q_{2}\log|(f^{z,4}_{u})^{\prime}|\\\ \mu^{2}_{h^{2}}\circ
f^{z,4}_{u}\\\ \end{pmatrix}_{u\geq 0,z\in\mathbb{Q}^{2}\cap\mathbb{D}},$
(3.12)

in terms of finite dimensional distributions, where the first coordinate is
endowed with the topology of the uniform convergence on compact sets, the
second one with the topology of the convergence of distributions and the third
one with the weak convergence of measures on $\mathbb{D}$.

###### Proof.

By Skorokhod’s representation theorem we can assume that the convergences
(3.1) and (3.9) hold a.s.. Let $z\in\mathbb{D}\cap\mathbb{Q}^{2}$ and $u\geq
0$. Let $f_{\kappa}=f^{z,\kappa}_{u}$. Then, from the Carathéodory convergence
of $D^{z,\kappa}_{u}$ as $\kappa\uparrow 4$, we know that $f_{\kappa}$
converges uniformly on compact sets towards $f_{4}$. We also know that for
every compact subset $K\subset D^{z,4}_{u}$, the conformal map
$f^{-1}_{\kappa}$ converges uniformly on $K$ towards $f_{4}^{-1}$ as
$\kappa\uparrow 4$. From Cauchy’s formula, we know that the derivatives of
$f^{-1}_{\kappa}$ also converge uniformly on $K$ to the derivatives of
$f_{4}^{-1}$. Let us first show that $h^{\sqrt{\kappa}}\circ f_{\kappa}$
converges to $h^{2}\circ f_{4}$ for the weak-$*$ topology. Let
$\varphi:\mathbb{D}\rightarrow\mathbb{R}$ be a $\mathcal{C}^{\infty}$ function
with compact support. Since $f_{\kappa}$ converges uniformly on compact sets
to $f_{4}$ as $\kappa\uparrow 4$, there exists some relatively compact open
subset $U\subset\mathbb{D}$ such that for $\kappa$ close enough to $4$, one
has $\operatorname{supp}(\varphi\circ
f_{\kappa}^{-1})=f_{\kappa}(\operatorname{supp}\varphi)\subset U.$ We thus
have

$\displaystyle|\langle h^{\sqrt{\kappa}}\circ f_{\kappa}-h^{2}\circ
f_{4},\varphi\rangle|=$ $\displaystyle|\langle h^{\sqrt{\kappa}},(\varphi\circ
f_{\kappa}^{-1})\cdot|(f_{\kappa}^{-1})^{\prime}|^{2}\rangle-\langle
h^{2},(\varphi\circ f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\rangle|$
$\displaystyle=$ $\displaystyle|\langle h^{\sqrt{\kappa}},(\varphi\circ
f_{\kappa}^{-1})\cdot|(f_{\kappa}^{-1})^{\prime}|^{2}-(\varphi\circ
f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\rangle$ $\displaystyle+\langle
h^{\sqrt{\kappa}}-h^{2},(\varphi\circ
f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\rangle|$ $\displaystyle\leq$
$\displaystyle\|h^{\sqrt{\kappa}}\|_{H^{-1}(U)}\|(\varphi\circ
f_{\kappa}^{-1})\cdot|(f_{\kappa}^{-1})^{\prime}|^{2}-(\varphi\circ
f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\|_{H^{1}(U)}$
$\displaystyle+\|h^{\sqrt{\kappa}}-h^{2}\|_{H^{-1}(U)}\|(\varphi\circ
f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\|_{H^{1}(U)}.$

But we know that $\|h^{\sqrt{\kappa}}-h^{2}\|_{H^{-1}(U)}\rightarrow 0$ as
$\kappa\uparrow 4$. Thus, to prove that $\langle h^{\sqrt{\kappa}}\circ
f_{\kappa}-h^{2}\circ f_{4},\varphi\rangle\to 0$, it remains to check that
$\|(\varphi\circ
f_{\kappa}^{-1})\cdot|(f_{\kappa}^{-1})^{\prime}|^{2}-(\varphi\circ
f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\|_{H^{1}(\mathbb{D})}\to 0$, in
other words that

$\|(\varphi\circ
f_{\kappa}^{-1})\cdot|(f_{\kappa}^{-1})^{\prime}|^{2}-(\varphi\circ
f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\|_{\mathrm{L}^{2}(\mathbb{D})}+\left\|\nabla\left((\varphi\circ
f_{\kappa}^{-1})\cdot|(f_{\kappa}^{-1})^{\prime}|^{2}-(\varphi\circ
f_{4}^{-1})\cdot|(f_{4}^{-1})^{\prime}|^{2}\right)\right\|_{\mathrm{L}^{2}(\mathbb{D})}$

tends to zero as $\kappa\uparrow 4$, since $\varphi$ is $\mathcal{C}^{\infty}$
with compact support and $f_{\kappa}^{-1}$ and its derivatives converge
uniformly on compact sets to $f_{4}^{-1}$ and its derivatives. Thus
$h^{\sqrt{\kappa}}\circ f_{\kappa}$ converges to $h^{2}\circ f_{4}$ for the
weak-$*$ topology, so $h^{\sqrt{\kappa}}\circ
f_{\kappa}+Q_{\sqrt{\kappa}}\log|f_{\kappa}^{\prime}|$ converges to
$h^{2}\circ f_{4}+Q_{2}\log|f_{4}^{\prime}|$ for the weak-$*$ topology. The
third convergence of (3.12) comes easily from the uniform convergence on
compact sets of $f_{\kappa}^{-1}$ and from the convergence of
$\mu^{\sqrt{\kappa}}_{h^{\sqrt{\kappa}}}$. ∎

Now let us prove Proposition 3.3 using the above lemmas.

###### Proof of Proposition 3.3.

We first introduce some notation. For all $i\geq 1$, let
$\tau_{\varepsilon,i}^{\kappa}$ be the time of the $i$-th jump of
$(S_{t}^{\kappa,\mu((4/\kappa)-1)})_{t\geq 0}$ of size larger than
$\varepsilon$. Then we know that $\tau_{\varepsilon,i}^{\kappa}$ and the size
of the jump at time $\tau_{\varepsilon,i}^{\kappa}$ converge in distribution
by (3.11) but we do not know yet that the limit corresponds to a jump of
$S^{4,\mu}$. Henceforth, in the rest of the subsection, we denote by
$\mathrm{ql}^{\kappa}(u)$ the quantum length of the trunk at time $u$ for the
original parametrization of the chordal
$\mathrm{SLE}^{\mu((4/\kappa)-1)}_{\kappa}(\kappa-6)$. The function
$\mathrm{ql}^{\kappa}$ is non-decreasing and such that its limit
$\lim_{u\to\infty}\mathrm{ql}^{\kappa}(u)$ corresponds to the total quantum
length of the trunk. Since the exploration runs for all $u>0$, we have
$\mathrm{ql}^{\kappa}(u)<\lim_{v\to\infty}\mathrm{ql}^{\kappa}(v)$ for all
$u\geq 0$. From Remark 2.3 of [MSW22], we know that $\mathrm{ql}^{\kappa}$ is
characterized as the limit in probability:

$\mathrm{ql}^{\kappa}(u)=C(\kappa)\lim_{\varepsilon\to
0}\varepsilon^{4/\kappa}\\#\left\\{s\in[0,u];\ -\Delta
S_{\mathrm{ql}^{\kappa}(s)}^{\kappa,\mu((4/\kappa)-1)}\in[\varepsilon,2\varepsilon]\right\\}$
(3.13)

where one can compute $C(\kappa)=(4/\kappa)/(1-1/2^{4/\kappa})$ with our
choice of the multiplicative constant below Proposition 3.2 (the value of
$C(\kappa)$ is not important for our purpose, we only want it to converge to a
positive number as $\kappa\uparrow 4$). It is clear from (3.13) that
$(\mathrm{ql}^{\kappa}(u))_{u\geq 0}$ is adapted with respect to the natural
filtration of
$\big{(}S_{\mathrm{ql}^{\kappa}(u)}^{\kappa,\mu((4/\kappa)-1)}\big{)}_{u\geq
0}$ and in particular to the filtration associated to the quantum disks
discovered during the chordal exploration. For all $t\geq 0$, we denote by
$\mathrm{Cut}^{\kappa}_{t}$ the domain which is cut out or encircled by a
discovered loop at (quantum length) time $t$. By convention, we set
$\mathrm{Cut}^{\kappa}_{t}=\partial$ where $\partial$ is a cemetery point if
no such domain is encircled or cut out at (quantum length) time $t$.

_Step 1._ We first prove the tightness of the number of cut out and encircled
domains of quantum boundary length larger than $\varepsilon$. By Skorokhod’s
representation theorem, we may assume that (3.1), (3.9) and (3.12) hold almost
surely. Let $u\geq 0$. For all $\varepsilon>0$, for all $\kappa\in(8/3,4]$,
let $N_{\varepsilon}^{\kappa}(u)$ be the number of cut out domains or domains
encircled by loops of quantum boundary length larger than $\varepsilon$ before
time $u$, i.e.

$N_{\varepsilon}^{\kappa}(u)=\\#\left\\{t\leq\mathrm{ql}^{\kappa}(u);\
\left|\Delta S_{t}^{\kappa,\mu((4/\kappa)-1)}\right|>\varepsilon\right\\}.$

Let us denote by $b_{i}^{\kappa}$ and $(b_{i}^{\kappa})^{2}a_{i}^{\kappa}$ the
quantum boundary lengths and quantum areas of the domains which are encircled
by loops or cut out by the trunk during the whole chordal exploration, ranked
in the decreasing order of quantum boundary lengths. Note that by Proposition
3.2, conditionally on the $b_{i}^{\kappa}$’s, the $a_{i}^{\kappa}$’s are
i.i.d. of the same law as
$\mu^{\sqrt{\kappa}}_{h^{\sqrt{\kappa}}}(\mathbb{D})$. We also have the
relation

$\mu^{\sqrt{\kappa}}_{h^{\sqrt{\kappa}}}(\mathbb{D})=\sum_{i\geq
1}(b_{i}^{\kappa})^{2}a_{i}^{\kappa}.$ (3.14)

Moreover, by Lemma 3.5, the quantum boundary lengths of the cut out and
encircled domains during the whole
$\mathrm{SLE}^{\mu((4/\kappa)-1)}_{\kappa}(\kappa-6)$ chordal exploration
ranked in the non-increasing order converge in distribution as $\kappa\uparrow
4$. We also have the convergence in distribution of the $a_{i}^{\kappa}$’s by
(3.9). Let $\kappa_{n}\uparrow 4$. We may find a subsequence again denoted by
$\kappa_{n}$ such that the families $(b_{i}^{\kappa_{n}})_{i\geq 1}$ and
$a_{i}^{\kappa_{n}}$ converge a.s. by Skorokhod’s theorem so that we have a.s.
with the above convergences

$\forall i\geq 1,\qquad
b_{i}^{\kappa_{n}}\mathop{\longrightarrow}\limits_{n\to\infty}b_{i}\qquad\text{and}\qquad
a_{i}^{\kappa_{n}}\mathop{\longrightarrow}\limits_{n\to\infty}a_{i}.$ (3.15)

Notice that this implies that $N^{\kappa_{n}}_{\varepsilon}(u)$ is tight.
Indeed,

$\sum_{i\geq 1}\varepsilon^{2}a_{i}^{\kappa_{n}}{\bf
1}_{b_{i}^{\kappa_{n}}>\varepsilon}\leq\sum_{i\geq
1}(b_{i}^{\kappa_{n}})^{2}a_{i}^{\kappa_{n}}{\bf
1}_{b_{i}^{\kappa_{n}}>\varepsilon}\leq\mu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}}(\mathbb{D})\mathop{\longrightarrow}\limits_{n\to\infty}\mu^{2}_{h^{2}}(\mathbb{D})$

and the $a_{i}^{\kappa_{n}}$’s are i.i.d. and converge as $n\to\infty$. So we
may take a subsequence such that jointly with the above convergences, for all
$u\in\mathbb{Q}\cap\mathbb{R}_{+}$, we have

$N^{\kappa_{n}}_{\varepsilon}(u)\mathop{\longrightarrow}\limits_{n\to\infty}N_{\varepsilon}(u).$
(3.16)

_Step 2._ Let us prove the Carathéodory convergence of the cut out and
encircled domains of quantum boundary length larger than $\varepsilon$ along
some subsequence. More precisely, let $u\in\mathbb{Q}\cap\mathbb{R}_{+}$. Let
$1\leq i\leq N_{\varepsilon}(u)$. We first prove that
$\mathrm{Cut}_{\tau_{\varepsilon,i}^{\kappa_{n}}}^{\kappa_{n}}$ converges
along some subsequence.

Assume by contradiction that
$\mathrm{Cut}_{\tau_{\varepsilon,i}^{\kappa_{n}}}^{\kappa_{n}}$ does not
converge along any subsequenc for the Carathéodory topology. In particular,
for all $z\in\mathbb{D}\cap\mathbb{Q}^{2}$, there exists $n_{z}\in\mathbb{N}$
such that for all $n\geq n_{z}$, we have $z\not\in
D^{\kappa_{n}}_{\varepsilon,i_{n}}$. Let $(D^{z_{i},4}_{u})_{i\geq 1}$ be an
enumeration of the domains of the form $D^{z,4}_{u}$ for some
$z\in\mathbb{D}\cap\mathbb{Q}^{2}$. By (3.9) we know that
$\mu^{\sqrt{\kappa}}_{h^{\sqrt{\kappa}}}(\mathbb{D})$ converges towards
$\mu^{2}_{h^{2}}(\mathbb{D})$ as $\kappa\uparrow 4$. Moreover, by (3.12), for
all $z\in\mathbb{D}\cap\mathbb{Q}^{2}$, we also have the convergence of
$\mu^{\sqrt{\kappa}}_{h^{\sqrt{\kappa}}}(D^{z,\kappa}_{u})$ towards
$\mu^{2}_{h^{2}}(D^{z,4}_{u})$ as $\kappa\uparrow 4$. As a consequence, we
have for all $k\geq 1$,

$\displaystyle\sum_{i=1}^{k}\mu_{h^{2}}^{2}(D^{z_{i},4}_{u})=\lim_{n\to\infty}\sum_{i=1}^{k}\mu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}}(D^{z_{i},\kappa_{n}}_{u})\leq$
$\displaystyle\lim_{n\to\infty}\mu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}}(\mathbb{D})-\min\\{(b^{{\kappa_{n}}}_{j})^{2}a^{{\kappa_{n}}}_{j};\
j\geq 1,b^{{\kappa_{n}}}_{j}>\varepsilon\\}$ (3.17)
$\displaystyle=\mu^{2}_{h^{2}}(\mathbb{D})-\min\\{(b_{j})^{2}a_{j};\ j\geq 1,\
b_{j}>\varepsilon\\}.$

The above inequality in (3.17) comes from the fact that for $n$ large enough,
the $D^{z_{i},\kappa_{n}}_{u}$’s for $i\in[\\![1,k]\\!]$ are distinct. Indeed,
if $i\neq j$, then we cannot have
$D^{z_{i},\kappa_{n}}_{u}=D^{z_{j},\kappa_{n}}_{u}$ infinitely often,
otherwise we would have $D^{z_{i},4}_{u}=D^{z_{j},4}_{u}$. Moreover, the left-
hand side of (3.17) converges to $\mu^{2}_{h^{2}}(\mathbb{D})$ as
$k\to\infty$, hence a contradiction.

_Step 3._ By taking another subsequence again denoted by $\kappa_{n}$, we
assume that a.s. for all $i\in[\\![1,N_{\varepsilon}(u)]\\!]$, the domain
$\mathrm{Cut}^{\kappa_{n}}_{\tau_{\varepsilon,i}^{\kappa_{n}}}$ converges in
the sense of Carathéodory. Let us prove that the convergence of the
$\mathrm{Cut}^{\kappa_{n}}_{\tau_{\varepsilon,i}^{\kappa_{n}}}$’s also holds
in terms of quantum disks, in the sense that if we take $z$ in the limit of
$\mathrm{Cut}^{\kappa_{n}}_{\tau_{\varepsilon,i}^{\kappa_{n}}}$, then, if
$f_{\kappa_{n}}$ (resp. $f_{4}$) is the unique conformal map from $\mathbb{D}$
to $D^{z,\kappa_{n}}_{u}$ (resp. $D^{z,4}_{u}$) such that
$f_{\kappa_{n}}(0)=z$ and $f^{\prime}_{\kappa_{n}}(0)>0$ (resp. $f_{4}(0)=z$
and $f^{\prime}_{4}(0)>0$), we have the convergence jointly with the above
convergences

$\begin{pmatrix}h^{\sqrt{\kappa_{n}}}\circ
f_{\kappa_{n}}+Q_{\sqrt{\kappa_{n}}}\log|f_{\kappa_{n}}^{\prime}|\\\
\mu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}}\circ f_{\kappa_{n}}\\\
\nu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}\circ
f_{\kappa_{n}}\end{pmatrix}\qquad\mathop{\longrightarrow}\limits_{\kappa\uparrow
4}^{(\mathrm{d})}\qquad\begin{pmatrix}h^{2}\circ
f_{4}+Q_{2}\log|f^{\prime}_{4}|\\\ \mu^{2}_{h^{2}}\circ f_{4}\\\
\nu^{2}_{h^{2}_{|D^{z,4}_{u}}}\circ f_{4}\end{pmatrix}$ (3.18)

where the first coordinate is endowed with the $H^{-1}_{\mathrm{loc}}$
topology and the two last ones with the weak convergence of measures
respectively on $\mathbb{D}$ and $\partial\mathbb{D}$.

Recall the convergence (3.12) and the fact that for $n$ large enough, by the
Carathéodory convergence of
$\mathrm{Cut}^{\kappa_{n}}_{\tau^{\kappa_{n}}_{\varepsilon,i}}$, we have
$D^{z,\kappa_{n}}_{u}=\mathrm{Cut}^{\kappa_{n}}_{\tau^{\kappa_{n}}_{\varepsilon,i}}$.
Moreover, $\mathrm{Cut}^{\kappa_{n}}_{\tau^{\kappa_{n}}_{\varepsilon,i}}$ is
one of the cut-out or encircled domains of quantum boundary length
$b_{i}^{\kappa_{n}}$ so that there exists a random integer $I_{n}$, which is
a.s. bounded due to (3.14), such that
$\mathrm{Cut}^{\kappa_{n}}_{\tau^{\kappa_{n}}_{\varepsilon,i}}$ is the quantum
disk of quantum boundary length $b_{I_{n}}^{\kappa_{n}}$. Since for all $i\geq
1$ such that $b_{i}^{\kappa_{n}}>\varepsilon$, conditionally on
$b_{i}^{\kappa_{n}}$, the corresponding domain is a
$\sqrt{\kappa_{n}}$-quantum disk, we deduce from (3.9) that

$\tilde{h}^{\sqrt{\kappa_{n}}}\coloneqq h^{\sqrt{\kappa_{n}}}\circ
f_{\kappa_{n}}+Q_{\sqrt{\kappa_{n}}}\log|f_{\kappa_{n}}^{\prime}|-\frac{2}{\sqrt{\kappa_{n}}}\log\left(\nu_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}^{\sqrt{\kappa_{n}}}(\partial
D^{z,\kappa_{n}}_{u})\right)$

is tight as $n\to\infty$ in $H^{-1}_{\mathrm{loc}}(\mathbb{D})$ (and also the
quantum area and length measures
$\mu_{\tilde{h}^{\sqrt{\kappa_{n}}}}^{\sqrt{\kappa_{n}}}$ and
$\nu_{\tilde{h}^{\sqrt{\kappa_{n}}}}^{\sqrt{\kappa_{n}}}$ for the weak
convergence of measures). Thus
$\log\nu_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}^{\sqrt{\kappa_{n}}}(\partial
D^{z,\kappa_{n}}_{u})$ is tight in $\mathbb{R}$ as $n\to\infty$. So
$h^{\sqrt{\kappa_{n}}}\circ
f_{\kappa_{n}}+Q_{\sqrt{\kappa_{n}}}\log|f_{\kappa_{n}}^{\prime}|$ is tight in
$H^{-1}_{\mathrm{loc}}(\mathbb{D})$ and hence converges in
$H^{-1}_{\mathrm{loc}}(\mathbb{D})$ towards $h^{2}\circ
f_{4}+Q_{2}\log|f_{4}^{\prime}|$. Let us take a subsequence again denoted by
$\kappa_{n}\uparrow 4$ such that

$\begin{pmatrix}h^{\sqrt{\kappa_{n}}}\circ
f_{\kappa_{n}}+Q_{\sqrt{\kappa_{n}}}\log|f_{\kappa}^{\prime}|\\\
(\tilde{h}^{\sqrt{\kappa_{n}}},\mu_{\tilde{h}^{\sqrt{\kappa_{n}}}}^{\sqrt{\kappa_{n}}},\nu_{\tilde{h}^{\sqrt{\kappa_{n}}}}^{\sqrt{\kappa_{n}}})\\\
\log\left(\nu_{h^{\sqrt{\kappa_{n}}}_{D^{z,\kappa_{n}}_{u}}}^{\sqrt{\kappa_{n}}}(\partial
D^{z,\kappa_{n}}_{u})\right)\end{pmatrix}\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{a.s.})}\begin{pmatrix}h^{2}\circ
f_{4}+Q_{2}\log|f_{4}^{\prime}|\\\
(\tilde{h}^{2},\mu^{2}_{\tilde{h}^{2}},\nu^{2}_{\tilde{h}^{2}})\\\
Z\end{pmatrix},$

where $Z$ is some real random variable and $\tilde{h}^{2}$ is the field
corresponding to the parametrization by the unit disk of the quantum disk
corresponding to $b^{4}_{I}$ for some random $I$ integer such that
$b^{4}_{I}\geq\varepsilon$. Then $\tilde{h}^{2}+Z=h^{2}\circ
f_{4}+Q_{2}\log|f_{4}^{\prime}|$. Therefore

$e^{Z}=\nu^{2}_{\tilde{h}^{2}+Z}(\partial\mathbb{D})=\nu^{2}_{h^{2}\circ
f_{4}+Q_{2}\log|f_{4}^{\prime}|}(\partial\mathbb{D})=\nu^{2}_{h^{2}_{|D^{z,4}_{u}}}(f_{4}(\partial\mathbb{D}))=\nu^{2}_{h^{2}_{|D^{z,4}_{u}}}(\partial
D^{z,4}_{u}),$

where the third equality comes by definition of the equivalence relation on
quantum surfaces. Thus, we have the convergence in law of
$\log\nu_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}^{\sqrt{\kappa_{n}}}(\partial
D^{z,\kappa_{n}}_{u})$ towards $\log\nu^{2}_{h^{2}_{|D^{z,4}_{u}}}(\partial
D^{z,4}_{u})$. As a consequence,

$\tilde{h}^{2}=h^{2}\circ
f_{4}+Q_{2}\log|f_{4}^{\prime}|-\log\nu^{2}_{h^{2}_{|D^{z,4}_{u}}}(\partial
D^{z,4}_{u}).$

The convergence of
$\nu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}\circ
f_{\kappa_{n}}$ comes from the convergence of
$\nu_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}^{\sqrt{\kappa_{n}}}(\partial
D^{z,\kappa_{n}}_{u})$ and from the fact that

$\nu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}\circ
f_{\kappa_{n}}=\nu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}_{|D^{z,\kappa_{n}}_{u}}}(\partial
D^{z,\kappa_{n}}_{u})\nu^{\sqrt{\kappa_{n}}}_{\tilde{h}^{\sqrt{\kappa_{n}}}}.$

This ends the proof of (3.18).

_Step 4._ Let us check that $N_{\varepsilon}(u)=N^{4}_{\varepsilon}(u)$ for
all $u\in\mathbb{Q}\cap\mathbb{R}_{+}$ and that the convergence of the cut out
and encircled domains holds in terms of quantum disks. Recall that we have a
sequence $\kappa_{n}\uparrow 4$ such that the convergences (3.1), (3.9),
(3.12), (3.15), (3.16) and (3.18) hold almost surely.

By the proof of Proposition 3.1, we know that a cut out (resp. encircled)
domain of the form $\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$ for some
$i\in N^{4}_{\varepsilon}$ is the limit in the sense of Carathéodory of some
cut out (resp. encircled) domain. Moreover, by (3.12) we know that if
$z\in\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$, then for $n$ large enough,
we have

$\mu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}}(D^{z,\kappa_{n}}_{u})\geq\mu^{2}_{h^{2}}(\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}})/2.$

To prove that $D^{z,\kappa_{n}}_{u}$ converges in terms of quantum disks
towrds $\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$, we distinguish two
cases:

  * –

If $\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$ is a domain encircled by a
loop, then we know by Proposition 3.15 of [AG23a] that the quantum boundary
length of $\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$ is the limit of the
quantum boundary length of $D^{z,\kappa_{n}}_{u}$ as $n\to\infty$.

  * –

If $\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$ is a cut out domain, then we
use Lemma 3.13 of [AG23a] which states the convergence in law of the quantum
boundary length of a loop surrounding a point chosen at random according to
$\mu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}}/\mu^{\sqrt{\kappa_{n}}}_{h^{\sqrt{\kappa_{n}}}}(\mathbb{D})$
towards a positive random variable, together with the convergence (3.16) to
deduce that a.s.

$\lim_{k\to\infty}\sup_{n\geq 0}\sum_{i\geq
k}(b_{i,+}^{\kappa_{n}})^{2}a_{i,+}^{\kappa_{n}}=0,$

where the $b_{i,+}^{\kappa_{n}}$’s (resp. the
$(b_{i,+}^{\kappa_{n}})^{2}a_{i,+}^{\kappa_{n}}$’s) stand for the
$b_{i}^{\kappa_{n}}$’s (resp. the
$(b_{i}^{\kappa_{n}})^{2}a_{i}^{\kappa_{n}}$’s) which correspond to a domain
which is encircled by a loop, i.e. the $b_{i,+}^{\kappa_{n}}$’s are the
positive jumps of $S^{\kappa,\mu((4/\kappa)-1)}$. As a consequence, by
Proposition 3.2, and since the intensities of the Poisson point process of the
positive and negative jumps of $\widetilde{S}^{1+4/\kappa_{n}}$ only differ by
a multiplicative constant $\cos((1+4/\kappa_{n})\pi)$ tending to $1$ as
$n\to\infty$, we also have a.s.

$\lim_{k\to\infty}\sup_{n\geq 0}\sum_{i\geq
k}(b_{i,-}^{\kappa_{n}})^{2}a_{i,-}^{\kappa_{n}}=0,$

where the $b_{i,-}^{\kappa_{n}}$’s (resp. the
$(b_{i,-}^{\kappa_{n}})^{2}a_{i,-}^{\kappa_{n}}$’s) stand for the
$b_{i}^{\kappa_{n}}$’s (resp. the
$(b_{i}^{\kappa_{n}})^{2}a_{i}^{\kappa_{n}}$’s) which correspond to a cut out
domain, i.e. the $b_{i,-}^{\kappa_{n}}$’s are the negative jumps of
$S^{\kappa,\mu((4/\kappa)-1)}$. Therefore, we can choose $k\geq 0$ large
enough so that for all $n\geq 0$,

$\sum_{i\geq
k}(b_{i}^{\kappa_{n}})^{2}a_{i}^{\kappa_{n}}<\mu^{2}_{h^{2}}(\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}})/2.$

Thus, for all $n$ large enough, the quantum boundary length of
$D^{z,\kappa_{n}}_{u}$ is larger than $b_{k}^{4}/2$.

In both cases, since
$\nu_{h^{\sqrt{\kappa_{n}}}_{D^{z,\kappa_{n}}_{u}}}^{\sqrt{\kappa_{n}}}(D^{z,\kappa_{n}}_{u})$
is a.s. lower-bounded by some positive r.v. as $n\to\infty$, by the
convergence (3.18) of the previous step, we deduce that

$D^{z,\kappa_{n}}_{u}\mathop{\longrightarrow}\limits_{n\to\infty}\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$

in terms of quantum disks. Since every cut out or encircled domain
$\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}}$ is such a limit, we get that
$N_{\varepsilon}(u)\geq N^{4}_{\varepsilon}(u)$. Moreover, since by the
previous step the domain
$\mathrm{Cut}^{\kappa_{n}}_{\tau_{i,\varepsilon}^{\kappa_{n}}}$ converges in
terms of quantum disks, and since the points of a Poisson point process of
intensity $dx/x^{2}$ are a.s. different from $\varepsilon$, we obtain that
$N_{\varepsilon}(u)\leq N^{4}_{\varepsilon}(u)$ and hence
$N_{\varepsilon}(u)=N^{4}_{\varepsilon}(u)$.

_Step 5._ By the two previous steps, we have the a.s. convergence of the
domains
$(\mathrm{Cut}^{\kappa_{n}}_{\tau_{i,\varepsilon}^{\kappa_{n}}})_{1\leq i\leq
N^{\kappa_{n}}_{\varepsilon}}$ towards some reordering of the family
$(\mathrm{Cut}^{4}_{\tau_{i,\varepsilon}^{4}})_{1\leq i\leq
N^{4}_{\varepsilon}}$ for the Carathéodory topology and in terms of quantum
disks. Let us check that the cut-out and encircled domains converge in the
right order, i.e. we have a.s.

$\left(\mathrm{Cut}^{\kappa_{n}}_{\tau^{\kappa_{n}}_{\varepsilon,i}}\right)_{i\in[\\![1,N^{\kappa_{n}}_{\varepsilon}(u)]\\!]}\mathop{\longrightarrow}\limits_{n\to\infty}\left(\mathrm{Cut}^{4}_{\tau^{4}_{\varepsilon,i}}\right)_{i\in[\\![1,N^{4}_{\varepsilon}(u)]\\!]},$
(3.19)

for the Carathéodory topology and in terms of quantum disks. This is easy to
see by looking at rational times $u_{i}$’s for $i\geq 1$ such that
$N^{4}_{\varepsilon}(u_{i})=i$ for all $i\leq N^{4}_{\varepsilon}(u)$: at time
$u_{1}$ there is only one possibility for the cut out or encircled domain, at
time $u_{2}$ we have already identified the limit of the first region to be
encircled or cut out, etc. Note that the convergence (3.19) holds for any
$\varepsilon\in\mathbb{R}_{+}^{*}\cap\mathbb{Q}$.

_Step 6._ Here we prove (3.10). Recall that we have the a.s. the convergences
(3.1), (3.9), (3.12), (3.15), (3.16), (3.18), (3.19). We may also assume that
the convergence (3.11) also holds almost surely by taking another subsequence
again denoted by $\kappa_{n}$. By the definition of
$(\overline{L}^{4}_{t}+\overline{R}^{4}_{t})_{t\geq 0}$ in Lemma 3.5, to prove
(3.10), it is enough to check that for all $t\geq 0$, on the event $E_{t}$ the
process $(S^{4,\mu}_{s})_{0\leq s\leq t}$ has the same law as
$(\overline{S}^{4}_{s})_{0\leq s\leq
t}\coloneqq(\overline{L}^{4}_{s}+\overline{R}^{4}_{s})_{0\leq s\leq t}$ on the
event $\\{\forall s\in[0,t],\ \overline{L}^{4}_{s},\overline{R}^{4}_{s}>0\\}$,
where $\overline{L}^{4}$ and $\overline{R}^{4}$ are defined in Lemma 3.5. In
what follows, we drop the exponents $\mu((4/\kappa_{n})-1)$ and $\mu$ above
$S^{\kappa_{n},\mu((4/\kappa_{n})-1)}$ and $S^{4,\mu}$ to simplify the
notation. Note that for all $i_{0}\geq 1$ for $\varepsilon>0$ small enough,
for all $i\leq i_{0}$ and $n\geq 1$, the time
$\tau^{\kappa_{n}}_{\varepsilon,i}$ is finite and converges in law as
$\kappa\uparrow 4$ by (3.11). Moreover, for all $u\geq 0$, by definition of
$N^{\kappa_{n}}_{\varepsilon}(u)$, we obtain that

$\tau^{\kappa_{n}}_{\varepsilon,N^{\kappa_{n}}_{\varepsilon}(u)}\leq\mathrm{ql}^{\kappa_{n}}(u)<\tau^{\kappa_{n}}_{\varepsilon,N^{\kappa_{n}}_{\varepsilon}(u)+1}.$
(3.20)

Thus, by (3.16) and (3.11) by letting $n\to\infty$ and then $\varepsilon\to 0$
for all rational $u\geq 0$, there is a non-decreasing process
$(\widetilde{\mathrm{ql}}(u))_{u\in\mathbb{Q}\cap\mathbb{R}_{+}}$ such that
for all $u\in\mathbb{Q}\cap\mathbb{R}_{+}$,

$\mathrm{ql}^{\kappa_{n}}(u)\mathop{\longrightarrow}\limits_{n\to\infty}\widetilde{\mathrm{ql}}(u).$

Actually, by density of the jumps of $\overline{S}^{4}$, one can see that
$\widetilde{\mathrm{ql}}$ extends to a continuous increasing function so that
we have a.s. uniformly on compact sets

$\mathrm{ql}^{\kappa_{n}}\mathop{\longrightarrow}\limits_{n\to\infty}\widetilde{\mathrm{ql}}.$

Then, by the definition of the Skorokhod $J_{1}$ convergence, the above
convergence implies that

$(S^{\kappa_{n}}_{\mathrm{ql}^{\kappa_{n}}(u)})_{u\geq
0}\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{a.s.})}(\overline{S}^{4}_{\widetilde{\mathrm{ql}}(u)})_{u\geq
0}$

in the sense of Skorokhod. In particular, since the Skorokhod convergence
implies the convergence of the jumps of size at least $\varepsilon$ for all
$\varepsilon>0$, we deduce that a.s. for all
$u\in\mathbb{Q}\cap\mathbb{R}_{+}$, for all $\varepsilon>0$, by (3.19) we have

$N^{4}_{\varepsilon,-}(u)\coloneqq\\#\\{v\in[0,u];\
-\Delta{S}^{4}_{{\mathrm{ql}}^{4}(v)}\geq\varepsilon\\}=\\#\\{v\in[0,u];\
-\Delta\overline{S}^{4}_{\widetilde{\mathrm{ql}}(v)}\geq\varepsilon\\}.$

Since the left and the right terms are càdlàg in $u$, the above equality holds
for all $u\geq 0$. We can check that $\widetilde{\mathrm{ql}}=\mathrm{ql}^{4}$
using the caracterization similar to (3.13) of the quantum length of the trunk
as a limit in probability (the constant $2$ is chosen for consistency): for
all $u>0$,

$\displaystyle\mathrm{ql}^{4}(u)$ $\displaystyle=2\lim_{\varepsilon\to
0}\varepsilon\\#\\{v\in[0,u];\ -\Delta
S^{4}_{\mathrm{ql}^{4}(v)}\in[\varepsilon,2\varepsilon]\\}$
$\displaystyle=2\lim_{\varepsilon\to
0}\varepsilon(N^{4}_{\varepsilon,-}(u)-N^{4}_{2\varepsilon,-}(u))$
$\displaystyle=2\lim_{\varepsilon\to
0}\varepsilon\\#\\{t\in[0,\widetilde{\mathrm{ql}}(u)];\
-\Delta\overline{S}^{4}_{t}\in[\varepsilon,2\varepsilon]\\}$
$\displaystyle=\widetilde{\mathrm{ql}}(u).$

The last equality of limits in probability stems from the definition of
$\overline{S}^{4}$ as a Doob transform of Cauchy processes. Thus a.s.
$\widetilde{\mathrm{ql}}=\mathrm{ql}^{4}$. Besides, by (3.19) and by (3.11),
the processes $(S^{4}_{\mathrm{ql}^{4}(u)})_{u\geq 0}$ and
$(\overline{S}^{4}_{\widetilde{\mathrm{ql}}(u)})_{u\geq 0}$ have the same
jumps (at the same times), and this is thus also true for the two processes
$S^{4}$ and $\overline{S}^{4}$. Using the fact that these processes are pure
jump processes, we conclude that $S^{4}=\overline{S}^{4}$, hence (3.10).

_Step 7._ In order to prove the statement on the cut out domains and on the
remaining to explore region, it suffices to check that for all
$\varepsilon>0$, if $F$ is a continuous bounded function on
$\mathbb{D}([0,t])$ and $G$ is a continuous bounded function on the space of
finite sequences of distributions on $\mathbb{D}$ seen modulo the
transformation (2.1) (formally it can be written as $\bigcup_{n\geq
0}(\mathcal{D}^{\prime}(\mathbb{D}))^{n}$ where
$\mathcal{D}^{\prime}(\mathbb{D})$ is the space of distributions on
$\mathbb{D}$ modulo the transformation (2.1) for all Möbius transformation
$\varphi$), then

$\mathbb{E}\left(F((S^{4}_{s})_{0\leq s\leq
t})G\left(h_{t},(h_{\varepsilon,i})_{i\geq
1,\tau^{4}_{\varepsilon,i}<t}\right)\right)=\mathbb{E}\left(F((S^{4}_{s})_{0\leq
s\leq t})G\left(h(S^{4}_{t}),(h(\Delta
S^{4}_{\tau^{4}_{\varepsilon,i}}))_{i\geq
1,\tau^{4}_{\varepsilon,i}<t}\right)\right),$

where a representative of $h_{t}$ is $h^{2}\circ f_{z}+2\log|f_{z}^{\prime}|$
where $z\in\mathcal{D}_{t}^{4}$ and $f_{z}:\mathbb{D}\to\mathcal{D}^{4}_{t}$
is the unique conformal map such that $f_{z}(0)=z$ and $f_{z}^{\prime}(0)>0$,
where for all $i\geq 1$, we have defined $h_{\varepsilon,i}$ as the
equivalence class of $h^{2}\circ f_{z_{i}}+2\log|f_{z_{i}}^{\prime}|$ where
$z_{i}\in\mathrm{Cut}^{4}_{\tau_{\varepsilon,i}^{4}}$ and
$f_{z_{i}}:\mathbb{D}\to\mathrm{Cut}^{4}_{\tau_{\varepsilon,i}^{4}}$ is the
unique conformal map such that $f_{z_{i}}(0)=z_{i}$ and
$f_{z_{i}}^{\prime}(0)>0$, and where $h(b)$ denotes the instance of the GFF of
an independent quantum disk of quantum boundary length $b$ (conditionally on
$b$). But the above equality holds clearly thanks to the convergences (3.11),
(3.12), (3.19), thanks to the convergence of $\mathrm{ql}^{\kappa_{n}}$
towards $\mathrm{ql}^{4}$, thanks to the second point of Proposition 3.2 which
states the same equality for $\kappa<4$ and thanks to the fact that the
unexplored domain is the Carathéodory limit of the unexplored domain by the
equation (3.3) in the proof of Proposition 3.1. ∎

##  4 The $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ converges to the uniform
exploration as $\mu\to\infty$

Recall that we proved Theorem 1.1 in the case of the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}$ exploration for $\mu\in\mathbb{R}$ in
Subsection 3.5. In this section, we prove Theorem 1.1 in the case of the
uniform exploration, which was defined in Subsection 2.1. To this end, we let
$\mu\to\infty$ to approximate the uniform exploration using the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration.

###  4.1 From the Carathéodory convergence to the Markov property for quantum
disks

The fact that the uniform exploration can be constructed by letting
$\mu\to\infty$ was first observed in the last paragraph of p.22 of [WW13].
However, we will need a convergence in terms of Carathéodory topology. Recall
the radial $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ defined at the end of
Subsection 3.2. For all $\mu\in\mathbb{R}$, for all $z\in\mathbb{D}$, let
$({\bf D}^{z,\mu}_{s})_{s\geq 0}$ be the radial exploration targeted at $z$
using the $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$. More precisely, for
every time $s\geq 0$, the domain ${\bf D}_{s}^{z,\mu}$ is the unexplored
region containing $z$ for the radial
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$. The parametrization is chosen so
that if $f_{s}:\mathbb{D}\to{\bf D}^{z,\mu}_{s}$ is the unique conformal map
such that $f_{s}(0)=z$ and $f_{s}^{\prime}(0)>0$, then
$f^{\prime}_{s}(0)=e^{-s}$. The positive number $f^{\prime}_{s}(0)$ is called
the conformal radius of ${\bf D}^{z,\mu}_{s}$ seen from $z$. For all
$w\in\mathbb{D}\cap\mathbb{Q}^{2}\setminus\\{z\\}$, we denote by
$D^{z,w,\mu}_{s}$ the connected component containing $w$ of the complement in
$\mathbb{D}$ of the path of the chordal
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ towards $z$ until time $s$. Note
that there is a time $\sigma^{\mu}_{z,w}$ at which the radial
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ towards $z$ cuts $w$ out from $z$
or encircles $w$ with a loop which does not encircle $z$. Therefore, for all
$s<\sigma^{\mu}_{z,w}$, we have $D^{z,w,\mu}_{s}={\bf D}^{z,\mu}_{s}$ and for
all $s\geq\sigma^{\mu}_{z,w}$, the domain
$D^{z,w,\mu}_{s}={D}^{z,w,\mu}_{\sigma^{\mu}_{z,w}}$ corresponds to the domain
which is cut out or encircled by a loop. Similarly, using the uniform
exploration defined in Subsection 2.1, let $({\bf D}^{z}_{s})_{s\geq 0}$ be
the exploration targeted at $z$, again parametrized so that the conformal
radius of ${\bf D}^{z}_{s}$ is $e^{-s}$. More precisely, ${\bf D}^{z}_{s}$ is
the unexplored region containing $z$ at time $s$. Note that for all
$w\in\mathbb{D}\cap\mathbb{Q}^{2}\setminus\\{z\\}$ there is a time
$\sigma_{z,w}$ at which the exploration cuts $w$ out from $z$ or encircles $w$
with a loop which does not encircle $z$. We then set $D^{z,w}_{s}={\bf
D}^{z}_{s}$ for all $s<\sigma_{z,w}$, and we define $D^{z,w}_{s}$ for
$s\geq\sigma_{z,w}$ as the domain which is cut out or encircled by a loop at
time $\sigma_{z,w}$.

######  Proposition 4.1.

One has the convergence indistribution of

$\left(({\bf D}^{z,\mu}_{s})_{s\geq
0,z\in\mathbb{D}\cap\mathbb{Q}^{2}},(D^{z,w,\mu}_{s})_{s\geq
0,z\in\mathbb{D}\cap\mathbb{Q}^{2},w\in\mathbb{D}\cap\mathbb{Q}^{2}\setminus\\{z\\}}\right)$

to

$\left(({\bf D}^{z}_{s})_{s\geq
0,z\in\mathbb{D}\cap\mathbb{Q}^{2}},(D^{z,w}_{s})_{s\geq
0,z\in\mathbb{D}\cap\mathbb{Q}^{2},w\in\mathbb{D}\cap\mathbb{Q}^{2}\setminus\\{z\\}}\right)$

as $\mu\to\infty$ in terms of finite dimensional distributions for the
Carathéodory topology. The convergence holds also in a stronger sense: if
$(s_{\mu})_{\mu\geq 0}$ converges towards $s$ as $\mu\to\infty$, then jointly
with the above convergence, for all $z\in\mathbb{D}\cap\mathbb{Q}^{2}$, we
have ${\bf D}^{z,\mu}_{s_{\mu}}\mathop{\longrightarrow}{\bf D}^{z}_{s}.$ as
$\mu\to\infty$ in distribution.

The proof of Theorem 1.1 for $(Y_{t})_{t\geq 0}$ then follows from roughly the
same lines as the proof of Proposition 3.3, although here we deal with the
radial exploration instead of the chordal one. Let us only give a sketch of
the proof:

###### Sketch of the proof of Theorem 1.1 for $(Y_{t})_{t\geq 0}$ using
Proposition 4.1.

For all $z\in\mathbb{D}$, for all $\mu\in\mathbb{R}$, let us write
$\mathrm{ql}_{z,\mu}(s)$ the quantum length of the trunk of the radial
exploration $({\bf D}^{z,\mu}_{s})_{s\geq 0}$. More precisely, it is
characterized (with our choice of the multiplicative constant) by

$\mathrm{ql}_{z,\mu}(s)=2\lim_{\varepsilon\to
0}\varepsilon\\#\\{v\in[0,s];-\Delta\nu^{2}_{h^{2}_{|{\bf
D}^{z,\mu}_{v}}}(\partial{\bf
D}^{z,\mu}_{v})\in[\varepsilon,2\varepsilon]\\}.$

Let us write $\mathrm{ql}_{z,\mu}^{-1}$ the right-continuous inverse of
$\mathrm{ql}_{z,\mu}$ and for all $t\geq 0$,

$\hat{S}^{z,\mu}_{t}\coloneqq\nu^{2}_{h^{2}_{|{\bf
D}^{z,\mu}_{\mathrm{ql}_{z,\mu}^{-1}(t)}}}(\partial{\bf
D}^{z,\mu}_{\mathrm{ql}_{z,\mu}^{-1}(t)}).$

Let $Z_{i}$ for $i\geq 1$ be i.i.d. random points of $\mathbb{D}$ of law
$\mu^{2}_{h^{2}}/\mu^{2}_{h^{2}}(\mathbb{D})$ which are independent of $h^{2}$
and of the explorations. One can see using Theorem 1.1 for
$(Y^{\mu}_{t})_{t\geq 0}$, the relation between the quantum area and the
quantum boundary length and the renewal property that the process
$(\hat{S}^{Z_{i},\mu}_{t})_{t\geq 0,i\geq 1}$ has the same distribution for
all $\mu\in\mathbb{R}$ and that for all $i\geq 1$, conditionally on their
quantum boundary lengths, ${\bf
D}^{Z_{i},\mu}_{\mathrm{ql}^{-1}_{Z_{i},\mu}(t)}$ is chosen at random among
the quantum disks corresponding to the unexplored regions of the exploration
tree at height $t$. More precisely, if we denote by $a_{j}^{\mu}(t)$ for
$j\geq 1$ the quantum areas of the unexplored quantum disks when we stop the
exploration tree at height $t$ ranked in the non-increasing order, then
conditionally on the exploration tree at height $t$ and on the
$a^{\mu}_{j}$’s, the quantum surface ${\bf
D}^{Z_{i},\mu}_{\mathrm{ql}^{-1}_{Z_{i},\mu}(t)}$ corresponds to the quantum
disk of area $a_{j}^{\mu}(t)$ with probability $a_{j}^{\mu}(t)/(\sum_{k\geq
1}a_{k}^{\mu}(t))$. In particular, the quantum surface ${\bf
D}^{Z_{i},\mu}_{\mathrm{ql}^{-1}_{Z_{i},\mu}(t)}$ is constant in distribution.

By the same reasoning as in the proof of Lemma 3.6, we know that the
convergence in Proposition 4.1 also holds for the underlying fields in terms
of the convergence in the sense of distribution and for the quantum area
measures with the weak convergence of measures.

Using that $\mu^{2}_{h^{2}}(\mathbb{D})<\infty$, it is easy to see that the
number of encircled or cut out disks of quantum boundary length larger than
some $\varepsilon>0$ during the exploration ${\bf D}^{Z_{i},\mu}$ until time
$s$ is tight, as in the first step of the proof of Proposition 3.3. As a
result, using that $\hat{S}^{Z_{i},\mu}$ is constant in law, we obtain as in
(3.20) that for all $u\in\mathbb{Q}\cap\mathbb{R}_{+}$, the family
$(\mathrm{ql}_{Z_{i},\mu}(u))_{\mu\geq 0}$ is tight. By letting
$\varepsilon\to 0$ and by the densitiy of the jumps of $\hat{S}^{Z_{i},\mu}$,
we get that $\mathrm{ql}_{Z_{i},\mu}$ is tight as $\mu\to\infty$ for the
topology of the uniform convergence on compact sets and that the limit must be
continuous and increasing.

Let $\mu_{n}\to\infty$;. By taking a subsequence again denoted by $\mu_{n}$
and by applying Skorokhod’s theorem, we may assume that the convergence of
Proposition 4.1 holds a.s. together with the convergences for all $i\geq 1$,

$\hat{S}^{Z_{i},\mu_{n}}\mathop{\longrightarrow}\limits_{n\to\infty}\overline{S}^{Z_{i}}\qquad\mathrm{ql}_{Z_{i},\mu_{n}}\mathop{\longrightarrow}\limits_{n\to\infty}\widetilde{\mathrm{ql}}_{Z_{i}},$

where the first convergence holds for the $J_{1}$ Skorokhod topology and the
second one holds uniformly on compact sets, where $\overline{S}^{Z_{i}}$ is a
càdlàg process which has the same law as $\hat{S}^{Z_{i},\mu_{n}}$ for all $n$
and where $\widetilde{\mathrm{ql}}_{Z_{i}}$ is a continuous increasing process
from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$.

As a consequence, by the convergence of Proposition 4.1, we have also a.s. for
all $t\in\mathbb{Q}\cap\mathbb{R}_{+}$,

${\bf
D}^{Z_{i},\mu_{n}}_{\mathrm{ql}^{-1}_{Z_{i},\mu_{n}}(t)}\mathop{\longrightarrow}\limits_{n\to\infty}{\bf
D}^{Z_{i}}_{\widetilde{\mathrm{ql}}^{-1}_{Z_{i}}(t)}$

in the sense of Carathéodory, and thus also in terms of quantum disks, since
it is constant in law. Therefore, for all $t\in\mathbb{Q}\cap\mathbb{R}_{+}$,

$\hat{S}^{Z_{i},\mu_{n}}_{t}\mathop{\longrightarrow}\limits_{n\to\infty}\nu^{2}_{h^{2}_{|{\bf
D}^{Z_{i}}_{\widetilde{\mathrm{ql}}^{-1}_{Z_{i}}(t)}}}\left(\partial{\bf
D}^{Z_{i}}_{\widetilde{\mathrm{ql}}^{-1}_{Z_{i}}(t)}\right).$

Hence for all $t\in\mathbb{Q}\cap\mathbb{R}_{+}$,

$\nu^{2}_{h^{2}_{|{\bf
D}^{Z_{i}}_{\widetilde{\mathrm{ql}}^{-1}_{Z_{i}}(t)}}}\left(\partial{\bf
D}^{Z_{i}}_{\widetilde{\mathrm{ql}}^{-1}_{Z_{i}}(t)}\right)=\overline{S}^{Z_{i}}_{t},$
(4.1)

and the equality holds in fact for all $t\geq 0$ since both processes are
càdlàg.

We see that the quantum length $\mathrm{ql}_{z,\mu_{n}}$ converges to the
quantum natural distance as $n\to\infty$ using the definition of the quantum
natural distance $\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},{\bf D}^{z}_{s})$
as the limit in probability

$\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},{\bf
D}^{z}_{s})=2\lim_{\varepsilon\to 0}\varepsilon\\#\\{v\in[0,s],\
-\Delta\nu^{2}_{h^{2}_{|{\bf D}^{z}_{v}}}(\partial{\bf
D}^{z}_{v})\in[\varepsilon,2\varepsilon]\\}.$

The above definition is indeed consistent with the definition 2.2 of the
distance to the locally largest component as long as $z$ stays in the locally
largest component since the Cauchy process $\widetilde{S}^{2}$ in Proposition
3.3 is symmetric. From the above formula, it is easy to show, as in the end of
the sixth step of the proof of Proposition 4.1 that for all $s\geq 0$, we have
$\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},{\bf
D}^{Z_{i}}_{s})=\widetilde{\mathrm{ql}}_{Z_{i}}(s)$.

For all $z\in\mathbb{D}$, if $\Psi_{z}$ is the inverse of
$s\mapsto\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},{\bf D}^{z}_{s})$, then we
set for all $t\geq 0$,

$\hat{S}^{z}_{t}\coloneqq\nu^{2}_{h^{2}_{|{\bf
D}^{z}_{\Psi_{z}(t)}}}(\partial{\bf D}^{z}_{\Psi_{z}(t)}).$

By (4.1) and since for all $s\geq 0$, we have
$\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},{\bf
D}^{Z_{i}}_{s})=\widetilde{\mathrm{ql}}_{Z_{i}}(s)$, we deduce that
$\overline{S}^{Z_{i}}=\hat{S}^{Z_{i}}$ and thus a.s.

$(\hat{S}^{Z_{i},\mu_{n}}_{t})_{t\geq
0}\mathop{\longrightarrow}\limits_{n\to\infty}(\hat{S}^{Z_{i}}_{t})_{t\geq 0}$
(4.2)

in the sense of Skorokhod.

Next, we consider the locally largest component of the
$\mathrm{SLE}^{\langle\mu_{n}\rangle}_{4}(-2)$ exploration parametrized by the
quantum length of the trunk, that we denote by
$(\mathcal{C}^{\mu_{n}}_{t})_{t\geq 0}$. Recall that its quantum boundary
length is denoted by $(Y^{\mu_{n}}_{t})_{t\geq 0}$. By Theorem 1.1 for
$Y^{\mu_{n}}$, we know that the law of $(Y^{\mu_{n}}_{t})_{t\geq 0}$ does not
depend on $n$ and that conditionally on $(Y^{\mu_{n}}_{t^{\prime}})_{0\leq
t^{\prime}\leq t}$, the cut out or encircled domains, together with
$\mathcal{C}_{t}^{\mu_{n}}$, are independent $2$-quantum disks of boundary
lengths corresponding to the negative and positive jumps of
$(Y^{\mu_{n}}_{t^{\prime}})_{0\leq t^{\prime}\leq t}$ and to
$Y^{\mu_{n}}_{t}$.

Let us recall from Subsection 2.3 that the locally largest component along the
uniform exploration of the $\mathrm{CLE}_{4}$ parametrized by the quantum
natural distance is denoted by $(C_{\sigma(t)})_{t\geq 0}$ and that its
quantum boundary length is denoted by $(\mathcal{Y}(\sigma(t)))_{t\geq 0}$. We
set $\mathcal{C}_{t}\coloneqq C_{\sigma(t)}$ and
$Y^{4}_{t}\coloneqq\mathcal{Y}(\sigma(t))$ for all $t\geq 0$.

Let $t\geq 0$. Let $i$ such that $Z_{i}\in\mathcal{C}_{t}$. Then
$\mathcal{C}_{t}={\bf D}^{Z_{i}}_{\Psi_{Z_{i}}(t)}$. Therefore, for all
$t^{\prime}\in[0,t]$, we have

${\bf
D}^{Z_{i}}_{\Psi_{Z_{i}}(t^{\prime})}=\mathcal{C}_{t^{\prime}}\qquad\text{and}\qquad\hat{S}^{Z_{i}}_{t^{\prime}}=Y^{4}_{t^{\prime}}.$

Besides, by the Skorokhod convergence (4.2), we know that for $\mu$ large
enough, the process $(\hat{S}^{z,\mu}_{t^{\prime}})_{0\leq t^{\prime}\leq t}$
does not have any negative jump at some time $t^{\prime}\leq t$ of size at
least half of $\hat{S}^{z,\mu}_{t^{\prime}}$. In other words, for $\mu$ large
enough, for all $t^{\prime}\in[0,t]$,

${\bf
D}^{z,\mu}_{\mathrm{ql}_{z,\mu}^{-1}(t^{\prime})}=\mathcal{C}^{\mu}_{t^{\prime}}\qquad\text{and}\qquad\hat{S}^{z,\mu}_{t^{\prime}}=Y^{\mu}_{t^{\prime}}.$

This provides the desired result by reasoning again as in the last steps of
the proof of Proposition 3.3. ∎

###  4.2 Convergence of the branch towards the origin

The rest of this subsection is devoted to the proof of Proposition 4.1 and
follows the same lines as Section 2 of [AHPS23]. For this purpose, we recall a
stronger topology on the space of families of domains as in [AHPS23] and
[MS16]. Let $\mathcal{D}$ be the space of families $(D_{s})_{s\geq 0}$ of
simply connected subdomains of $\mathbb{D}$ which are increasing for the
inclusion, such that for every $s\geq 0$, we have $0\in D_{s}$, and if
$f_{s}=f[D_{s}]$ is the unique conformal map from $\mathbb{D}$ to $D_{s}$
which sends $0$ to $0$ and such that $f^{\prime}_{s}(0)>0$, then
$f^{\prime}_{s}(0)=e^{-s}$. The space $\mathcal{D}$ is endowed with the
topology such that a sequence of domains $(D^{n}_{s})_{s\geq 0}\in\mathcal{D}$
converges to $(D_{s})_{s\geq 0}$ if and only if for all compact subset
$K\subset\mathbb{D}$, for all $T>0$,

$\sup_{s\in[0,T]}\sup_{z\in K}\left\lvert
f^{n}_{s}(z)-f_{s}(z)\right\rvert\mathop{\longrightarrow}\limits_{n\to\infty}0,$

where $f^{n}_{s}=f[D^{n}_{s}]$ and $f_{s}=f[D_{s}]$. The space $\mathcal{D}$
is then metrizable and separable. Note that the convergence of
$(D^{n}_{s})_{s\geq 0}$ to $(D_{s})_{s\geq 0}$ in $\mathcal{D}$ implies the
convergence of $D^{n}_{s}$ towards $D_{s}$ in the Carathéodory topology for
every $s\geq 0$. As in [AHPS23], we will sometimes consider evolving domains
$(D_{s})_{s\geq 0}$ such that the conformal radius of $D_{s}$ is $e^{-s}$ for
all $s\in[0,S]$ for some finite $S>0$ and such that for all $s\geq S$,
$D_{s}=D_{S}$. In order to see $(D_{s})_{s\geq 0}$ as an element of
$\mathcal{D}$, we replace $D_{s}$ for $s\geq S$ by
$f_{S}(e^{-(s-S)}\mathbb{D})$ where $f_{S}=f[D_{S}]$.

Let us recall the definition of (measure-driven) radial Loewner chains. If
$\lambda$ is a measure on $[0,\infty)\times\partial\mathbb{D}$ whose marginal
on $[0,\infty)$ is the Lebesgue measure, we define the radial Loewner equation
driven by $\lambda$ by setting for all $z\in\mathbb{D}$ and for all $s\geq 0$,

$g_{s}(z)=\int_{[0,s]\times\partial\mathbb{D}}g_{u}(z)\frac{w+g_{u}(z)}{w-g_{u}(z)}d\lambda(u,w);\qquad
g_{0}(z)=z.$

It is known (see Proposition 6.1 of [MS16]) that for any such $\lambda$, this
ODE has a unique solution $s\mapsto g_{s}(z)$ for each $z\in\mathbb{D}$,
defined until time $s_{z}\coloneqq\sup\\{s\geq 0;\ g_{s}(z)\in\mathbb{D}\\}$.
Moreover, if one defines $D_{s}=\\{z\in\mathbb{D};\ s_{z}<s\\}$ for all $s\geq
0$, then $(D_{s})_{s\geq 0}$ is called the radial Loewner chain driven by
$\lambda$.

If one restricts to a measure of the form $\lambda(A,ds)=\delta_{W(s)}(A)ds$
with $W:\mathbb{R}_{+}\rightarrow\partial\mathbb{D}$ a piecewise continuous
function, then one recovers the more classical notion of Loewner chains.

######  Remark 4.2.

(Remark 2.1 of [AHPS23]) Using Proposition 6.1 of [MS16], we know that the
weak convergence of the driving measure implies the convergence in
$\mathcal{D}$ of the radial Loewner chain. In particular, the convergence of
the radial Loewner chain holds if we assume the convergence in the Skorokhod
$J_{1}$ topology of the driving functions.

Let us then recall the direct construction of the radial
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ (see [AHPS23] Subsection 2.1.3 for
the analogous definition for $\kappa^{\prime}>4$ and see Subsection 4.1 of
[ASW19] for the case $\mu=0$). Let $(B_{s})_{s\geq 0}$ be a standard Brownian
motion and let $(\ell^{0}_{s})_{s\geq 0}$ be its local time at zero. Set, for
all $s\geq 0$,

$U_{s}=\int_{0}^{s}\cot(B_{u})du+\mu\ell^{0}_{s},$

where the integral is not absolutely convergent but can be defined as the
limit in probability of integrals of the form $\int_{0}^{s}\cot(B_{u}){\bf
1}_{d(B_{u},\pi\mathbb{Z})\geq\varepsilon}du$ as $\varepsilon\to 0$, in the
same way as in [WW13] in the chordal case. Let
$\left(W_{0}^{\mu}\right)_{s}=e^{i(2B_{s}-U_{s})}$. Then, if one considers the
radial Loewner chain $({\bf D}^{0,\mu}_{s})_{s\geq 0}$ in $\mathbb{D}$ driven
by $W^{\mu}_{0}$, it corresponds to the radial
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ targeted at zero (and starting from
$1$). Let $\tau_{0}$ be the first time that $B$ reaches $\pm\pi$, which
corresponds to the first time that $0$ gets encircled by a loop during the
$\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration (see Subsection 4.1 of
[ASW19] and Subsection 4.3 of [She09] for more details). Let

$\left({D}^{0,\mu}_{s}\right)_{s\geq 0}=\left({\bf
D}^{0,\mu}_{s\wedge\tau_{0}}\right)_{s\geq 0}.$

We also recall that, during each excursion of $B$ away from $\pi\mathbb{Z}$,
the radial Loewner chain draws a loop of the $\mathrm{CLE}_{4}$ which does not
contain $0$ (see again for example Subsection 4.1 of [ASW19] in the less
general case $\mu=0$ and [She09] for more details). See Figure 3.

Figure 3: Left: the Brownian motion $(B_{s})_{s\geq 0}$. Center: a radial
exploration of the $\mathrm{CLE}_{4}$. The red and blue excursions of height
at least $\varepsilon$ give rise to two loops which do not encircle zero and
the green part of the Brownian motion generates a loop which encircles zero.
Right: approximation of the radial exploration.

Let us present an approximation of this radial
$\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$. Let $n\in\mathbb{N}$. The idea is
to remove intervals of time where $B$ is making tiny excursions away from
zero. We set $T_{0}^{n}\coloneqq 0$ and for all $k\geq 0$,

$\left\\{\begin{array}[]{cl}R_{k+1}^{n}&=\inf\\{s\geq T_{k}^{n};\ \lvert
B_{s}\rvert\geq 1/2^{n}\\};\\\ S^{n}_{k+1}&=\sup\\{s\leq R^{n}_{k+1};\
B_{s}=0\\};\\\ T^{n}_{k+1}&=\inf\\{s\geq R^{n}_{k+1};\
B_{s}=0\\}.\end{array}\right.$

Call $e^{n}_{k}$ the $k$-th excursion associated to the interval
$[S^{n}_{k},T^{n}_{k}]$. Set
$\Lambda^{n}\coloneqq\sup\\{k;S^{n}_{k}\leq\tau_{0}\\}$ and

$l_{k}^{n}=T^{n}_{k}-S^{n}_{k}\ \forall k<\Lambda^{n}\ ;\
l^{n}_{\Lambda^{n}}=\tau_{0}-S^{n}_{\Lambda^{n}}\ ;\ L_{k}^{n}=\sum_{1\leq
j\leq k}l^{n}_{j}\ \forall k\leq\Lambda^{n}.$

Now we define for all $k\in[\\![1,\Lambda^{n}]\\!]$,

$\forall k\in[\\![1,\Lambda^{n}]\\!],\ \forall
s\in[L^{n}_{k-1},L^{n}_{k}),\qquad(W^{\mu,n}_{0})_{s}=(W^{\mu}_{0})_{S^{n}_{k}+(s-L^{n}_{k-1})},$

and set $D^{0,\mu,n}$ to be the radial Loewner chain with driving function
$W^{\mu,n}_{0}$. This is defined up to time
$\tau^{n}_{0}=L^{n}_{\Lambda^{n}}$.

Let us recall an approximation of the uniform exploration of the
$\mathrm{CLE}_{4}$. Sample $(X_{k}^{n})_{k\geq 1}$ uniformly and independently
on $\partial\mathbb{D}$. For all $s\in[0,\tau^{n}_{0})$, set

$\forall k\in[\\![1,\Lambda^{n}]\\!],\ \forall
s\in[L^{n}_{k-1},L^{n}_{k}),\qquad(W^{n}_{0})_{s}=X^{n}_{k}\exp\left(i\left(2B_{s+S^{n}_{k}}-\int_{S_{k}^{n}}^{s+S_{k}^{n}}\cot(B_{u})du\right)\right).$
(4.3)

Let $D^{0,n}$ be the radial Loewner chain with driving function $W^{n}_{0}$.
By the same argument as in Section 4 of [WW13] we know that $D^{0,n}$
converges in $\mathcal{D}$ in probability towards a uniform exploration
$D^{0}$ targeted at $0$ in $\mathcal{D}$ and stopped at the time $\tau_{0}$