defined by $D^{0}_{s}={\bf D}^{0}_{s\wedge\tau_{0}}$ for all $s\geq 0$. Note
that the time $\tau_{0}$ thus also corresponds to the first time at which $0$
gets encircled by a loop for the uniform exploration ${\bf D}^{0}$. Moreover,
as in the beginning of Section 4 of [WW13], each loop drawn by ${\bf D}^{0}$
again corresponds to the excursion of $B$ away from $\pi\mathbb{Z}$ at the
same time.

Let us show the convergence in law jointly in
$z\in\mathbb{Q}^{2}\cap\mathbb{D}$ as $\mu\to\infty$ of the
$\mathrm{SLE}^{\mu}_{4}(-2)$ branch towards $z$ to the uniform exploration
branch towards $z$, together with the convergence of the $\mathrm{CLE}_{4}$
loops surrounding $z$ to the $\mathrm{CLE}_{4}$ loops surrounding $z$. We
first show the convergence of the branch $({\bf D}^{0,\mu}_{s})_{s\geq 0}$
targeted at $0$. To show this, it suffices to check the following statement:

######  Proposition 4.3.

The stopped exploration branch towards zero converges:

$(D^{0,\mu},\tau_{0},B)\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}(D^{0},\tau_{0},B),$

where the first coordinate converges in the space $\mathcal{D}$ and where $B$
is the Brownian motion from which the $D^{0,\mu}$’s are defined (and also
$D^{0}$ via the approximation).

By the iterative definition of the exploration of the (nested)
$\mathrm{CLE}_{4}$ towards zero, the convergence for all time follows
immediately from Proposition 4.3:

######  Proposition 4.4.

The whole exploration branch towards zero converges:

$({\bf
D}^{0,\mu},\tau_{0},B)\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}({\bf
D}^{0},\tau_{0},B),$

where the first coordinate converges in $\mathcal{D}$ and $B$ is the Brownian
motion with which the $D^{0,\mu}$’s are defined. Notice that we also get in
particular the convergence in the sense of Carathéodory of the domain ${\bf
D}^{0,\mu}_{\tau_{0}}$ encircled by the outermost loop encircling $0$ towards
${\bf D}^{0}_{\tau_{0}}$.

For all $z\in\mathbb{D}\cap\mathbb{Q}^{2}$, we can define the space
$\mathcal{D}_{z}$ of families of domains containing $z$ which are in
$\mathcal{D}$ after applying the conformal map $w\mapsto(w-z)/(1-\bar{z}w)$,
equipped by the topology induced by the topology of $\mathcal{D}$. We also
denote by $\tau_{z}^{\mu}$ (resp. $\tau_{z}$) the first time at which $z$ is
encircled by a loop in the radial $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$
targeted at $z$ (resp. in the uniform exploration $({\bf D}^{z}_{s})_{s\geq
0}$). We also denote by $B^{z,\mu}$ (resp. $B^{z}$) the Brownian motion which
defines $D^{z,\mu}$ (resp. $D^{z}$) composed with the conformal map
$w\mapsto(w-z)/(1-\bar{z}w)$. The time $\tau^{\mu}_{z}$ (resp. $\tau_{z}$) is
then the first time at which $B^{z,\mu}$ (resp. $B^{z}$) reaches $\pm\pi$.
Note that as in the case $z=0$, we can build a coupling between the
explorations using the same underlying Brownian motion so that
$B^{z,\mu}=B^{z}$ and $\tau^{\mu}_{z}=\tau_{z}$ for all $\mu\in\mathbb{R}$.
The convergence of the other branches then follows by conformal invariance as
stated just below. We endow the space of continuous real functions
$\mathcal{C}(\mathbb{R}_{+},\mathbb{R})$ with the topology of the uniform
convergence on compact sets.

######  Corollary 4.5.

For all $z\in\mathbb{D}\cap\mathbb{Q}^{2}$,

$({\bf
D}^{z,\mu},\tau^{\mu}_{z},B^{z,\mu})\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}({\bf
D}^{z},\tau_{z},B^{z})\qquad\text{in
}\mathcal{D}_{z}\times\mathbb{R}\times\mathcal{C}(\mathbb{R}_{+},\mathbb{R}),$
(4.4)

Moreover, we also have the convergence of the domain encircled by the
outermost loop $D^{z,\mu}_{\tau^{\mu}_{z}}$ containing $z$ towards
$D^{z}_{\tau_{z}}$ in the sense of Carathéodory jointly with the above
convergence.

Exactly as in [AHPS23], our proof of Proposition 4.3 goes through the
approximations $D^{0,\mu,n}$ and $D^{0,n}$.

######  Lemma 4.6.

Let $d_{\mathcal{D}}$ be the distance on $\mathcal{D}$. We have, a.s.
$\tau^{n}_{0}\rightarrow\tau_{0}$ as $n\to\infty$ and

$\sup_{\mu\in\mathbb{R}}d_{\mathcal{D}}(D^{0,\mu,n},D^{0,\mu})\mathop{\longrightarrow}\limits_{n\to\infty}^{\text{a.s}}0.$

###### Proof.

The convergence of $\tau^{n}_{0}$ stems from elementary properties of the
Brownian motion. To prove the uniform convergence in $\mathcal{D}$, let us
define the driving functions using the same Brownian motion $B$. Let $T>0$. By
Remark 4.2, it suffices to check that if
$F:\partial\mathbb{D}\times[0,T]\to\mathbb{R}$ is a continuous function,
bounded in absolute value by some constant $M>0$, then a.s.

$\int_{0}^{T}F((W^{\mu,n}_{0})_{s\wedge\tau_{0}^{n}},s)ds\mathop{\longrightarrow}\limits_{n\to\infty}\int_{0}^{T}F((W^{\mu}_{0})_{s\wedge\tau_{0}},s)ds,$

uniformly on $\mu\in\mathbb{R}$. This follows from the convergence of
$\tau^{n}_{0}$ since

$\displaystyle\left\lvert\int_{0}^{T}F((W^{\mu,n}_{0})_{s\wedge\tau_{0}^{n}},s)ds-\int_{0}^{T}F((W^{\mu}_{0})_{s\wedge\tau_{0}},s)ds\right\rvert\leq
2M(\tau_{0}-\tau^{n}_{0}).$

∎

######  Lemma 4.7.

Fix $n\in\mathbb{N}$. Then

$(D^{0,\mu,n},\tau^{n}_{0},B)\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}(D^{0,n},\tau^{n}_{0},B),$

where the first coordinate converges in $\mathcal{D}$.

###### Proof.

We set for all $k\in[\\![1,\Lambda^{n}]\\!]$,
$X_{k}^{\mu,n}=(W^{\mu}_{0})_{S_{k}^{n}},$ and ${\bf
X}^{\mu,n}=(X^{\mu,n}_{k})_{1\leq k\leq\Lambda^{n}}$. For the uniform
exploration we set ${\bf X}^{n}=(X^{n}_{k})_{1\leq k\leq\Lambda^{n}}$. We also
write ${\bf e}^{n}\coloneqq(e^{n}_{k})_{1\leq k\leq\Lambda^{n}}$ the family of
the excursions of height larger than $1/2^{n}$ before time $\tau_{0}$. To
prove the above lemma, it is enough to verify that

${\bf
X}^{\mu,n}\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}{\bf
X}^{n}.$ (4.5)

Indeed, we know that $W_{0}^{n}$ is defined by (4.3) and moreover for all
$k\in[\\![1,\Lambda^{n}]\\!]$, for all $s\in[L^{n}_{k-1},L^{n}_{k})$,

$\displaystyle(W^{\mu,n}_{0})_{s}$
$\displaystyle=(W^{\mu}_{0})_{S^{n}_{k}}\exp\left(i\left(2B_{s+S^{n}_{k}}-2B_{S^{n}_{k}}-\int_{S^{n}_{k}}^{s+S^{n}_{k}}\cot(B_{u})du-\mu(\ell^{0}_{s}-\ell^{0}_{S^{n}_{k}})\right)\right)$
$\displaystyle=X^{\mu,n}_{k}\exp\left(i\left(2B_{s+S^{n}_{k}}-\int_{S^{n}_{k}}^{s+S^{n}_{k}}\cot(B_{u})du\right)\right).$

Thus, if ones assumes (4.5), then we have clearly the convergence of the
driving function $W^{\mu,n}_{0}$ towards $W^{n}_{0}$ in the Skorokhod $J_{1}$
topology. This entails the convergence in $\mathcal{D}$ of
$(D^{0,\mu,n}_{s})_{s\geq 0}$ by Remark 4.2. Let us now prove the convergence
(4.5). One can see that

$(W_{0}^{\mu})_{S^{n}_{1}}=\exp\left(i\left(2B_{S^{n}_{1}}-\int_{0}^{S^{n}_{1}}\cot(B_{s})ds-\mu\ell^{0}_{S_{1}^{n}}\right)\right),$

and that for all $k\geq 2$,

$(W_{0}^{\mu})_{S^{n}_{k}}=(W_{0}^{\mu})_{S^{n}_{k-1}}\exp\left(i\left(2B_{S_{k}^{n}}-\int_{S^{n}_{k-1}}^{S^{n}_{k}}\cot(B_{s})ds-\mu(\ell^{0}_{S^{n}_{k}}-\ell^{0}_{T^{n}_{k-1}})\right)\right).$

But the $\ell^{0}_{S^{n}_{k}}-\ell^{0}_{T^{n}_{k-1}}$’s are i.i.d. exponential
random variables whose parameter $\lambda_{n}>0$ only depends on $n$ (and not
on $\mu$). Thus, to conclude it suffices to see that if $E$ is an exponential
random variable of parameter $\lambda_{n}>0$, then

$(E,e^{-i\mu
E})\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}(E,X^{n}_{1}),$

where $X^{n}_{1}$ is independent from $E$, which is elementary. ∎

###### Proof of Proposition 4.3.

This follows by combining Lemma 4.6, Lemma 4.7 and the fact that $D^{0,n}$
converges towards the uniform exploration of the $\mathrm{CLE}_{4}$ targeted
at zero. ∎

###  4.3 Joint convergence of the exploration branches

Before proving Proposition 4.1, let us show the convergence of the interiors
of the loops drawn along the exploration.

######  Lemma 4.8.

Let $\varepsilon>0$. For all $i\geq 1$, let $e_{\varepsilon,i}$ be the $i$-th
excursion of $B$ of height at least $\varepsilon$ away from $\pi\mathbb{Z}$
during an interval $[S_{\varepsilon,i},T_{\varepsilon,i}]$. Let
$\mathcal{L}^{0,\mu}_{\varepsilon,i}$ (resp.
$\mathcal{L}^{0}_{\varepsilon,i}$) denote the loop drawn by ${\bf D}^{0,\mu}$
(resp. ${\bf D}^{0}$) during the interval
$[S_{\varepsilon,i},T_{\varepsilon,i}]$, so that the domain inside the loop is
$\mathcal{B}^{0,\mu}_{\varepsilon,i}={\bf
D}^{0,\mu}_{S_{\varepsilon,i}}\setminus\overline{{\bf
D}^{0,\mu}_{T_{\varepsilon,i}}}$ (resp. $\mathcal{B}^{0}_{\varepsilon,i}={\bf
D}^{0}_{S_{\varepsilon,i}}\setminus\overline{{\bf
D}^{0}_{T_{\varepsilon,i}}}$). Then we have, jointly with the convergence of
Proposition 4.4, for all $\varepsilon>0$,

$\left(\mathcal{B}^{0,\mu}_{\varepsilon,i}\right)_{i\geq
1}\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}\left(\mathcal{B}^{0}_{\varepsilon,i}\right)_{i\geq
1}$

in terms of finite dimensional distributions for the Carathéodory topology and
for the Hausdorff topology for their closure.

###### Proof.

It suffices to prove the convergence in the sense of Carathéodory since the
outermost loop surrounding a point $z$ is constant in law as $\mu\to\infty$.
Let $\mu_{n}\to\infty$. By Skorokhod’s representation theorem, we assume that
the convergence of Proposition 4.3 holds almost surely along $\mu_{n}$.

Let $\varepsilon>0$. It is enough to prove the convergence for the excursions
before the time $\tau_{0}$. Let $i\geq 1$ such that
$T_{\varepsilon,i}<\tau_{0}$. By Proposition 4.3, we know that a.s.

$D^{0,\mu_{n}}_{S_{\varepsilon,i}}\mathop{\longrightarrow}\limits_{n\to\infty}D^{0}_{S_{\varepsilon,i}}$
(4.6)

in the sense of Carathéodory.

Besides, let us denote by
$g^{0,\mu_{n}}_{S_{\varepsilon,i}}:D^{0,\mu_{n}}_{S_{\varepsilon,i}}\to\mathbb{D}$
(resp. $g^{0}_{S_{\varepsilon,i}}:D^{0}_{S_{\varepsilon,i}}\to\mathbb{D}$) the
unique conformal map such that $g^{0,\mu_{n}}_{S_{\varepsilon,i}}(0)=0$ and
$(g^{0,\mu_{n}}_{S_{\varepsilon,i}})^{\prime}(0)=e^{-s}$ (resp.
$g^{0}_{S_{\varepsilon,i}}(0)=0$ and
$(g^{0}_{S_{\varepsilon,i}})^{\prime}(0)=e^{-s}$). Then by definition of the
radial Loewner chain, we have

$\frac{1}{(W^{\mu_{n}}_{0})_{S_{\varepsilon,i}}}g_{S_{\varepsilon,i}}^{0,\mu_{n}}\left(\mathcal{L}^{0,\mu_{n}}_{\varepsilon,i}\right)=\frac{1}{X_{i,\varepsilon}}g^{0}_{S_{\varepsilon,i}}\left(\mathcal{L}^{0}_{\varepsilon,i}\right),$

where $X_{i,\varepsilon}$ is a uniform random variable in $\partial\mathbb{D}$
which comes from the approximation of the uniform exploration. One concludes
using the fact that the $(W^{\mu_{n}}_{0})_{S_{\varepsilon,i}}$’s for $i\geq
1$ converge to independent uniform random variables on $\partial\mathbb{D}$ as
in the end of the proof of Lemma 4.7, so that

$g_{S_{\varepsilon,i}}^{0,\mu_{n}}\left(\mathcal{B}^{0,\mu_{n}}_{\varepsilon,i}\right)\mathop{\longrightarrow}\limits_{n\to\infty}g_{S_{\varepsilon,i}}^{0}\left(\mathcal{B}^{0}_{\varepsilon,i}\right)$

in the sense of Carathéodory in distribution jointly in $i\geq 1$. By taking
(4.6) into account, this ends the proof. ∎

Let us conclude this section by ending the proof of Proposition 4.1.

###### Proof of Proposition 4.1.

Let us first prove the convergence of the exploration branches. More
precisely, we want the convergences (4.4) to hold jointly in
$z\in\mathbb{D}\cap\mathbb{Q}^{2}$.

For all $\mu\in\mathbb{R}$, for all $z\neq w\in\mathbb{D}\cap\mathbb{Q}^{2}$,
we define the time $\sigma_{z,w}^{\mu}$ (resp. $\sigma_{z,w}$) at which the
radial $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$ exploration (resp. uniform
exploration) towards $z$ separates $w$ from $z$ by

$\sigma^{\mu}_{z,w}=\inf\\{s\geq 0;\ w\not\in{\bf
D}^{z,\mu}_{s}\\}\qquad\text{(resp.}\enskip\sigma_{z,w}=\inf\\{s\geq 0;\
w\not\in{\bf D}^{z}_{s}\\}\text{).}$

By the target invariance of the explorations and by the renewal property, it
is enough to show that for all $z\neq w\in\mathbb{D}\cap\mathbb{Q}^{2}$ we
have the convergence of the separation time

$\sigma^{\mu}_{z,w}\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}\sigma_{z,w},$

jointly with the convergence of the branch ${\bf D}^{z,\mu}$. Actually, by
conformal invariance, it is enough to prove that for all
$w\in(\mathbb{D}\cap\mathbb{Q}^{2})\setminus\\{0\\}$,

$\sigma^{\mu}_{0,w}\mathop{\longrightarrow}\limits_{\mu\to\infty}^{(\mathrm{d})}\sigma_{0,w},$
(4.7)

jointly with the convergence of the branch ${\bf D}^{0,\mu}$. The convergence
of the separation time (4.7) will be easier to prove in our context than in
[AHPS23]. One can first see that $\sigma^{\mu}_{0,w}$ is tight as
$\mu\to\infty$. Indeed, for all $\mu\in\mathbb{R}$, let $I^{\mu}$ be the
smallest integer $i\geq 1$ such that the $i$-th loop surrounding $0$ does not
contain $w$. Then $\sigma^{\mu}_{0,w}$ is smaller than the time at which we
draw the $I^{\mu}$-th loop. If we assumed by contradiction that
$\sigma^{\mu_{n}}_{0,z}\to\infty$ as $n\to\infty$ for some sequence
$\mu_{n}\to\infty$, then we would have $I^{\mu_{n}}\rightarrow\infty$ and then
$w$ would be at distance zero from $0$ by the Hausdorff convergence of the
domains inside the loops encircling $0$ of Lemma 4.8, absurd. Similarly,
$\sigma_{w,0}^{\mu}$ is also tight as $\mu\to\infty$. Now let
$\mu_{n}\to\infty$. By taking a subsequence, again denoted by $\mu_{n}$, we
may assume that

$({\bf D}^{0,\mu_{n}},\tau^{\mu_{n}}_{0},B^{0,\mu_{n}},{\bf
D}^{w,\mu_{n}},\tau_{w}^{\mu_{n}},B^{w,\mu_{n}})\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{d})}({\bf
D}^{0},\tau_{0},B^{0},\widetilde{\bf
D}^{w},\widetilde{\tau}_{w},\widetilde{B}^{w})$ (4.8)

in
$\mathcal{D}\times\mathbb{R}\times\mathcal{C}(\mathbb{R}_{+},\mathbb{R})\times\mathcal{D}_{w}\times\mathbb{R}\times\mathcal{C}(\mathbb{R}_{+},\mathbb{R})$,
where $(\widetilde{\bf D}^{w},\widetilde{\tau}_{w},\widetilde{B}^{w})$ has the
same law as $({\bf D}^{w},\tau_{w},B^{w})$, jointly with the convergence of
the separation times

$(\sigma^{\mu_{n}}_{0,w},\sigma^{\mu_{n}}_{w,0})\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{d})}(\sigma^{*}_{0,w},\sigma^{*}_{w,0})$
(4.9)

towards some random variables $\sigma^{*}_{0,w},\sigma^{*}_{w,0}$. For all
$n\geq 0$, let
$\varphi_{n}:[0,\sigma^{\mu_{n}}_{0,w}]\to[0,\sigma^{\mu_{n}}_{w,0}]$ be the
continuous increasing function such that for all
$s\in[0,\sigma^{\mu_{n}}_{0,w}]$, we have
$B_{s}^{0,\mu_{n}}=B^{w,\mu_{n}}_{\varphi_{n}(s)}$ (given by construction of
the branching exploration). One can also express $\varphi_{n}(s)$ as minus the
logarithm of the conformal radius of $D^{0,\mu_{n}}_{s}$ seen from $w$. Since
$\sigma^{\mu_{n}}_{w,0}$ converges in distribution, and thanks to (4.8), we
may take another subsequence again denoted by $\mu_{n}$ such that for all
$s\in[0,\sigma^{*}_{0,w})\cap\mathbb{Q}$, we have

$\varphi_{n}(s)\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{d})}\varphi^{*}(s)$
(4.10)

for some increasing process
$(\varphi^{*}(t))_{t\in[0,\sigma^{*}_{0,w})\cap\mathbb{Q}}$ jointly with the
convergences (4.8) and (4.9). This implies that
$B^{0}_{t}=\widetilde{B}^{w}_{\varphi^{*}(t)}$ for all
$t\in[0,\sigma^{*}_{0,w})\cap\mathbb{Q}$, hence $\varphi^{*}$ is strictly
increasing. Similarly, we can also assume that for all
$s\in[0,\sigma^{*}_{w,0})\cap\mathbb{Q}$, jointly with (4.8), (4.9) and
(4.10), we have

$\varphi_{n}^{-1}(s)\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{d})}\psi^{*}(s)$
(4.11)

for some increasing process
$(\psi^{*}(s))_{s\in[0,\sigma^{*}_{w,0})\cap\mathbb{Q}}$. For all
$s\in[0,\sigma^{*}_{w,0})\cap\mathbb{Q}$, we also have
$B^{0}_{\psi^{*}(s)}=\widetilde{B}^{w}_{s}$. So
$B^{0}_{|[0,\sigma^{*}_{0,w}]}$ is obtained as a time change of
$\widetilde{B}^{z}_{|[0,\sigma^{*}_{w,0}]}$, which is itself a time change of
$B^{0}_{|[0,\sigma^{*}_{0,w}]}$ Thus, $\varphi^{*}$ and $\psi^{*}$ can be
extended to continuous increasing bijections between $[0,\sigma^{*}_{0,w}]$
and $[0,\sigma^{*}_{w,0}]$ such that $\varphi^{*}=(\psi^{*})^{-1}$. As a
result, the convergences (4.10) and (4.11) hold uniformly on the corresponding
intervals.

For all $\varepsilon>0$, for all $i,n\geq 1$, let
$[S^{0,\mu_{n}}_{\varepsilon,i},T^{0,\mu_{n}}_{\varepsilon,i}]$ (resp.
$[S^{0}_{\varepsilon,i},T^{0}_{\varepsilon,i}]$,
$[S^{w,\mu_{n}}_{\varepsilon,i},T^{w,\mu_{n}}_{\varepsilon,i}]$,
$[\widetilde{S}^{w}_{\varepsilon,i},\widetilde{T}^{w}_{\varepsilon,i}]$) be
the interval of the $i$-th excursion of $B^{0,\mu_{n}}$ (resp. $B^{0}$,
$B^{w,\mu_{n}}$, $\widetilde{B}^{w}$) of height larger than $\varepsilon$,
during which the exploration ${\bf D}^{0,\mu_{n}}$ (resp. ${\bf D}^{0}$, ${\bf
D}^{w,\mu_{n}}$, ${\bf D}^{w}$) draws a loop encircling a domain
$\mathcal{B}^{0,\mu_{n}}_{\varepsilon,i}$ (resp.
$\mathcal{B}^{0}_{\varepsilon,i}$, $\mathcal{B}^{w,\mu_{n}}_{\varepsilon,i}$,
$\widetilde{\mathcal{B}}^{w}_{\varepsilon,i}$). We may assume by Lemma 4.8
that these domains converge: for all $\varepsilon>0$, jointly with the
previous convergences,

$\left(\mathcal{B}^{0,\mu_{n}}_{\varepsilon,i},\mathcal{B}^{w,\mu_{n}}_{\varepsilon,i}\right)_{i\geq
1}\mathop{\longrightarrow}\limits_{n\to\infty}^{(\mathrm{d})}\left(\mathcal{B}^{0}_{\varepsilon,i},\widetilde{\mathcal{B}}^{w}_{\varepsilon,i}\right)_{i\geq
1}$

in the sense of Carathéodory and also for the Hausdorff distance when we
consider their closure. Notice that, by the convergence of the Brownian
motions, we also have the convergence of the times of start and end of the
excursions of height larger than $\varepsilon$. By Skorokhod’s representation
theorem, we assume that all these convergences hold almost surely.

If $s<\sigma_{0,w}$, then $w\in{\bf D}^{0}_{s}$, so that for $n$ large enough
we have $w\in{\bf D}^{0,\mu_{n}}_{s}$, hence $s<\sigma_{0,w}^{\mu_{n}}$. Thus
we have plainly $\sigma^{*}_{0,w}\geq\sigma_{0,w}$. Let us now prove that
$\sigma^{*}_{0,w}\leq\sigma_{0,w}$. Let $\varepsilon>0$ small enough so that
we can find $i\geq 1$ such that
$S^{0}_{\varepsilon,i}<T^{0}_{\varepsilon,i}<\sigma^{*}_{0,w}$. Let
$s<S^{0}_{\varepsilon,i}$. Then we have by definition of the exploration
$\mathcal{B}^{0}_{\varepsilon,i}\subset{\bf D}^{0}_{s}$ and
$\widetilde{\mathcal{B}}^{w}_{\varepsilon,i}\subset\widetilde{\bf
D}^{w}_{\varepsilon,i}$. But since $T^{0}_{\varepsilon,i}<\sigma^{*}_{0,w}$ we
have for all $n$ large enough
$T^{0,\mu_{n}}_{\varepsilon,i}<\sigma^{\mu_{n}}_{0,w}$. Thus, for all $n$
large enough,
$\mathcal{B}^{0,\mu_{n}}_{\varepsilon,i}=\mathcal{B}^{w,\mu_{n}}_{\varepsilon,i}$
(given that $B^{0,\mu_{n}}_{s}=B^{w,\mu_{n}}_{\varphi_{n}(s)}$ for all
$s\leq\sigma^{\mu_{n}}_{0,w}$). Hence
$\mathcal{B}^{0}_{\varepsilon,i}=\widetilde{\mathcal{B}}^{w}_{\varepsilon,i}$
a.s. by the Hausdorff convergence. Therefore ${\bf
D}^{0}_{s}\cap\widetilde{\bf D}^{w}_{\varphi^{*}(s)}\neq\emptyset$ almost
surely. But then by taking some $z\in{\bf D}^{0}_{s}\cap\widetilde{\bf
D}^{w}_{\varphi^{*}(s)}$, we see by the Carathéodory convergence that the
limits of ${\bf D}^{0,\mu_{n}}_{s}={\bf D}^{w,\mu_{n}}_{\varphi_{n}(s)}$
coincide, i.e. that ${\bf D}^{0}_{s}=\widetilde{\bf D}^{w}_{\varphi^{*}(s)}$.
As a consequence, $s<\sigma_{0,w}$. By taking $\varepsilon$ small enough and
$i$ such that $S_{\varepsilon,i}^{0}$ is close enough to $\sigma^{*}_{0,w}$ we
obtain that $\sigma^{*}_{0,w}\leq\sigma_{0,w}$, hence (4.7). This concludes
the proof since the domains $D^{z,w,\mu_{n}}_{s}$ for $s\geq 0$ correspond up
to a time-change to the branch ${\bf D}^{w,\mu_{n}}$ stopped when $z$ and $w$
are separated. ∎

##  5 Conformally invariant distance and quantum natural distance

This section is devoted to the relationships between the distance
$\mathrm{d}_{\mathrm{WW}}$ to the boundary of [WW13] and the quantum natural
distance $\mathrm{d}_{\mathrm{q}}$ introduced by [AHPS23]. We prove Theorem
1.2 using Theorem 1.1, which relates these two distances via a Lamperti type
transform modulo a process with stationary increments.

###### Proof of Theorem 1.2.

Let $\tau$ be defined by (1.2) with $\alpha=-1$. We know from Theorem 1.1 that

$({\mathcal{Y}}({\sigma(t)}))_{t\geq
0}\mathop{=}\limits^{(\mathrm{d})}(e^{\xi(\tau(t))})_{t\geq 0},$

so that $(\log(\mathcal{Y}(\sigma(\tau^{-1}(t)))))_{t\geq 0}$ has the same law
as $\xi$ and we identify the two processes using an appropriate coupling. By
(2.2) and by the fact that the Poisson point process (PPP) of loops
$(\hat{\gamma}_{u})_{u\geq 0}$ of intensity $M$ introduced in Subsection 2.1
has an infinite intensity, we get that
$u\mapsto\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},C_{u})$ is increasing, so
that $t\mapsto\sigma(t)$ is continuous. Similarly, since the Lévy measure of
$\xi$ is infinite, $t\mapsto\sigma(t)$ is increasing.

To end the proof, it is enough to show that the process
$(\sigma(\tau^{-1}(t)))_{t\geq 0}$ has stationary increments. Let us denote by
$(\mathcal{E}_{t})_{t\geq 0}$ (resp. $(\mathcal{F}_{u})_{u\geq 0}$) the
augmented natural filtration of the càdlàg process
$(\mathcal{Y}(\sigma(t)))_{t\geq 0}$ (resp. $(\mathcal{Y}(u))_{u\geq 0}$). One
can write

$(\Delta_{+}\log(\mathcal{Y}(u)))_{u\geq
0}=(\Delta_{+}\log(\nu^{2}_{h^{2}}(\partial C_{u})))_{u\geq 0}.$

Each positive jump at some time $u$ corresponds to the discovery of a simple
loop $F_{u}^{-1}(\hat{\gamma}_{u})$ where $(\hat{\gamma}_{u})_{u\geq 0}$ is a
PPP which has the same law as described in Subsection 2.1 and
$F_{u}:C_{u}\to\mathbb{D}$ is the conformal transformation which is given by
$F_{u}=\Phi\circ\hat{F}_{u}^{z_{j_{k}}}\circ\Phi^{-1}$ when
$u\in[\tau_{k-1},\tau_{k})$ using the notation introduced in Subsections 2.1
and 2.3, and where $\Phi:\mathbb{H}\to\mathbb{D}$ is a fixed conformal map.

Furthermore, by Theorem 1.1, for all $t\geq 0$, conditionally on
$(\mathcal{Y}(\sigma(s)))_{0\leq s\leq t}$, the unexplored region
$(C_{\sigma(t)},h^{2}_{|C_{\sigma(t)}})$ is a $2$-quantum disk of quantum
boundary length $\nu^{2}_{h^{2}}(\partial C_{\sigma(t)})$. In other words,
conditionally on $(\mathcal{Y}(\sigma(s)))_{0\leq s\leq t}$, the quantum
surface

$\left(\mathbb{D},h\circ
F_{\sigma(t)}^{-1}+2\log|(F_{\sigma(t)}^{-1})^{\prime}|-\log(\nu^{2}_{h^{2}}(\partial
C_{\sigma(t)}))\right)$

is a unit boundary $2$-quantum disk. But using the fact that $\tau^{-1}(t)$ is
an $(\mathcal{E}_{t})_{t\geq 0}$-stopping time, we get that for all $t\geq 0$,
conditionally on $\mathcal{E}_{\tau^{-1}(t)}$, the quantum surface

$\left(\mathbb{D},h\circ
F_{\sigma(\tau^{-1}(t))}^{-1}+2\log|(F_{\sigma(\tau^{-1}(t))}^{-1})^{\prime}|-\log(\nu^{2}_{h^{2}}(\partial
C_{\sigma(\tau^{-1}(t))}))\right)$ (5.1)

is a unit boundary $2$-quantum disk.

Besides, by (2.2) we know that
$u\mapsto\sigma^{-1}(u)=\mathrm{d}_{\mathrm{q}}(\partial\mathbb{D},C_{u})$ is
adapted with respect to $(\mathcal{F}_{u})_{u\geq 0}$. Moreover, for all
$u\geq 0$,

$\tau(\sigma^{-1}(u))=\int_{0}^{\sigma^{-1}(u)}\frac{dt}{\mathcal{Y}(\sigma(t))}=\int_{0}^{u}\frac{\sigma^{-1}(dv)}{\mathcal{Y}(v)}.$

So $(\tau(\sigma^{-1}(u)))_{u\geq 0}$ is $(\mathcal{F}_{u})_{u\geq
0}$-adapted. In particular, we obtain that for all $t\geq 0$, the random
variable $\sigma(\tau^{-1}(t))$ is an $(\mathcal{F}_{u})_{u\geq 0}$-stopping
time. As a result, given that $(\hat{\gamma}_{u})_{u\geq 0}$ is independent of
$h^{2}$, we get that for all $t\geq 0$, conditionally on
$\mathcal{F}_{\sigma(\tau^{-1}(t))}$, the point process
$(\hat{\gamma}_{\sigma(\tau^{-1}(t))+u})_{u\geq 0}$ has the same law as
$(\hat{\gamma}_{u})_{u\geq 0}$ i.e. it is a PPP of intensity $M$. In addition
to that, by definition of $(F_{u})_{u\geq 0}$ from $(\hat{\gamma}_{u})_{u\geq
0}$, we have that conditionally on $\mathcal{F}_{\sigma(\tau^{-1}(t))}$, the
pair $((\hat{\gamma}_{\sigma(\tau^{-1}(t))+u})_{u\geq
0},(F_{\sigma(\tau^{-1}(t))}\circ F_{\sigma(\tau^{-1}(t))+u}^{-1})_{u\geq 0})$
has the same law as $((\hat{\gamma}_{u})_{u\geq 0},(F_{u}^{-1})_{u\geq 0})$.

Thus, by taking the above observation on the quantum disk (5.1) into account,
we have that conditionally on
$\mathcal{E}_{\tau^{-1}(t)}\subseteq\mathcal{F}_{\sigma(\tau^{-1}(t))}$, the
quantum disk (5.1) is a unit boundary length $2$-LQG disk independent of the
pair $((\hat{\gamma}_{\sigma(\tau^{-1}(t))+u})_{u\geq
0},(F_{\sigma(\tau^{-1}(t))}\circ F_{\sigma(\tau^{-1}(t))+u}^{-1})_{u\geq
0})$, which has the same law as $((\hat{\gamma}_{u})_{u\geq
0},(F_{u}^{-1})_{u\geq 0})$. In other words, conditionally on
$\mathcal{E}_{\tau^{-1}(t)}$, the quantum disk (5.1) is a unit boundary length
$2$-LQG disk independent of
$(F_{\sigma(\tau^{-1}(t))}(C_{\sigma(\tau^{-1}(t))+u}))_{u\geq 0}$, which has
the same law as $(C_{u})_{u\geq 0}$.

Besides, for all $u\geq 0$,

$\displaystyle\log\mathcal{Y}_{\sigma(\tau^{-1}(t))+u}-\log\mathcal{Y}_{\sigma(\tau^{-1}(t))}$
$\displaystyle=\log\left(\frac{\nu^{2}_{h^{2}}(F_{\sigma(\tau^{-1}(t))}^{-1}(F_{\sigma(\tau^{-1}(t))}(\partial
C_{\sigma(\tau^{-1}(t))+u})))}{\nu^{2}_{h^{2}}(\partial
C_{\sigma(\tau^{-1}(t))})}\right)$ $\displaystyle=\log\left(\nu^{2}_{h\circ
F_{\sigma(\tau^{-1}(t))}^{-1}+2\log|(F_{\sigma(\tau^{-1}(t))}^{-1})^{\prime}|-\log(\nu^{2}_{h^{2}}(\partial
C_{\sigma(\tau^{-1}(t))})}(F_{\sigma(\tau^{-1}(t))}(\partial
C_{\sigma(\tau^{-1}(t))+u}))\right),$

where we used (2.1) and the fact that $\nu^{2}_{h^{2}-\log
b}=\nu^{2}_{h^{2}}/b$ for all $b>0$. So conditionally on
$\mathcal{E}_{\tau^{-1}(t)}$, the process
$(\log\mathcal{Y}_{\sigma(\tau^{-1}(t))+u}-\log\mathcal{Y}_{\sigma(\tau^{-1}(t))})_{u\geq
0}$ has the same distribution as $(\log\mathcal{Y}_{u})_{u\geq 0}$.

It is now a simple matter to check that $(\sigma(\tau^{-1}(t)))_{t\geq 0}$ has
stationary increments. Indeed, let $t\geq 0$. For all $u,s\geq 0$, set

$\widetilde{\mathcal{Y}}_{u}\coloneqq\frac{\mathcal{Y}({\sigma(\tau^{-1}(t))+u})}{\mathcal{Y}({\sigma(\tau^{-1}(t))})}\qquad\text{and}\qquad\widetilde{\xi}(s)\coloneqq\xi({t+s})-\xi(t).$

Define $\widetilde{\sigma}$ from $\widetilde{\mathcal{Y}}$ as $\sigma$ is
defined from $\mathcal{Y}$. Notice that $\widetilde{\xi}$ is characterized by
the equality

$\forall s\geq
0,\qquad\log\left(\widetilde{\mathcal{Y}}\left({\widetilde{\sigma}\left(\int_{0}^{s}e^{\widetilde{\xi}(r)}dr\right)}\right)\right)=\widetilde{\xi}(s).$
(5.2)

Since
$(\log\mathcal{Y}_{\sigma(\tau^{-1}(t))+u}-\log\mathcal{Y}_{\sigma(\tau^{-1}(t))})_{u\geq
0}$ has the same distribution as $(\log\mathcal{Y}_{u})_{u\geq 0}$, we obtain
that

$\left((\log\mathcal{Y}_{\sigma(\tau^{-1}(t))+u}-\log\mathcal{Y}_{\sigma(\tau^{-1}(t))})_{u\geq
0},\widetilde{\sigma},\widetilde{\xi}\right)$

has the same law as $(\mathcal{Y},\sigma,\xi)$. Besides, by (5.2), we have for
all $s\geq 0$,

$\displaystyle\log\left(\widetilde{\mathcal{Y}}\left({\widetilde{\sigma}\left(\int_{0}^{s}e^{\widetilde{\xi}(r)}dr\right)}\right)\right)$
$\displaystyle=\widetilde{\xi}(s)$ $\displaystyle=\xi(t+s)-\xi(t)$
$\displaystyle=\log\mathcal{Y}({\sigma(\tau^{-1}(t+s))})-\log\mathcal{Y}({\sigma(\tau^{-1}(t))})$
$\displaystyle=\log\widetilde{Y}({\sigma(\tau^{-1}(t+s))-\sigma(\tau^{-1}(t))}).$

As a consequence, by density of the jumps of $\mathcal{Y}$ that for all $s\geq
0$,

$\sigma(\tau^{-1}(t+s))-\sigma(\tau^{-1}(t))={\widetilde{\sigma}\left(\int_{0}^{s}e^{\widetilde{\xi}(r)}dr\right)}$

Using the fact that for all $s\geq 0$, we have
$\tau^{-1}(s)=\int_{0}^{s}e^{\xi(r)}dr$, we conclude that
$\sigma\circ\tau^{-1}$ has stationary increments. ∎

##  6 Scaling limit of the distances to the boundary in $3/2$-stable maps

In this section, we describe the scaling limit of the distance between large
faces and the boundary in $3/2$ stable maps.

###  6.1 Background on $3/2$-stable maps and the peeling exploration

Recall the definition of finite $3/2$-stable maps from Subsection 1.1. If one
conditions a $3/2$-stable map of perimeter $2\ell$ on having at least $n$
edges and lets $n\to\infty$, then one obtains as a local limit an infinite
$3/2$-stable map of perimeter $2\ell$, whose law is written
$\mathbb{P}^{(\ell)}_{\infty}$. The infinite $3/2$-stable map was studied in
[BCM18] and we will rely on this work in the next subsection.

To study the distances $\mathrm{d}^{\dagger}_{\mathrm{gr}}$ and
$\mathrm{d}^{\dagger}_{\mathrm{fpp}}$ on Boltzmann planar maps, a convenient
tool is the peeling exploration. We only give a brief overview of its
definition and we refer the interested reader to the lecture notes [Cur23] and
[Bud]. It consists in discovering the faces of the map, starting from the root
face $f_{r}$ by looking at each step what is behind an edge on the boundary of
the explored region which is chosen according to a peeling algorithm. More
precisely, we say that a map $\mathfrak{e}$ with a distinguished simple face
different from $f_{r}$ called a hole is a submap (with one hole) of
$\mathfrak{m}$ if one can obtain $\mathfrak{m}$ by gluing a well chosen planar
map in the hole of $\mathfrak{e}$ (see Figure 4).

Figure 4: Left: a map $\mathfrak{e}$ with one hole which is a submap of the
map $\mathfrak{m}$ on the right. The map $\mathfrak{u}$ which is in the hole
of $\mathfrak{e}$ in the centre enables to recover $\mathfrak{m}$ after gluing
the boundary of $\mathfrak{u}$ to the boundary of the hole.

A peeling algorithm $\mathcal{A}$ is a function which takes a submap
$\mathfrak{e}$ and gives back an edge in the boundary $\partial\mathfrak{e}$
of the hole of $\mathfrak{e}$. A (filled-in) peeling exploration of
$\mathfrak{m}$ is an increasing sequence $(\overline{\mathfrak{e}}_{n})_{n\geq
0}$ of submaps of $\mathfrak{m}$ starting from the submap
$\overline{\mathfrak{e}}_{0}$ with only two faces which are the root face
$f_{r}$ and a hole with the same degree, such that for all $n\geq 0$ the
submap $\overline{\mathfrak{e}}_{n+1}$ is obtained from
$\overline{\mathfrak{e}}_{n}$ as follows:

  1. 1.

If the face $f$ which is on the other side of the peeled edge
$\mathcal{A}(\overline{\mathfrak{e}}_{n})$ is not in
$\overline{\mathfrak{e}}_{n}$, then $\overline{\mathfrak{e}}_{n+1}$ is
obtained by gluing $f$ to $\overline{\mathfrak{e}}_{n}$ onto the edge
$\mathcal{A}(\overline{\mathfrak{e}}_{n})$. We denote this case by $C_{k}$,
where $k=\deg(f)/2$.

  2. 2.

If the face $f$ on the other side of
$\mathcal{A}(\overline{\mathfrak{e}}_{n})$ is already in
$\overline{\mathfrak{e}}_{n}$, then $f$ must be on the boundary of the hole of
$\overline{\mathfrak{e}}_{n}$ and in this case $\overline{\mathfrak{e}}_{n+1}$
is obtained by identifying the two edges on the boundary of the hole. This
creates at most two holes. If $\mathfrak{m}$ is infinite, then we choose to
fill in the hole which contains a finite part of the map. If $\mathfrak{m}$ is
finite, we choose here to fill in the hole which has the smallest degree and
in case of equality we choose to fill in a hole chosen arbitrarily (so that in
the finite case we explore the locally largest component). This case is
written $G_{k,*}$ or $G_{*,k}$, where $2k$ is the degree of the hole which is
filled-in, depending on whether the filled-in hole is on the left or on the
right of the peeled edge. If there is not any hole, then the exploration
stops.

Figure 5: From left to right: two peeling steps of the map $\mathfrak{m}$ from
Figure 4 starting from the submap $\mathfrak{e}$ on the left. The red edge on
the left is the edge chosen by the peeling algorithm. The peeling of the red
edge causes an event of type $C_{4}$. The peeling algorithm then chooses
another edge on the boundary which is again in red. When we peel this edge, an
event of type $G_{3,1}$ happens and we fill-in the hole which has the smallest
perimeter with a map made in this case of only one edge.

When performing the exploration, one can keep track of the perimeter process
$(P(n))_{n\geq 0}$, where for all $n\geq 0$, the integer $P(n)$ is the half
degree of the hole of $\overline{\mathfrak{e}}_{n}$. Notice that on an event
of type $C_{k}$ for $k\geq 1$, the perimeter process increases by $k-1$ while
on an event of type $G_{*,k}$ or $G_{k,*}$ for some $k\geq 0$, the perimeter
process decreases by $k+1$.

A peeling algorithm which is particularly well suited to the study of the
distance $\mathrm{d}_{\mathrm{fpp}}^{\dagger}$ is the uniform peeling
algorithm. Given a submap $\mathfrak{e}$, it chooses uniformly at random an
edge in $\partial{e}$. Using this algorithm properly coupled with the
exponential random lengths defining $\mathrm{d}_{\mathrm{fpp}}^{\dagger}$, by
the elementary properties of exponential random variables, one can see that
for all $n\geq 0$, assuming that the exploration has not stopped before, the
fpp distance from $f_{r}$ to the hole $\mathfrak{u}_{n}$ of
$\overline{\mathfrak{e}}_{n}$ can be written

$T({n+1})\coloneqq\mathrm{d}^{\dagger}_{\mathrm{fpp}}(f_{r},\mathfrak{u}_{n})=\sum_{i=0}^{n}\frac{\mathcal{E}_{i}}{2P(i)},$
(6.1)

where the $\mathcal{E}_{i}$’s are i.i.d. exponential random variables of
parameter $1$. More precisely, $\overline{\mathfrak{e}}_{n}$ corresponds
exactly to the closed ball around $f_{r}$ of radius $T(n)$ where we filled-in
the holes which were filled-in during the exploration, and where $T(0)=0$ by
convention. In particular, a face $f$ which is discovered at time $n$ (i.e.
$f\in\mathrm{Faces}(\overline{\mathfrak{e}}_{n})\setminus\mathrm{Faces}(\overline{\mathfrak{e}}_{n-1}$))
is at distance $T(n)$ from the root face $f_{r}$. When the explored map has
the law $\mathbb{P}^{(\ell)}$ or $\mathbb{P}^{(\ell)}_{\infty}$, the
$\mathcal{E}_{i}$’s are independent from the perimeter process. See e.g.
Section 13.1 of [Cur23] or Subsection 3.2 of [Kam23] for details.

For the dual graph distance $\mathrm{d}^{\dagger}_{\mathrm{gr}}$, an
appropriate peeling algorithm is the peeling by layers algorithm
$\mathcal{A}_{\mathrm{layers}}$. Given a submap $\mathfrak{e}$ of
$\mathfrak{m}$, we define the height of an edge $e\in\partial\mathfrak{e}$ as
the dual graph distance between $f_{r}$ and the face next to $e$ which is not
in the hole. Assume that there exists an integer $h\geq 0$ such that all the
edges of $\partial\mathfrak{e}$ have heights in $\left\\{h,h+1\right\\}$ and
that the edges at height $h$ form a connected interval of
$\partial{\mathfrak{e}}$ (we denote this hypothesis by (H)). Then
$\mathcal{A}_{\mathrm{layers}}(\mathfrak{e})$ is defined as the leftmost edge
in $\partial\mathfrak{e}$ of height $h$ (see Figure 6) and when all the edges
of $\partial\mathfrak{e}$ are at the same height we choose one of them
arbitrarily. Notice that $\overline{\mathfrak{e}}_{0}$ satisfies (H) and that
if $\overline{\mathfrak{e}}_{n}$ satisfies (H) then with
$\mathcal{A}_{\mathrm{layers}}$ the submap $\overline{\mathfrak{e}}_{n+1}$
also satisfies (H) (with $h$ increasing by zero or one). For all $n\geq 0$,
let us denote by $H(n)$ the smallest height of the edges in
$\partial\overline{\mathfrak{e}}_{n}$ when performing a peeling by layers
exploration. The sequence $(H(n))_{n\geq 0}$ is called the height process.
Notice that $H(0)=0$ and that for all $n\geq 0$, we have
$H(n+1)-H(n)\in\\{0,1\\}$. See e.g. Section 13.2 of [Cur23] for more details.

Figure 6: Illustration of the peeling by layers algorithm. The faces are
colored according to their distance to the root face $f_{r}$. The edge chosen
by the algorithm $\mathcal{A}_{\mathrm{layers}}$ is in green.

To be consistent with the notation of [Cur23], [BBCK18] and others, we add a
subscript $\infty$ (resp. $*$) to the processes $(P(n))_{n\geq
0},(T(n))_{n\geq 0},(H(n))_{n\geq 0}$ under the law
$\mathbb{P}_{\infty}^{(\ell)}$ (resp. $\mathbb{P}^{(\ell)}$). Recall that the
peeling exploration under $\mathbb{P}^{(\ell)}$ satisfies a spatial Markov
property in the sense that for all $n\geq 0$, conditionally on the explored
region until time $n$, the maps which fill in the holes are independent
Boltzmann maps, see e.g. Proposition 4.6 of [Cur23] for more details.

We finally recall from [BBCK18] the scaling limit of the perimeter process
under $\mathbb{P}^{(\ell)}$ and $\mathbb{P}^{(\ell)}_{\infty}$ as
$\ell\to\infty$. Let $(\Upsilon^{\uparrow}(t))_{t\geq 0}$ be a symmetric
Cauchy process conditioned to stay positive and starting at $1$. More
precisely, $(\Upsilon^{\uparrow}(t))_{t\geq 0}$ is the Doob $h$-transform of
the Lévy process with no drift and no Brownian part, of Lévy measure
$(dx/|x|){\bf 1}_{x\neq 0}$ starting at one, using the function
$h(x)=\sqrt{x}$. We recall from Proposition 6.6 of [BBCK18] the following
convergences in distribution for the $J_{1}$ topology of Skorokhod:

$\displaystyle\text{under
}\mathbb{P}^{(\ell)}_{\infty},\qquad\left(\frac{P_{\infty}(\lfloor\ell
t\rfloor)}{\ell}\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left(\Upsilon^{\uparrow}(p_{\bf
q}t)\right)_{t\geq 0};$ (6.2) $\displaystyle\text{under
}\mathbb{P}^{(\ell)},\qquad\left(\frac{P_{*}(\lfloor\ell
t\rfloor)}{\ell}\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left(X^{(-1)}(\pi
p_{\bf q}t)\right)_{t\geq 0}.$ (6.3)

Notice here that the scaling limit of the perimeter of the locally largest
component (parametrized by the exploration time) corresponds already, up to a
linear time change, to the quantum boundary length of the locally largest
component during a uniform or $\mathrm{SLE}^{\langle\mu\rangle}_{4}(-2)$
exploration of a $\mathrm{CLE}_{4}$-decorated quantum disk, parametrized by
the quantum natural distance or the quantum length of the trunk, by Theorem
1.1.

###  6.2 Metric growth for infinite $3/2$-stable maps with increasing
perimeter

Here we establish the scaling limit of $T_{\infty}$ and $H_{\infty}$ under
$\mathbb{P}_{\infty}^{(\ell)}$ as $\ell\to\infty$. The scaling limit of
$T_{\infty}$ is very easy to obtain:

######  Proposition 6.1.

Using the uniform peeling exploration, we have jointly with (6.2) the
convergence in distribution for the topology of the uniform convergence on
compact sets:

$\text{under }\mathbb{P}^{(\ell)}_{\infty},\qquad\left(T_{\infty}(\lfloor\ell
t\rfloor)\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\left(\int_{0}^{t}\frac{1}{2\Upsilon^{\uparrow}(p_{\bf
q}s)}ds\right)_{t\geq 0}.$

###### Proof.

Using (6.1) and the fact that the $\mathcal{E}_{n}$’s are independent from
$P_{\infty}$, we have for all $t\geq 0$,

$\mathbb{E}^{(\ell)}_{\infty}\left(\left.\sum_{n=0}^{\lfloor\ell
t\rfloor-1}\left(\frac{\mathcal{E}_{n}-1}{2P_{\infty}(n)}\right)^{2}\right|P_{\infty}\right)=\sum_{n=0}^{\lfloor\ell
t\rfloor-1}\frac{1}{\left(2P_{\infty}(n)\right)^{2}}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathbb{P})}0,$

where the convergence comes from (6.2). Moreover, again by (6.2),

$\text{under }\mathbb{P}^{(\ell)}_{\infty},\qquad\left(\sum_{n=0}^{\lfloor\ell
t\rfloor-1}\frac{1}{2P_{\infty}(n)}\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left(\int_{0}^{t}\frac{1}{2\Upsilon^{\uparrow}(p_{\bf
q}s)}ds\right)_{t\geq 0}.$

This ends the proof since $t\mapsto\int_{0}^{t}1/(2\Upsilon^{\uparrow}(p_{\bf
q}s))ds$ is a continuous and increasing process. ∎

For the scaling limit of $H_{\infty}$, a $\log\ell$ factor appears:

######  Proposition 6.2.

For the peeling by layers exploration, we have the convergence in distribution
jointly with (6.2) for the topology of the uniform convergence on compact
sets:

$\text{under
}\mathbb{P}^{(\ell)}_{\infty},\qquad\left(\frac{H_{\infty}(\lfloor\ell
t\rfloor)}{\log\ell}\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\left(\int_{0}^{t}\frac{p_{\bf
q}}{2\Upsilon^{\uparrow}(p_{\bf q}s)}ds\right)_{t\geq 0}.$

To prove the above result, we will use the inverse $\theta$ of $H_{\infty}$
defined by

$\forall k\geq 0,\qquad\theta_{k}=\inf\left\\{n\geq 0,H_{\infty}(n)\geq
k\right\\}.$

We will rely on the following lemma from [BCM18]:

######  Lemma 6.3.

(Corollary 2 of [BCM18]) For all $\varepsilon\in(0,1)$, for all
$k\in\mathbb{N}$,

$\lim_{\ell\to\infty}\mathbb{P}^{(\ell)}_{\infty}\left(\frac{p_{\bf
q}\log\ell}{2\ell}\frac{\theta_{k}}{k}\leq 1+\varepsilon\right)=1.$

We then prove Proposition 6.2.

###### Proof of Proposition 6.2.

This proof uses the same techniques as the proof of Proposition 6 in [BCM18],
which gives a lower bound for $H_{\infty}(n)/(\log n)^{2}$ under
$\mathbb{P}^{(1)}_{\infty}$ as $n\to\infty$, and will also rely on Lemma 6 of
[BCM18], which bounds from above the mean increase of an interpolated version
of $H_{\infty}$. Let $C>0$. For all $\ell\geq 1$ and $k\geq 0$, set
$t_{k}^{(\ell)}=\lfloor C\frac{k\ell}{\log\ell}\rfloor$. Notice that

$\Delta t^{(\ell)}_{k}\coloneqq
t_{k+1}^{(\ell)}-t^{(\ell)}_{k}=(C+o(1))\frac{\ell}{\log\ell}$ (6.4)

when $\ell\to\infty$ where the $o(1)$ is uniform in $k$. Note also that for
all real $s>0$ we have

$t^{(\ell)}_{\lfloor\frac{s\log\ell}{C}\rfloor}=\lfloor\ell s\rfloor(1+o(1))$
(6.5)

as $\ell\to\infty$ where the $o(1)$ is non-positive.

Let $\delta\in(0,1)$. For all $\ell\geq 1$ and $k\geq 0$, let

$\eta^{(\ell)}_{k}=\left(H_{\infty}(t^{(\ell)}_{k+1})-H_{\infty}(t^{(\ell)}_{k})-C\frac{p_{\bf
q}\ell}{2P_{\infty}(t_{k}^{(\ell)})}\right){\bf
1}_{P_{\infty}(t^{(\ell)}_{k})/\ell\in[\delta,1/\delta]}.$

Take $\varepsilon\in(0,2\delta/Cp_{\bf q})$. Let
${N}^{(\ell)}_{k}=\lfloor(1-\varepsilon)(Cp_{\bf
q}\ell)/(2P_{\infty}(t^{(\ell)}_{k}))\rfloor$. Then one can see that

$\left\\{\frac{P_{\infty}(t^{(\ell)}_{k})}{\ell}\not\in[\delta,1/\delta]\right\\}\cup\left(\left\\{\frac{P_{\infty}(t^{(\ell)}_{k})}{\ell}\in[\delta,1/\delta]\right\\}\cap\left\\{{N}^{(\ell)}_{k}\leq
H_{\infty}(t^{(\ell)}_{k+1})-H_{\infty}(t^{(\ell)}_{k})\right\\}\right)\\\
\subset\left\\{\eta^{(\ell)}_{k}\geq-2\right\\}.$

Besides, by (6.4), for all $\ell$ large enough, for all $k\geq 0$, on the
event $\\{{P_{\infty}(t^{(\ell)}_{k})}/{\ell}\in[\delta,1/\delta]\\}$, we have

$\frac{\Delta
t^{(\ell)}_{k}}{{N}^{(\ell)}_{k}}\geq\frac{2P_{\infty}(t^{(\ell)}_{k})}{(1-\varepsilon/2)p_{\bf
q}\log P_{\infty}({t^{(\ell)}_{k}})}.$ (6.6)

We define for all $\ell\geq 1$ and $k\geq 0$ the stopping times
$\theta_{t_{k}^{(\ell)},0}=t_{k}^{(\ell)}$ and for all $i\geq 1$,

$\theta_{t_{k}^{(\ell)},i}=\inf\left\\{n\geq 0;\
H_{\infty}(n+t_{k}^{(\ell)})\geq H_{\infty}({t_{k}^{(\ell)}})+i\right\\}.$

Then by Lemma 6.3, for all $\gamma\in(0,1)$, for $\ell$ large enough, for all
$k\geq 0$,

$\displaystyle\mathbb{P}^{(\ell)}_{\infty}$
$\displaystyle\left(\left\\{\frac{P_{\infty}(t^{(\ell)}_{k})}{\ell}\in[\delta,1/\delta]\right\\}\cap\left\\{{N}^{(\ell)}_{k}\leq
H_{\infty}(t^{(\ell)}_{k+1})-H_{\infty}(t^{(\ell)}_{k})\right\\}\right)$
$\displaystyle=\mathbb{P}^{(\ell)}_{\infty}\left(\left\\{\frac{P_{\infty}(t^{(\ell)}_{k})}{\ell}\in[\delta,1/\delta]\right\\}\cap\left\\{\frac{\theta_{t_{k}^{(\ell)},{N}^{(\ell)}_{k}}}{{N}^{(\ell)}_{k}}\leq\frac{\Delta
t_{k}^{(\ell)}}{{N}^{(\ell)}_{k}}\right\\}\right)$
$\displaystyle\geq\mathbb{P}^{(\ell)}_{\infty}\left(\left\\{\frac{P_{\infty}(t^{(\ell)}_{k})}{\ell}\in[\delta,1/\delta]\right\\}\cap\left\\{\frac{\theta_{t_{k}^{(\ell)},{N}^{(\ell)}_{k}}}{{N}^{(\ell)}_{k}}\leq\frac{2P_{\infty}(t^{(\ell)}_{k})}{(1-\varepsilon/2)p_{\bf
q}\log P_{\infty}({t^{(\ell)}_{k}})}\right\\}\right)\qquad\text{by (\ref{eq
delta t k sur N})}$
$\displaystyle\geq(1-\gamma)\mathbb{P}^{(\ell)}_{\infty}\left(\frac{P_{\infty}(t^{(\ell)}_{k})}{\ell}\in[\delta,1/\delta]\right),$

the last inequality being uniform in $k$ since, under the event
$\\{{P_{\infty}(t^{(\ell)}_{k})}/{\ell}\in[\delta,1/\delta]\\}$, the random
variable ${N}^{(\ell)}_{k}$ is bounded above by $(Cp_{q})/(2\delta)$. Thus, by
taking $\gamma$ arbitrarily small,

$\lim_{\ell\to\infty}\inf_{k\geq
0}\mathbb{P}^{(\ell)}_{\infty}\left(\eta^{(\ell)}_{k}\geq-2\right)=1.$ (6.7)

Let $s\geq 0$. Let

$\mathcal{H}_{\ell,\delta}(s)=\frac{1}{\log\ell}\sum_{k=0}^{\lfloor(s\log\ell)/C\rfloor}(H_{\infty}(t^{(\ell)}_{k+1})-H_{\infty}(t^{(\ell)}_{k})){\bf
1}_{\forall n\in[\\![t_{k}^{(\ell)},t_{k+1}^{(\ell)}]\\!],\
P_{\infty}(n)/\ell\in[\delta,1/\delta]}.$

By (6.5), we have

$\mathcal{H}_{\delta,\ell}(s)\leq\frac{1}{\log\ell}H_{\infty}(\lfloor\ell
s\rfloor)$ (6.8)

Moreover, by (6.5) and (6.2), for all $\upsilon>0$, with probability $1-o(1)$
as $\delta\to 0$, for all $\ell$ large enough,

$\frac{1}{\log\ell}H_{\infty}(\lfloor\ell
s\rfloor)\leq\mathcal{H}_{\delta,\ell}(s+\upsilon).$ (6.9)

Notice that with probability $1-o(1)$

One can write

$\displaystyle\mathcal{H}_{\delta,\ell}(s)=$
$\displaystyle\frac{1}{\log\ell}\sum_{k=0}^{\lfloor(s\log\ell)/C\rfloor}C\frac{p_{\bf
q}\ell}{2P_{\infty}(t_{k}^{(\ell)})}{\bf 1}_{\forall
n\in[\\![t^{(\ell)}_{k},t^{(\ell)}_{k+1}]\\!],\
P_{\infty}(n)/\ell\in[\delta,1/\delta]}$
$\displaystyle+\frac{1}{\log\ell}\sum_{k=0}^{\lfloor(s\log\ell)/C\rfloor}\eta^{(\ell)}_{k}{\bf
1}_{\forall n\in[\\![t^{(\ell)}_{k},t^{(\ell)}_{k+1}]\\!],\
P_{\infty}(n)/\ell\in[\delta,1/\delta]}.$

Consider the first term.

$\displaystyle\frac{1}{\log\ell}\sum_{k=0}^{\lfloor(s\log\ell)/C\rfloor}C$
$\displaystyle\frac{p_{q}\ell}{2P_{\infty}({t_{k}^{(\ell)}})}{\bf 1}_{\forall
n\in[\\![t^{(\ell)}_{k},t^{(\ell)}_{k+1}]\\!],\
P_{\infty}(n)/\ell\in[\delta,1/\delta]}$
$\displaystyle=\frac{1}{\log\ell}\int_{0}^{\lfloor({s\log\ell})/{C}\rfloor+1}C\frac{p_{q}\ell}{2P_{\infty}({t_{\lfloor
v\rfloor}^{(\ell)}})}{\bf 1}_{\forall n\in[\\![t^{(\ell)}_{\lfloor
v\rfloor},t^{(\ell)}_{\lfloor v\rfloor+1}]\\!],\
{P_{\infty}(n)}/{\ell}\in[\delta,\frac{1}{\delta}]}dv$
$\displaystyle=O\left(\frac{1}{\log\ell}\right)+\int_{0}^{s}\frac{p_{q}\ell}{2P_{\infty}\left({t_{\lfloor({u\log\ell})/{C}\rfloor}^{(\ell)}}\right)}{\bf
1}_{\forall
n\in[\\![t_{\lfloor({u\log\ell})/{C}\rfloor}^{(\ell)},t_{\lfloor({u\log\ell})/{C}\rfloor+1}^{(\ell)}]\\!],\
{P_{\infty}\left(n\right)}/{\ell}\in[\delta,\frac{1}{\delta}]}du$
$\displaystyle\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\int_{0}^{s}\frac{p_{\bf
q}}{2\Upsilon^{\uparrow}(p_{\bf q}u)}{\bf 1}_{\Upsilon^{\uparrow}(p_{\bf
q}u)\in[\delta,1/\delta]}du\qquad\text{by (\ref{eq t s log ell}) and by
dominated convergence.}$

For the second term, one can see that the expectation of the negative part of
$\eta^{(\ell)}_{k}$ is bounded uniformly in $\ell\geq 1,k\geq 0$:

$\mathbb{E}^{(\ell)}_{\infty}\left((\eta^{(\ell)}_{k})_{-}\right)\leq
2+\mathbb{P}^{(\ell)}_{\infty}\left(\eta^{(\ell)}_{k}<-2\right)C\frac{p_{\bf
q}}{2\delta}.$

Hence, by (6.7),

$\lim_{\ell\to\infty}\sup_{k\geq
0}\mathbb{E}^{(\ell)}_{\infty}\left((\eta^{(\ell)}_{k})_{-}\right)\leq 2.$

Thus by taking $C\to\infty$, there exists a sequence of random variables
$X_{\ell}$ which are stochastically dominated by
$\mathcal{H}_{\delta,\ell}(s)$ such that $X_{\ell}$ converges in distribution
towards $\int_{0}^{s}({p_{\bf q}}/({2\Upsilon^{\uparrow}(p_{\bf q}u)})){\bf
1}_{\Upsilon^{\uparrow}(p_{\bf q}u)\in[\delta,1/\delta]}du$.

Let us then give an upper bound in expectation of
$\mathcal{H}_{\delta,\ell}(s)$. By Lemma 6 of [BCM18], for all
$\varepsilon^{\prime}>0$, there exists a constant $C^{\prime}>0$ such that for
all $\ell\geq 1$ and $k\geq 0$,

$\displaystyle\mathbb{E}^{(\ell)}_{\infty}$
$\displaystyle\left(\left(H_{\infty}(t^{(\ell)}_{k+1})-H_{\infty}(t^{(\ell)}_{k})\right){\bf
1}_{\forall n\in[\\![t_{k}^{(\ell)},t_{k+1}^{(\ell)}]\\!],\
P_{\infty}(n)/\ell\in[\delta,1/\delta]}\right)$ $\displaystyle\leq
2+(1+\varepsilon^{\prime})^{3}\frac{p_{\bf
q}}{2}\sum_{n=t^{(\ell)}_{k}}^{t^{(\ell)}_{k+1}-1}\mathbb{E}^{(\ell)}_{\infty}\left(\frac{\log
P_{\infty}(n)+C^{\prime}}{P_{\infty}(n)}{\bf
1}_{P_{\infty}(n)/\ell\in[\delta,1/\delta]}\right).$

As a result, by (6.5),

$\mathbb{E}^{(\ell)}_{\infty}\left(\mathcal{H}_{\delta,\ell}(s)\right)\leq\frac{1}{\log\ell}\left(2\left\lfloor\frac{s\log\ell}{C}\right\rfloor+2+(1+\varepsilon^{\prime})^{3}\frac{p_{\bf
q}}{2}\sum_{n=0}^{\lfloor
s\ell\rfloor-1}\mathbb{E}^{(\ell)}_{\infty}\left(\frac{\log
P_{\infty}(n)+C^{\prime}}{P_{\infty}(n)}{\bf
1}_{P_{\infty}(n)/\ell\in[\delta,1/\delta]}\right)\right).$

Thus, using (6.2), by dominated convergence as $\ell\to\infty$, and then by
letting $\varepsilon^{\prime}\to 0$ and $C\to\infty$, we get that for all
$\delta>0$,

$\limsup_{\ell\to\infty}\mathbb{E}^{(\ell)}_{\infty}\left(\mathcal{H}_{\delta,\ell}(s)\right)\leq\mathbb{E}\left(\int_{0}^{s}\frac{p_{\bf
q}}{2\Upsilon^{\uparrow}(p_{\bf q}v)}{\bf 1}_{\Upsilon^{\uparrow}(p_{\bf
q}v)\in[\delta,1/\delta]}dv\right).$ (6.10)

Besides, by the lower bound on $\mathcal{H}_{\delta,\ell}(s)$ and by
Skorokhod’s representation theorem, we may assume that the
$\mathcal{H}_{\delta,\ell}(s)$’s are defined on the same probability space and
that a.s.

$\liminf_{\ell\to\infty}\mathcal{H}_{\delta,\ell}(s)\geq\mathcal{H}_{\delta}(s)\coloneqq\int_{0}^{s}\frac{p_{\bf
q}}{2\Upsilon^{\uparrow}(p_{\bf q}v)}{\bf 1}_{\Upsilon^{\uparrow}(p_{\bf
q}v)\in[\delta,1/\delta]}dv.$ (6.11)

But by Fatou’s lemma, we deduce that

$\mathbb{E}\mathcal{H}_{\delta}(s)\leq\liminf_{\ell\to\infty}\mathbb{E}\mathcal{H}_{\delta,\ell}(s)\leq\limsup_{\ell\to\infty}\mathbb{E}\mathcal{H}_{\delta,\ell}(s)\leq\mathbb{E}\mathcal{H}_{\delta}(s),$

where the last inequality stems from (6.10), hence
$\lim_{\ell\to\infty}\mathbb{E}\mathcal{H}_{\delta,\ell}(s)=\mathbb{E}\mathcal{H}_{\delta}(s)$.
Moreover, notice that for all $\varepsilon^{\prime}>0$, by dominated
convergence and by (6.11),

$\displaystyle\mathbb{E}\left((\mathcal{H}_{\delta}(s)-\mathcal{H}_{\delta,\ell}(s))_{+}\right)$
$\displaystyle\leq\varepsilon^{\prime}+\mathbb{E}\left((\mathcal{H}_{\delta}(s)-\mathcal{H}_{\delta,\ell}(s)){\bf
1}_{\mathcal{H}_{\delta}(s)-\mathcal{H}_{\delta,\ell}(s)>\varepsilon^{\prime}}\right)$
$\displaystyle\leq\varepsilon^{\prime}+\mathbb{E}\left(\mathcal{H}_{\delta}(s){\bf
1}_{\mathcal{H}_{\delta}(s)>\mathcal{H}_{\delta,\ell}(s)+\varepsilon^{\prime}}\right)\mathop{\longrightarrow}\limits_{\ell\to+\infty}\varepsilon^{\prime}.$

Now, since for all $x\in\mathbb{R}$, $|x|=-x+2x_{+}$, we get that
$\mathbb{E}|\mathcal{H}_{\delta}(s)-\mathcal{H}_{\delta,\ell}(s)|\longrightarrow
0$ as $\ell\to\infty$. Using (6.8) and (6.9), this concludes the proof.

∎

###  6.3 Large $3/2$-stable maps and $\mathrm{CLE}_{4}$-decorated quantum
disks

We first derive the analogues of Propositions 6.1 and 6.2 for finite
$3/2$-stable maps by absolute continuity.

######  Corollary 6.4.

For the uniform peeling, jointly with (6.2) we have the convergence

$\text{under }\mathbb{P}^{(\ell)},\qquad\left(T_{*}(\ell t)\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\left(\int_{0}^{t}{1\over
2X^{(-1)}(\pi p_{\bf q}s)}ds\right)_{t\geq 0}.$

For the peeling by layers, jointly with (6.2) we have the convergence

$\text{under }\mathbb{P}^{(\ell)},\qquad\left({H_{*}(\ell
t)\over\log\ell}\right)_{t\geq
0}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\left(\int_{0}^{t}{p_{\bf
q}\over 2X^{(-1)}(\pi p_{\bf q}s)}ds\right)_{t\geq 0}.$

Then the convergences (1.3) and (1.4) are clearly a consequence of the above
corollary.

###### Proof.

It follows by mimicking the proof of Proposition 6.6 of [BBCK18]: by
Proposition 6.3 of [BBCK18], for all $\ell\geq 1$, the Radon-Nikodym
derivative of the law of the processes $(P_{*},T_{*},H_{*})$ until some time
$n\geq 0$ under $\mathbb{P}^{(\ell)}$ with respect to the law of
$(P_{\infty},T_{\infty},H_{\infty})$ until time $n$ under
$\mathbb{P}^{(\ell)}_{\infty}$ is given by
$f^{\uparrow}(\ell)/f^{\uparrow}(P_{\infty}(n))$ times the indicator function
that $P_{\infty}$ does not make any negative jump of size larger than its
half, where $f^{\uparrow}(\ell)\coloneqq h^{\uparrow}(\ell)/(W^{(\ell)}c_{\bf
q}^{-\ell})$ and $h^{\uparrow}(\ell)=2\ell 2^{-\ell}\binom{2\ell}{\ell}$. One
concludes using Propositions 6.1 and 6.2, together with the convergence in
distribution of $f^{\uparrow}(\ell)/f^{\uparrow}(P_{\infty}(\ell t))$ towards
$\Upsilon^{\uparrow}(p_{\bf q}t)^{-5/2}$ which comes from (1.1), and
identifying the Radon-Nikodym derivative of the law of $(X^{(-1)}(\pi p_{\bf
q}s))_{0\leq s\leq t}$ with respect to the law of $(\Upsilon^{\uparrow}(p_{\bf
q}s))_{0\leq s\leq t}$ by Proposition 4.1 of [BBCK18]. ∎

Let us now consider the whole exploration tree in order to prove Theorem 1.3:
instead of filling-in the holes, we run new explorations inside. Let us focus
on the dual graph distance $\mathrm{d}^{\dagger}_{\mathrm{gr}}$, using the
peeling by layers algorithm.

Recall the recursive definition of the discrete perimeter cell-system
$\left((P_{u}(k))_{k\geq 0}\right)_{u\in\mathbb{U}}$ associated with a
branching peeling exploration of a finite map $\mathfrak{m}$, where
$\mathbb{U}=\bigcup_{n\geq 0}(\mathbb{N}^{*})^{n}$ is the Ulam tree. The
process $P_{\emptyset}$ is the (half-)perimeter process of the locally largest
cycle. We denote by $(N_{j})_{j\geq 1}$ the times of negative jumps of
$P_{\emptyset}$ (usually ranked in the non-increasing order of jump sizes and
in the decreasing order of jump times if the jump sizes are equal). Each one
corresponds to an event of type $G_{k_{1},k_{2}}$, i.e. to a splitting of the
distinguished hole of half-perimeter $P_{\emptyset}(N_{j})$ into two holes of
half-perimeters $P_{\emptyset}(N_{j}+1)$ and
$\Delta_{-}P_{\emptyset}(N_{j}):=P_{\emptyset}(N_{j})-P_{\emptyset}(N_{j}+1)-1\leq
P_{\emptyset}(N_{j}+1)$. We then define $(P_{j}(k))_{k\geq 0}$ as the
perimeter process of the locally largest cycle during the exploration of the
submap filling the second hole. We then define $P_{uj}$ from $P_{u}$ and the
time of negative jump $N_{uj}$ in the same way for all $u\in\mathbb{U}$. The
birth time $B_{u}$ of the cell $u\in\mathbb{U}$ is defined recursively by
$B_{\emptyset}=0$ and for all $u\in\mathbb{U},j\in\mathbb{N}^{*}$, we set
$B_{uj}=B_{u}+N_{uj}$. For all $u\in\mathbb{U}$, let $\mathcal{C}_{u}$ be the
cycle (i.e. the boundary of the hole) created during the birth of the cell
$u$, with by convention $\mathcal{C}_{\emptyset}$ being the boundary of the
root face.

Let $X$ be a self-similar Markov process such that $0$ is an absorbing state
which tends a.s. towards zero or is a.s. absorbed at zero after a finite time.
Let $P_{x}$ be the law of $X$ starting from $x\in\mathbb{R}_{+}$. We say that
$(X_{u})_{u\in\mathbb{U}}$ is a (continuous) cell-system associated with $X$
starting at $x\geq 0$ if $X_{\emptyset}$ follows the law $P_{x}$ and for all
$u\in\mathbb{U}$, conditionally on $X_{u}$, if we write
$((x_{j},\beta_{uj}))_{j\geq 1}$ the family of negative jumps of $X_{u}$,
where $x_{j}>0$ is the size of the jump and $\beta_{uj}$ is the time of jump,
ordered in the non increasing lexicographic order ($x_{j}\geq x_{j+1}$ and if
$x_{j}=x_{j+1}$ then $\beta_{j}>\beta_{j+1}$), then the $X_{uj}$ for $j\geq 1$
are independent of laws $P_{x_{j}}$. Let $\left((\mathcal{X}_{u}(t))_{t\geq
0},(\beta_{uj})_{j\geq 1},b_{u},\zeta_{u}\right)_{u\in\mathbb{U}}$ be the
continuous cell-system associated with the self-similar Markov process
$X^{(-1)}$.

Recall the definition of the $J_{1}$ Skorokhod topology on the space of
families of càdlàg functions indexed by $\mathbb{U}$ denoted by
$\mathbb{D}(\mathbb{R}_{+},\mathbb{R})^{\mathbb{U}}$ :
$(f_{u}^{(n)}(t))_{t\geq 0,u\in\mathbb{U}}$ converges towards
$(f_{u}(t))_{t\geq 0,u\in\mathbb{U}}$ if for all finite subset
$F\subset\mathbb{U}$, $(f_{u}^{(n)}(t))_{t\geq 0,u\in F}$ converges towards
$(f_{u}(t))_{t\geq 0,u\in F}$ in the space
$\mathbb{D}(\mathbb{R}_{+},\mathbb{R}^{F})$ for the $J_{1}$ Skorokhod
topology.

######  Lemma 6.5.

The rescaled dicrete cell-system associated with $\mathfrak{M}^{(\ell)}$
converges in distribution to the continuous cell-system as a random variable
with values in $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})^{\mathbb{U}}$ equipped
with the $J_{1}$ Skorokhod topology:

$\text{under }\mathbb{P}^{(\ell)},\qquad\left(\left(\frac{P_{u}(\ell
t)}{\ell}\right)_{t\geq
0}\right)_{u\in\mathbb{U}}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\left(\left(\mathcal{X}_{u}(\pi
p_{\bf q}t)\right)_{t\geq 0}\right)_{u\in\mathbb{U}}.$ (6.12)

###### Proof.

The proof is the same as the proof of Lemma 17 in [BCK18]. The first step is
to show that the convergence (6.3) holds in the space $\mathbb{D}([0,\infty])$
equipped with Skorokhod’s $J_{1}$ topology. To prove this, it suffices to
control $P_{*}(\ell t)/\ell$ for large $t\geq 0$ uniformly in $\ell$ using the
supermartingale $(f^{\uparrow}(P_{*}(n)))_{n\geq 0}$ under
$\mathbb{P}^{(\ell)}$ coming from Proposition 6.4 of [BBCK18], where
$f^{\uparrow}$ is defined in the proof of Corollary 6.4. Then (6.12) stems
from the spatial Markov property and from the fact that the
$\mathcal{X}_{u}$’s for $u\in\mathbb{U}$ have a.s. no common jump times, using
Proposition 2.2 page 338 of [JS87]. ∎

At this point, we are in position to prove the main result of this section.

###### Proof of Theorem 1.3.

We first extend the convergence (6.12) to the height processes. For every
$u\in\mathbb{U}$, we can define $H_{u}$ the height process of the locally
largest cycle during the (filled-in) exploration of the submap of
$\mathfrak{m}$ contained in the hole created at the birth time of the cell
$u$. Notice that $H_{\emptyset}=H_{*}$. Analogously, in the continuous
setting, we write for all $u\in\mathbb{U}$, for all $t\geq 0$,

$\mathcal{H}_{u}(t)=\int_{0}^{t}\frac{p_{\bf q}}{2\mathcal{X}_{u}(\pi p_{\bf
q}s)}ds.$

A direct consequence of Corollary 6.4 and Lemma 6.5 is the following
convergence towards a family of continuous increasing processes which holds
jointly with (6.12):

$\text{under }\mathbb{P}^{(\ell)},\qquad\left(\left(\frac{H_{u}(\ell
t)}{\log\ell}\right)_{t\geq
0}\right)_{u\in\mathbb{U}}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\left(\left(\mathcal{H}_{u}(t)\right)_{t\geq
0}\right)_{u\in\mathbb{U}}.$ (6.13)

The topology for which the above convergence holds is the product topology
over $\mathbb{U}$ where each factor is equipped with the topology of the
uniform convergence on compact sets.

Yet, the processes $H_{u}$ for $u\in\mathbb{U}$ only describe the height in
the map encircled by $\mathcal{C}_{u}$. In order to describe the height in the
whole map $\mathfrak{m}$, we introduce some additional notation. In the
discrete setting, for all $u\in\mathbb{U}$, let $H(\mathcal{C}_{u})$ be the
minimum of the heights (i.e. the distances to the root face) of the edges of
the cycle $\mathcal{C}_{u}$. We also define $\widetilde{H}_{u}$ by induction:
$\widetilde{H}_{\emptyset}=0$ and for all $u\in\mathbb{U},j\geq 1$,
$\widetilde{H}_{uj}=\widetilde{H}^{(\ell)}_{u}+H_{u}(N_{uj})$. In the
continuous setting, we define the birth heights
$(b^{\mathcal{H}}_{u})_{u\in\mathbb{U}}$ by $b^{\mathcal{H}}_{\emptyset}=0$
and for all $u\in\mathbb{U}$, for all $j\geq 1$,
$b_{uj}^{\mathcal{H}}=b^{\mathcal{H}}_{u}+\mathcal{H}_{u}\left({\beta_{uj}}/({\pi
p_{\bf q}})\right)$.

Notice that by (6.12) and (6.13), we have jointly with (6.12) and (6.13) the
convergence for the product topology

$\text{under
}\mathbb{P}^{(\ell)},\qquad\left(\frac{H_{u}(N_{uj})}{\log\ell}\right)_{u\in\mathbb{U},j\geq
1}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left(\mathcal{H}_{u}\left(\frac{\beta_{uj}}{\pi
p_{\bf q}}\right)\right)_{u\in\mathbb{U},j\geq 1}.$ (6.14)

Since the peeling algorithm which is used in the exploration is the peeling by
layers, for all $u\in\mathbb{U},j\geq 1$, every edge of $\mathcal{C}_{uj}$ is
at distance $H_{u}(N_{uj})$ or $H_{u}(N_{uj})+1$ from $\mathcal{C}_{u}$.
Therefore, for all $u\in\mathbb{U}$, we have $\widetilde{H}^{(\ell)}_{u}\leq
H^{(\ell)}(\mathcal{C}_{u})\leq\widetilde{H}^{(\ell)}_{u}+|u|,$, where $|u|$
denotes the length of the word $u$. Thus (6.14) entails that, jointly with
(6.12) and (6.13),

$\left(\frac{H^{(\ell)}(\mathcal{C}_{u})}{\log\ell}\right)_{u\in\mathbb{U}}\mathop{\longrightarrow}\limits_{\ell\to+\infty}^{(\mathrm{d})}\left(b^{\mathcal{H}}_{u}\right)_{u\in\mathbb{U}}$
(6.15)

Note that the positive jumps of the $P_{u}$’s for $u\in\mathbb{U}$ correspond
to the (half) perimeters of the faces of $\mathfrak{m}$ minus one (without the
root face).

Besides, notice that by the Markov property of Theorem 1.1, the cell system

$\left((Y_{u}(t))_{t\geq
0}\right)_{u\in\mathbb{U}}\coloneqq\left(\left(\mathcal{X}_{u}(\pi
t)\right)_{t\geq 0}\right)_{u\in\mathbb{U}}$

describes the quantum boundary lengths along the exploration tree of the
$\mathrm{CLE}_{4}$ decorated $2$-LQG disk, both for the uniform exploration
and for the $\mathrm{SLE}_{4}^{\langle\mu\rangle}(-2)$ exploration,
parametrized by the quantum natural distance for the first one and by the
quantum length of the trunk for the second one. In particular, the positive
jumps correspond the quantum boundary lengths of the loops, so that the
quantum boundary lengths of the loops are all distinct (note that one can also
see the fact that the quantum boundary lengths of the loops are distinct as a
direct consequence of Theorem 1.2 of [AG23a] which gives the law of the
quantum boundary lengths of the loops). Moreover, one can see that for all
$\varepsilon>0$ there is a finite number of (outermost) loops of quantum
boundary length larger than $\varepsilon$. It can be seen for instance as a
consequence of Theorem 1.1 which shows that one can write
$\mu^{2}_{h^{2}}(\mathbb{D})\geq\sum_{j\geq 1}b_{j}^{2}a_{j},$ where the
$a_{j}$’s are i.i.d. of the same law as $\mu^{2}_{h^{2}}(\mathbb{D})$ and
independent of the $b_{j}$’s which are the quantum boundary lengths of the
(outermost) loops.

Then the desired result is a straightforward consequence of the convergences
(6.12), (6.13) and (6.15). Indeed, by Skorokhod’s representation theorem, take
a probability space where these convergences hold almost surely. Take
$\varepsilon>0$. Let $F\subset\mathbb{U}$ be the finite set of $u$’s such that
$Y_{u}$ has at least one jump of size larger than $\varepsilon$. The
convergence (6.12) entails the scaling limit of the perimeter to the quantum
boundary length and the scaling limit of the exploration time towards the
quantum natural distance or quantum length of the trunk. Finally, the
convergences (6.13) and (6.15) give the scaling limit of the height towards
the Lamperti transform. ∎

Note that one can also obtain a “slicing at heights” of $3/2$-stable maps as
in [BCK18] or [BBCK18]. For all $r\geq 1$, let
$\mathrm{Ball}^{\dagger}_{r}(\mathfrak{m})$ be the ball of radius $r$ centred
at $f_{r}$ in $\mathfrak{m}$. This ball is the submap of $\mathfrak{m}$ made
by keeping the faces of $\mathfrak{m}$ which are at height at most $r$ and
cutting along the edges which are adjacent on both sides to faces at distance
$r$ to the root face. This ball has several holes of boundaries
$\mathfrak{h}_{1}(\mathfrak{m},r),\mathfrak{h}_{2}(\mathfrak{m},r),\ldots$
ranked in the non-increasing order of perimeter. For all $i\geq 1$ and $r\geq
0$, let $L_{i}(r)$ be the perimeter of the hole
$\mathfrak{h}_{i}(\mathfrak{m},r)$.

Let us recall the notions of self-similar growth-fragmentation. See
Subsections 2.2 and 3.1 of [BBCK18] or Section 2 of [Ber17] for more details.
If $(\mathcal{X}_{u},b_{u},\zeta_{u})_{u\in\mathbb{U}}$ is a continuous cell
system associated with a self-similar Markov process $X$, for all $t\geq 0$ we
set

${\bf X}(t)=\left\\{\\!\\!\\{\mathcal{X}_{u}(t-b_{u});\ u\in\mathbb{U},\
b_{u}\leq t<b_{u}+\zeta_{u}\right\\}\\!\\!\\}$

where the notation $\left\\{\\!\\!\\{\cdots\right\\}\\!\\!\\}$ refers to
multiset. By Theorem 2 in [Ber17], the elements of ${\bf X}(t)$ can be ranked
in the non-increasing order and form a sequence that tends to $0$, say
$X_{1}(t)\geq X_{2}(t)\geq\ldots\geq 0$. If ${\bf X}(t)$ has only finitely
many elements, say n, then we set $X_{n+1}(t)=X_{n+2}(t)=\ldots=0$. The
process $({\bf X}_{t})_{t\geq 0}$ is called the growth-fragmentation process
associated with $X$.

Let $\ell_{5/2}^{\downarrow}$ be the space of non-increasing sequences
$(x_{i})_{i\geq 0}$ such that $\sum_{i\geq 1}x_{i}^{5/2}<\infty$. Let
$\mathbb{D}(\mathbb{R}_{+},\ell_{5/2}^{\downarrow})$ be the space of càdlàg
functions from $\mathbb{R}_{+}$ to $\ell^{\downarrow}_{5/2}$ equipped with the
$J_{1}$ Skorokhod topology. Let ${\bf X}^{(0)}=((X^{(0)}_{i}(t))_{i\geq
1})_{t\geq 0}$ be the growth-fragmentation associated to the self-similar
process $X^{(0)}$. Then, using exactly the same proof as the proof of Theorem
6.8 of [BBCK18], we obtain the following scaling limit:

.

######  Proposition 6.6.

Then we have the scaling limit in
$\mathbb{D}(\mathbb{R}_{+},\ell_{5/2}^{\downarrow})$

$\text{Under
}\mathbb{P}^{(\ell)},\qquad\left(\left(\frac{L_{i}(s\log\ell)}{\ell}\right)_{i\geq
1}\right)_{s\geq
0}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left({\bf
X}^{(0)}(2\pi s)\right)_{s\geq 0}.$

##  7 Discussion

###  7.1 Application to $O(2)$-decorated maps

Here we describe briefly how our results transfer to some critical $O(2)$
loop-decorated planar maps. Let us first recall the definition of a rigid
loop-decorated map and the definition of Boltzmann $O(n)$ loop-decorated
random planar maps for $n\in(0,2]$.

A loop-decorated map $(\mathfrak{m},{\bf\mathcal{L}})$ is a (rooted bipartite
planar) map $\mathfrak{m}$ together with a loop configuration ${\bf
L}=(\mathcal{L}_{1},\ldots,\mathcal{L}_{k})$ of disjoint unoriented simple
closed paths on the dual map $\mathfrak{m}^{\dagger}$. Furthermore, the loops
$\mathcal{L}_{i}$ do not go through the root face of $\mathfrak{m}$ and are
rigid in the sense that they only visit quadrangles and they enter and exit
the quadrangles through opposite sides. See Figure 7.

Figure 7: Left: a loop-decorated planar map. The loops, which cross
quadrangles are in red. Right: its gasket, obtained by removing the interior
of the outermost loops.

By convention, the distances $\mathrm{d}_{\mathrm{gr}}^{\dagger}$ and
$\mathrm{d}_{\mathrm{fpp}}^{\dagger}$ are defined on a loop-decorated map by
adding the rule that all the quadrangles of each loop are identified to one
point, i.e. are at distance zero.

We denote the set of loop-decorated maps by $\mathcal{LM}$. For all $\ell\geq
1$, we also introduce the subset $\mathcal{LM}^{(\ell)}$ of loop-decorated
maps whose root face has degree $2\ell$. Recall that ${\bf q}=(q_{k})_{k\geq
1}$ is a critical non-generic weight sequence of type $2$ in the sense that
the partition function of (non-decorated rooted bipartite planar) maps
$W^{(\ell)}$ behaves as in $(\ref{eq critique de type deux})$. We assume
moreover, that for all $k\geq 1$,

$q_{k}\geq\frac{2}{c_{\bf q}^{2k}}W^{(k)}.$ (7.1)

Such a sequence does exist, as shown in Remark 1 of [Bud18] one can take for
all $k\geq 1$

$q_{k}=\frac{1}{(3\pi)^{k-1}}\left(\frac{2}{\pi}\frac{1}{(2k-3)(2k-1)}+{\bf
1}_{k=1}\right).$

Using the so-called gasket decomposition introduced in [BBG12b, BBG12a], one
can define a critical $O(2)$ Boltzmann measure
$\widetilde{\mathbb{P}}^{(\ell)}$ on loop-decorated random maps of perimeter
$\ell$ for all $\ell\geq 1$ in a recursive way as follows. Let
$\mathfrak{M}^{(\ell)}$ be a random variable in $\mathcal{M}^{(\ell)}$ of law
$\mathbb{P}^{(\ell)}$. Then, independently, on each face $f$ of
$\mathfrak{M}^{(\ell)}$, with probability $(2W^{(k)})/(c_{\bf q}^{2k}q_{k})$
where $k=\deg(f)/2$, we glue quadrangles along the edges around $f$ hence
forming a rigid loop and we glue in this loop an independent loop-decorated
map of law $\widetilde{\mathbb{P}}^{(k)}$. The fact that
$\widetilde{\mathbb{P}}^{(\ell)}$ is well defined comes from the admissibility
of $\bf q$, from (7.1) and from [BBG12b] or from Theorem 1 of [Bud18]. The
random loop-decorated map thus obtained is called a non-generic critical
$O(n)$ loop-decorated map for $n=2$ of perimeter $\ell$ as defined at the
bottom of page 4 of [Bud18]. The map $\mathfrak{M}^{(\ell)}$ is called the
gasket of the obtained loop-decorated map. See Figure 7.

Using this gasket decomposition, one can see that Theorem 1.3 and Proposition
6.6 have their counterparts in terms of critical $O(2)$ loop-decorated planar
maps:

  * -

For Theorem 1.3, it suffices to replace the faces of the $3/2$-stable map by
the loops of the loop-decorated map and the non-nested $\mathrm{CLE}_{4}$ by a
nested $\mathrm{CLE}_{4}$. The peeling exploration becomes the peeling
exploration on loop-decorated maps defined in [Bud18].

  * -

For Proposition 6.6, one has to add a sign $(-1)^{N(\mathfrak{h})}$ to the
perimeter of each hole $\mathfrak{h}$, where $N(\mathfrak{h})$ is the number
of loops surrounding $\mathfrak{h}$ and one has to replace the growth-
fragmentation ${\bf X}^{(0)}$ by a signed growth fragmentation as defined in
[DS23], where the positive jumps give rise to cells of negative size. Notice
that the obtained signed growth-fragmentation corresponds up to a Lamperti
transform to the signed growth-fragmentation studied in [AdS22]. The
convergence towards the growth-fragmentation holds in
$\mathbb{D}(\mathbb{R}_{+},\ell^{\downarrow}_{2})$ since the harmonic
functions associated to the exploration of Boltzmann planar maps which give
rise to the martingales of Proposition 6.4 in [BBCK18] are replaced by the
harmonic function $h^{\downarrow}_{\mathfrak{p}}$ for $\mathfrak{p}=1$ in
Equation (5) of [Bud18], which is constant equal to $1$.

Besides, one can note that a similar adaptation answers Problem 4.5 of [AG23b]
in the case where the gasket is a $3/2$-stable map.

However, the question of whether every critical $O(n)$ loop-decorated map’s
gasket for $n=2$ is a $3/2$-stable map remains open inasmuch as the asymptotic
behavior of the partition function given by Equation (3) of [Bud18] (which
comes from [BBD23]) is only proven for $n\in(0,2)$.

###  7.2 The conformally invariant distance is not the Lamperti transform of
the quantum distance

We finally argue why the conformally invariant distance to the boundary
$\mathrm{d}_{\mathrm{WW}}$ of [WW13] is not the Lamperti transform
$\mathrm{d}_{\mathrm{Lamperti}}$ of the quantum distance.

Let us consider, as before, a unit boundary length $2$-quantum disk
$(\mathbb{D},h^{2})$ on which we draw an independent $\mathrm{CLE}_{4}$ using
the uniform exploration. Let $Z$ be an independent random point of law
$\mu^{2}_{h^{2}}/\mu^{2}_{h^{2}}(\mathbb{D})$ and let $\mathcal{L}$ be the
(outermost) loop surrounding $Z$. By definition of the uniform exploration,
conditionally on $h^{2}$ and on $Z$, the distance
$\mathrm{d}_{\mathrm{WW}}(\partial\mathbb{D},\mathcal{L})$ is an exponential
random variable of parameter $\lambda$ for some constant $\lambda>0$. In
particular, if we stop the exploration at distance $u>0$, conditionally on not
having encircled $Z$ before time $u$, the distance
$\mathrm{d}_{\mathrm{WW}}(\partial\mathbb{D},\mathcal{L})-u$ from
$\mathcal{L}$ to the boundary of the unexplored region is still an exponential
random variable of parameter $\lambda$. On the contrary, if we stop the
exploration at Lamperti transformed quantum distance $t>0$, then,
conditionally on not having encircled $Z$ yet, the point $Z$ lies in each
unexplored quantum disk with probability proportional to its quantum area. So
the quantum disk containing $Z$ is biased by its area, so that the absence of
memory for $\mathrm{d}_{\mathrm{Lamperti}}(\partial\mathbb{D},\mathcal{L})$ is
ruled out.

Nevertheless, we believe that the Lamperti transform
$\mathrm{d}_{\mathrm{Lamperti}}$ of the quantum natural distance could be
extended to some conformally invariant distance on the
$\mathrm{CLE}_{4}$-decorated $2$-LQG disk.

#### Acknowledgements.

I am deeply indebted to Matthis Lehmkuehler for his guidance in the
exploration of CLEs as well as in the powder snow of Les Diablerets,
especially for his help for the proof of Proposition 3.1. I also thank
Wendelin Werner for his advice. I am grateful to my supervisors Cyril Marzouk
and Nicolas Curien for their invaluable support.

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