# The moduli of annuli in random conformal geometry

Morris Ang and Guillaume Remy and Xin Sun

###### Abstract.

We obtain exact formulae for three basic quantities in random conformal
geometry that depend on the modulus of an annulus. The first is for the law of
the modulus of the Brownian annulus describing the scaling limit of uniformly
sampled planar maps with annular topology, which is as predicted from the
ghost partition function in bosonic string theory. The second is for the law
of the modulus of the annulus bounded by a loop of a simple conformal loop
ensemble (CLE) on a disk and the disk boundary. The formula is as conjectured
from the partition function of the O$(n)$ loop model on the annulus derived by
Saleur-Bauer (1989) and Cardy (2006). The third is for the annulus partition
function of the SLE8/3 loop introduced by Werner (2008), confirming another
prediction of Cardy (2006). The physics principle underlying our proofs is
that 2D quantum gravity coupled with conformal matters can be decomposed into
three conformal field theories (CFT): the matter CFT, the Liouville CFT, and
the ghost CFT. At the technical level, we rely on two types of integrability
in Liouville quantum gravity, one from the scaling limit of random planar
maps, the other from the Liouville CFT. We expect our method to be applicable
to a variety of questions related to the random moduli of non-simply-connected
random surfaces.

##  1\. Introduction

Over the last two decades, tremendous progress has been made in understanding
random surfaces of simply connected topology. By [Le 13, Mie13, BM17] and
their extensions, the uniformly sampled random planar maps on the sphere and
the disk converge as metric-measure spaces in the scaling limit. The limiting
surfaces are known as the Brownian sphere and disk, respectively. Moreover, as
proved in various senses [MS20, MS21a, MS21b, GMS20, HSar], once these
surfaces are conformally embedded in the complex plane, the embedded random
geometry is given by Liouville quantum gravity (LQG) [DS11, DDDF20, GM21b]
with $\gamma=\sqrt{8/3}$. Finally, as shown in [AHS17, Cer21, AHS21], the law
of the random field inducing the random geometry is given by the Liouville
conformal field theory on the sphere [DKRV16] and the disk [HRV18]. For random
planar maps decorated with statistical physics models, the scaling limit is
described by LQG surfaces decorated with Schramm Loewner evolution (SLE) and
conformal loop ensemble (CLE). Although there are still major open questions
concerning the scaling limit, the picture in the continuum is well understood,
thanks to the theory of the quantum zipper [She16] and the mating-of-trees
[DMS21]. For more background on this subject, see the surveys [GHS19, Gwy20,
She22] and the notes [LGM12, Var17, BP].

Figure 1. Conformal embedding of the Brownian annulus to the annulus
$\mathbb{A}_{\tau}=\\{z\>:\>|z|\in(e^{-2\pi\tau},1)\\}$ with (random) modulus
$\tau$.

For non-simply-connected surfaces, the key new feature is the non-uniqueness
of the conformal structure. For example, if we conformally embed the Brownian
annulus to the complex plane (Figure 1), the modulus of the planar annulus is
random. Precise descriptions of these surfaces, including the laws of the
random moduli, were conjectured in [DRV16] for the torus, in [GRV19] for
higher genus surfaces, and in [Rem18] by the second named author for the
annulus. The conjecture is based on a fundamental principle that dates back to
Polyakov’s seminal work [Pol81] on bosonic string theory. Namely, 2D quantum
gravity coupled with conformal matter can be decomposed into three conformal
field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. In
this paper, we prove this conjecture for the annulus in the case of pure
gravity which corresponds to the Brownian annulus, and the cases where the
matter corresponds to the simple CLE [She09, SW12] (i.e.
$\mathrm{CLE}_{\kappa}$ with $\kappa\in(8/3,4]$) or Werner’s
$\mathrm{SLE}_{8/3}$ loop [Wer08].

We also derive some exact formulae for SLE/CLE conjectured from physics. Since
the discovery of SLE [Sch00], it has been proved or widely conjectured that
SLE/CLE describe the scaling limits of interfaces in 2D lattice models such as
percolation [Smi01, CN08] and the Ising model [CDCH+14, CS12, BH19]. Many
scaling exponents/dimensions of such models predicted by CFT [BPZ84] were
derived rigorously from SLE/CLE (e.g. [LSW02, MSW14]). Moreover, some exact
formulae predicted via boundary CFT, such as Cardy’s formula for percolation
crossing, were rigorously derived from $\mathrm{SLE}$ (see e.g. [Wer04]).
Saleur-Bauer [SB89] and Cardy [Car06] derived a formula for the partition
function of the $O(n)$ loop model on the annulus, which provides crucial
information on the CFT description of its scaling limit. Based on the
conjectural relation between CLE and the $O(n)$ loop model (see e.g. [She09]),
this partition function corresponds to an exact formula for the loop
statistics of CLE. Viewing the self avoiding loop as the $O(0)$ model, Cardy
[Car06] further conjectured a formula for the annulus partition function of
the $\mathrm{SLE}_{8/3}$ loop. Deriving these formulae for SLE/CLE has been an
open question. In this paper we prove them for simple CLE and the SLE8/3 loop.

The key to our proofs is an explicit relation between the partition function
of LQG surfaces and the law of their moduli. This is obtained from the
conformal bootstrap of Liouville CFT on the annulus due to Wu [Wu22]. It can
be viewed as a Knizhnik-Polyakov-Zamolodchikov (KPZ) relation [KPZ88] between
the quantum and the Euclidean geometry at the partition function level,
whereas the traditional KPZ relation established in probability [DS11, BGRV16,
GHM15, GP19] is at the level of scaling exponents/dimensions. Our KPZ relation
reduces the computation of an annular quantity depending on the modulus to the
same quantity for LQG surfaces, which is easier to obtain from the random
planar map perspective.

Our work represents the first step towards understanding the relation between
LQG in the random geometry framework and measures on the moduli space of
Riemann surfaces, although quantum gravity intuition has inspired important
developments of the latter subject since 1980s [BK86, Wit91, Kon92, Mir07].
Our work is also a further step towards uncovering the CFT content of CLE,
after the proof of the imaginary DOZZ formula by the first and third named
authors [AS21]. It continues to demonstrate the rich interplay between various
kinds of integrability in conformal probability, a theme recently explored in
[AHS21, ARS21, AS21]. We expect that methods in our paper together with the
remarkable developments in the integrability of Liouville CFT [KRV20, GKRV20,
GKRV21] will result in further progress in this direction.

In the rest of the introduction, we first state our result on the modulus of
the Brownian annulus in Section 1.1. Then we present the matter-Liouville-
ghost decomposition for the Brownian annulus and the CLE decorated quantum
annulus in Section 1.2. We state our KPZ relation for annulus partition
functions in Section 1.3 and results for simple CLE and the
$\mathrm{SLE}_{8/3}$ loop in Sections 1.4 and 1.5, respectively. In Section
1.6 we discuss some subsequent projects and open questions concerning other
models or topologies.

Notation for the modulus. Let $\mathbb{C}$ be the complex plane. For
$\tau\in(0,\infty)$, let
$\mathbb{A}_{\tau}=\\{z\in\mathbb{C}:|z|\in(e^{-2\pi\tau},1)\\}$. For a
Riemann surface $A$ of the annular topology, there exists a unique $\tau$ such
that $A$ and $\mathbb{A}_{\tau}$ are conformally equivalent. We call $\tau$
the _modulus_ of $A$.

###  1.1. Random moduli for the Brownian annulus

There are various equivalent ways of introducing the Brownian annulus. For the
convenience of describing its conformal structure, we will introduce it via
the Brownian disk. For $a>0$, the _Brownian disk with boundary length $a$_ is
a random metric-measure space with the disk topology, which is the Gromov-
Hausdorff-Prokhorov (GHP) scaling limit of random quadrangulations or
triangulations of a large polygon under the critical Boltzmann weight [BM17,
GM19, AHS20a, BCFS21]; see Proposition B.2 for a precise scaling limit result.
If we reweight the law of a Brownian disk by its total area and add an
interior point according to the area measure, then we get a Brownian disk with
one interior marked point which we denote by $(\mathcal{D},p,d,\mu)$. Here
$p\in\mathcal{D}$ is the marked point, $d$ is the metric and $\mu$ is the area
measure. For $0<r<d(p,\partial\mathcal{D})$, let
$\mathcal{B}(p,r)=\\{x\in\mathcal{D}:d(p,x)\leq r\\}$ be the metric ball of
radius $r$ centered at $p$. Then each connected component of
$\mathcal{D}\setminus B_{r}(p,r)$ is locally absolutely continuous to a
Brownian disk near the boundary, hence has a notion of boundary length; see
[LG19, Theorem 3].

######  Definition 1.1.

Fix $a>0$ and $b>0$. Let $(\mathcal{D},p,d,\mu)$ be a Brownian disk with
boundary length $a$ and an interior marked point. On the event
$d(p,\partial\mathcal{D})>1$, let $\mathcal{A}$ be the (annular) connected
component of $\mathcal{D}\backslash\mathcal{B}(p,1)$ whose boundary contains
$\partial\mathcal{D}$. Let $\mathcal{L}$ be the length of the boundary
component of $\partial\mathcal{A}$ other than $\partial\mathcal{D}$. Let the
probability measure $\mathrm{BA}(a,b)^{\\#}$ be the conditional law of the
metric-measure space $(\mathcal{A},d,\mu)$ conditioning on $\mathcal{L}=b$. We
call a sample from $\mathrm{BA}(a,b)^{\\#}$ a _Brownian annulus of boundary
lengths $a$ and $b$_.

We give the full details for Definition 1.1 in Section 6.1. Although we define
$\mathrm{BA}(a,b)^{\\#}$ through conditional probability, which is only
specified for almost every $b$, there are various ways to remove this
ambiguity. For example, using techniques in [BM17, GM19, AHS20a], it is
possible to show that $\mathrm{BA}(a,b)^{\\#}$ is the $n\to\infty$ GHP scaling
limit of the critical Boltzmann weighted quadrangulation or triangulation of
an annulus with $\lfloor an\rfloor$ and $\lfloor bn\rfloor$ edges on the two
boundaries, respectively. Moreover, it is possible to describe
$\mathrm{BA}(a,b)^{\\#}$ directly in the continuum by some conditioned
variants of the Brownian snake as in the sphere [Le 13, Mie13] and disk [BM17]
cases. One can also give a mating-of-trees [DMS21] description of
$\mathrm{BA}(a,b)^{\\#}$ based on the bijection in [BHS18]. Since our main
curiosity lies in understanding the law of the modulus, we will not elaborate
on these aspects of the Brownian annulus.

According to [MS20, MS21a] which we recall in Section 6.1, given
$(\mathcal{D},p,d,\mu)$ in Definition 1.1, we can construct a random metric
$d_{\mathbb{H}}$ and a random measure $\mu_{\mathbb{H}}$ on the upper half
plane $\mathbb{H}$ using $\sqrt{8/3}$-LQG such that viewed as a (marked)
metric-measure space, $(\mathbb{H},i,d_{\mathbb{H}},\mu_{\mathbb{H}})$ equals
$(\mathcal{D},p,d,\mu)$. We call
$(\mathbb{H},i,d_{\mathbb{H}},\mu_{\mathbb{H}})$ a conformally embedded
Brownian disk with boundary length $a$ and an interior marked point at $i$. On
the event $d(p,\partial D)>1$, let
$B_{d_{\mathbb{H}}}(i,1)=\\{x:d_{\mathbb{H}}(i,x)\leq 1\\}$ and let $A$ be the
connected component of $\mathbb{H}\setminus B_{d_{\mathbb{H}}}(i,1)$ with
$\partial\mathbb{H}$ on its boundary. Then
$(A,d_{\mathbb{H}},\mu_{\mathbb{H}})$ is a conformal embedding of
$(\mathcal{A},d,\mu)$. Let $\mathrm{Mod}(\mathcal{A})$ be the modulus of the
annulus $A$. We call $\mathrm{Mod}(\mathcal{A})$ the _modulus_ of
$(\mathcal{A},d,\mu)$. By [MS21b, GMS20], $(d_{\mathbb{H}},\mu_{\mathbb{H}})$
is determined by $(\mathcal{D},p,d,\mu)$ modulo conformal symmetries. From
this we can see that $\mathrm{Mod}(\mathcal{A})$ is determined by the metric-
measure space $(\mathcal{A},d,\mu)$; see Lemma 6.7. Our theorem below
describes the law of $\mathrm{Mod}(\mathcal{A})$ under
$\mathrm{BA}(a,b)^{\\#}$.

######  Theorem 1.2.

For $\tau>0$, let $X_{\tau}$ be a random variable such that
$\mathbb{E}[X^{it}_{\tau}]=\frac{2\pi te^{-\frac{2\pi\tau
t^{2}}{3}}}{3\sinh(\frac{2}{3}\pi t)}$ for $t\in\mathbb{R}$. Let
$\rho_{\tau}(\cdot)$ be the probability density function of $X_{\tau}$. Then
the law of $\mathrm{Mod}(\mathcal{A})$ under $\mathrm{BA}(a,b)^{\\#}$ is the
probability measure on $(0,\infty)$ proportional to
$1_{\tau>0}\eta(i2\tau)\rho_{\tau}(\frac{b}{a})\mathrm{d}\tau$, where
$\eta(i\tau)=e^{-\frac{\pi\tau}{12}}\prod_{k=1}^{\infty}(1-e^{-2\pi k\tau})$
is the Dedekind eta function.

Theorem 1.2 yields statements on the scaling limit of the conformal structure
of uniformly random planar maps on the annulus. For example, the law of
$\mathrm{Mod}(\mathcal{A})$ should describe the limiting distribution of the
effective resistance of the critical Boltzmann triangulation with the given
boundary lengths, viewed as an electric network. Proving this rigorously
amounts to establishing the quenched scaling limit of random walks on such
maps. Analogous statements should hold for other quantities that encode the
modulus of the annulus, such as the annulus crossing probability for the
critical percolation or Ising model. The quenched scaling limit needed to
prove such statements is only known for site percolation on triangulations
[HSar].

Theorem 1.2 is a consequence of the matter-Liouville-ghost decomposition for
the Brownian annulus (Theorem 1.3), as we will explain in the next subsection.

###  1.2. Matter-Liouville-ghost decomposition for 2D quantum gravity

The decomposition of 2D quantum gravity coupled with conformal matter was
originally due to Polyakov [Pol81] where the matter is the free boson. David
[Dav88] and Distler-Kawai [DK89] generalized it to conformal matter with
central charge $c_{\mathrm{M}}\leq 1$. The ghost part of this decomposition
only manifests itself for non-simply connected surfaces. In the probabilistic
framework of LQG, this was first explained in [DRV16] for the torus case. The
annulus case was treated by the second named author [Rem18] in parallel to the
torus case. For higher genus closed surfaces, it was shown in [GRV19] that for
free boson matter this decomposition indeed produces a convergent theory
envisioned in [Pol81]. Based on the belief that random planar maps decorated
with statistical physics models are discretizations of matter-coupled 2D
quantum gravity, precise scaling limit conjectures were formulated in each of
the three papers [DRV16, GRV19, Rem18]. In particular, applying the
decomposition to the pure gravity on the annulus, a precise LQG description of
the Brownian annulus was provided in [Rem18, Section 4.3], following the
physics paper [Mar03]. We now recall this conjecture, which we prove as
Theorem 1.3 below.

We first introduce the free Brownian annulus with no constraint on the
boundary lengths. Let

(1.1)
$\mathrm{BA}=\iint_{0}^{\infty}\frac{1}{2\sqrt{ab}(a+b)}\mathrm{BA}(a,b)^{\\#}\mathrm{d}a\,\mathrm{d}b.$

We call a sample from the infinite measure $\mathrm{BA}$ a _free Brownian
annulus_. Here the factor $\frac{1}{\sqrt{ab}(a+b)}$ is according to
enumeration results for annular planar maps [BF18]; see Lemma B.3. From the
relation between $\sqrt{8/3}$-LQG and Brownian surfaces established in [MS20,
MS21a, MS21b], a free Brownian annulus can be represented as $(\phi,\tau)$
where $\tau\in(0,\infty)$ gives the modulus of the annulus where the Brownian
annulus is conformally embedded, and $\phi$ describes the variant of Gaussian
free field on the annulus that determines the $\sqrt{8/3}$-LQG geometry.

The first part of the conjecture in [Rem18] is that the law of $\phi$ given
$\tau$ can be described by the Liouville conformal field theory. To make the
statement as clean as possible, we work on the horizontal cylinder
$\mathcal{C}_{\tau}:=[0,\tau]\times[0,1]/{\sim}$ of unit circumference and
length $\tau$, where $\sim$ means identifying $[0,\tau]\times\\{0\\}$ with
$[0,\tau]\times\\{1\\}$; see Figure 2 (left). This way, the modulus of
$\mathcal{C}_{\tau}$ is $\tau$.111Our modular parameter $\tau$ is denoted by
$\ell$ in [Rem18]. Let $\mathbb{P}_{\tau}$ be the law of the free boundary
Gaussian free field on $\mathcal{C}_{\tau}$ defined as in [She07]; see Section
2.1. The _Liouville field measure_ $\mathrm{LF}_{\tau}$ on
$\mathcal{C}_{\tau}$ is the pushforward of $\mathbb{P}_{\tau}\times dc$ under
$(h,c)\mapsto h+c$, where $dc$ is the Lebesgue measure on $\mathbb{R}$. The
first part of the conjecture can be stated as follows. There exists a measure
$m(\mathrm{d}\tau)$ on $(0,\infty)$ such that $(\phi,\tau)$ sampled from
$\mathrm{LF}_{\tau}(d\phi)m(\mathrm{d}\tau)$ gives a free Brownian annulus
conformally embedded onto a horizontal cylinder.

The second and more mysterious part of the conjecture is the exact formula for
$m(\mathrm{d}\tau)$. According to matter-Liouville-ghost decomposition of pure
gravity, the free Brownian annulus can be written as

(1.2)
$\mathcal{Z}_{\textrm{GFF}}(\tau)\mathrm{LF}_{\tau}(d\phi)\times\mathcal{Z}_{\textrm{ghost}}(\tau)\mathrm{d}\tau,\quad\textrm{with
}\mathcal{Z}_{\textrm{GFF}}(\tau)=\frac{1}{\sqrt{2}\eta(2i\tau)}\textrm{ and
}Z_{\textrm{ghost}}(\tau)=\eta(2i\tau)^{2}.$

The measure $\mathcal{Z}_{\textrm{GFF}}(\tau)\mathrm{LF}_{\tau}(d\phi)$ is the
base measure for the path integral of the Liouville action on
$\mathcal{C}_{\tau}$, where $\mathcal{Z}_{\textrm{GFF}}(\tau)$ is the free
field partition function coming from this path integral. The function
$\mathcal{Z}_{\textrm{ghost}}(\tau)$ is the partition function on
$\mathcal{C}_{\tau}$ of the ghost field for bosonic string theory, which is
the non-physical conformal field theory with central charge
$c_{\mathrm{ghost}}=-26$ coming from conformal gauge fixing. See [Rem18,
Section 5.2] for more background on $\mathcal{Z}_{\textrm{GFF}}(\tau)$ and
$\mathcal{Z}_{\textrm{ghost}}(\tau)$. Putting the two parts of the conjecture
together we get:

######  Conjecture 1 ([Rem18]).

We have
$\mathrm{BA}=1_{\tau>0}2^{-1/2}\eta(2i\tau)\cdot\mathrm{LF}_{\tau}(d\phi)\mathrm{d}\tau$
in the sense that a pair $(\phi,\tau)$ sampled from (1.2) gives a free
Brownian annulus conformally embedded on $\mathcal{C}_{\tau}$.

Our formulation of Conjecture 1 is an identity on two infinite measures while
the formulation in [Rem18] is in terms of convergent functional integrals over
these measures, where the convergence is achieved by introducing cosmological
constants. In this paper we prove the following theorem.

######  Theorem 1.3.

Conjecture 1 holds.

Once we couple 2D quantum gravity with a conformal matter, the influence on
the random geometry can be intuitively understood from the scaling limit of
random planar maps. In this context, a conformal matter corresponds to a
statistical physics model $M$ with a conformal invariant scaling limit. The 2D
quantum gravity coupled with $M$ can be approximated by random planar maps
decorated by model $M$, where the law of the planar map is the uniform measure
weighted by the partition function of $M$. Now the conjectural description of
the conformally embedded geometry for the scaling limit with annular topology
becomes

(1.3)
$\mathcal{Z}_{M}(\tau)\times\mathcal{Z}_{\textrm{GFF}}(\tau)\mathrm{LF}_{\tau}(d\phi)\times\mathcal{Z}_{\textrm{ghost}}(\tau)\mathrm{d}\tau,$

where $Z_{M}(\tau)$ is the partition function of the matter $M$ on
$\mathcal{C}_{\tau}$. As explained in [DRV16, GRV19, Rem18], suppose the CFT
describing the scaling limit of $M$ has central charge $c_{\mathrm{M}}\leq 1$.
Then the random geometry is induced by the Liouville CFT with central charge
$c_{\mathrm{L}}=-c_{\mathrm{ghost}}-c_{\mathrm{M}}=26-c_{\mathrm{M}}$. Here
$c_{\mathrm{L}}$ is related to the background charge $Q$ and the coupling
parameter $\gamma$ via $c_{\mathrm{L}}=1+6Q^{2}$ and
$Q=\frac{\gamma}{2}+\frac{2}{\gamma}$.

For each $\gamma\in(\sqrt{8/3},2)$ (i.e. $c_{\mathrm{M}}\in(0,1)$), a quantum
surface of annular topology called the _quantum annulus_ was introduced by the
first and third named authors in [AS21]. The definition is purely in the
continuum in terms of $\gamma$-LQG coupled with $\mathrm{CLE}_{\gamma^{2}}$ as
we recall in Section 4. Intuitively, the quantum annulus describes the scaling
limit of the random planar maps of annular topology decorated with a
particular matter $M$. For example, for $\gamma=\sqrt{3}$, the matter $M$ is
the Ising model on the annulus with $+$ boundary condition on both sides and
the extra condition that there is a $+$ arm crossing the annulus. Using the
relation between the Ising model and $\mathrm{CLE}_{3}$ [BH19], the scaling
limit of $M$ is the $\mathrm{CLE}_{3}$ on the annulus without non-contractible
loops. Similarly, for $\gamma\in(\sqrt{8/3},2)$, the _quantum annulus_
$\mathrm{QA}^{\gamma}$ describes the natural LQG surface of annular topology
decorated by $\mathrm{CLE}_{\gamma^{2}}$ without non-contractible loops222CLE
on an annulus without non-contractible loops was studied in [SWW17].. Assuming
the relation between $\mathrm{CLE}$ and the scaling limit of O$(n)$-loop
model, the conjectural formula from [SB89, Car06] for the O$(n)$-loop model
(see (1.12) and Corollary 1.10) yields a formula for the annulus partition
function $Z_{\mathrm{M}}(\tau)$. Combined with (1.3) we arrive at a
conjectural description of $\mathrm{QA}^{\gamma}$, which we prove as the next
theorem.

######  Theorem 1.4.

In the same sense as in Conjecture 1, for $\gamma\in(\sqrt{8/3},2)$,
$\mathrm{QA}^{\gamma}$ is given by

(1.4)
$\mathrm{QA}^{\gamma}=1_{\tau>0}\cdot{\cos(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{\gamma}{2\pi}\theta_{1}(\frac{\gamma^{2}}{8},\frac{\gamma^{2}}{4}i\tau)\mathrm{LF}_{\tau}(\mathrm{d}\phi)\mathrm{d}\tau$

where
$\theta_{1}\\!\left(z,i\tau\right):=-i{e}^{-\pi\tau/4}\sum_{n=-\infty}^{\infty}{\left(-1\right)}^{n}{e}^{-n\left(n+1\right)\pi\tau}{e}^{(2n+1)\pi
iz}$ is the Jacobi theta function.

Theorems 1.3 and 1.4 are proved using the same and widely applicable method as
we outline below.

###  1.3. A KPZ relation for annulus partition functions

Fix $\tau>0$. Let $h$ be the free boundary Gaussian free field (see e.g.
[She07]) on the cylinder $\mathcal{C}_{\tau}$. As an example of Gaussian
multiplicative chaos [Kah85, DS11, RV14], for $\gamma\in(0,2)$ the
$\gamma$-LQG length measure on $\partial\mathcal{C}_{\tau}$ is defined by
$\mathcal{L}^{\gamma}_{h}=\lim_{\epsilon\to
0}\epsilon^{\frac{\gamma^{2}}{4}}e^{\frac{\gamma}{2}h_{\epsilon}(z)}dz$, where
$dz$ is the Lebesgue measure on $\partial\mathcal{C}_{\tau}$, and
$h_{\epsilon}(z)$ is the average of $h$ over
$\\{w\in\mathcal{C}_{\tau}:|w-z|=\epsilon\\}$. Similarly, the $\gamma$-LQG
area measure on $\mathcal{C}_{\tau}$ is
$\mathcal{A}^{\gamma}_{\phi}=\lim_{\epsilon\to
0}\epsilon^{\frac{\gamma^{2}}{2}}e^{\gamma h_{\epsilon}(z)}dz$. Let
$\partial_{0}\mathcal{C}_{\tau}=\\{0\\}\times[0,1]/{\sim}$ and
$\partial_{1}\mathcal{C}_{\tau}=\\{\tau\\}\times[0,1]/{\sim}$. In our previous
work [ARS21], based on the conformal bootstrap of boundary Liouville CFT, we
gave a conjecture ([ARS21, Conjecture 1.4]) on the joint distribution of
$\mathcal{L}^{\gamma}_{\phi}(\partial_{0}\mathcal{C}_{\tau})$,
$\mathcal{L}^{\gamma}_{\phi}(\partial_{1}\mathcal{C}_{\tau})$ and
$\mathcal{A}^{\gamma}_{\phi}(\mathcal{C}_{\tau})$ under the Liouville field
measure $\mathrm{LF}_{\tau}$. This conjecture was recently proved by Wu [Wu22]
based on the rigorous bootstrap framework for Liouville CFT developed in
[GKRV20, GKRV21]. The starting point of our proof of Theorems 1.3 and 1.4 is a
corollary of Wu’s theorem, which is interesting in its own right. It gives a
simple description of the law of
$\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}$
where $h$ is a free boundary GFF. (Although the definition of free boundary
GFF involves an additive constant as we recall in Section 2, the ratio
$\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}$
is canonically defined.)

######  Theorem 1.5.

Let $\gamma\in(0,2)$ and $\tau>0$, the characteristic function of
$\log\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})-\log\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})$
is

(1.5)
$\mathbb{E}[e^{ix(\log\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})-\log\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau}))}]=\frac{\pi\gamma^{2}xe^{-\frac{\pi\gamma^{2}\tau
x^{2}}{4}}}{4\sinh(\frac{\gamma^{2}}{4}\pi x)}\qquad\textrm{for
}x\in\mathbb{R}.$

Theorem 1.5 further demonstrates the elegant integrability of 1D Gaussian
multiplicative chaos, which was explored in [Rem20, RZ20b, RZ20a, GRSS20]. We
recall Wu’s theorem and prove Theorem 1.5 in Section 3. From this we derive
what we call the _KPZ relation at the annulus partition function level_.

######  Theorem 1.6.

Fix $\gamma\in(0,2)$. Let $m(\mathrm{d}\tau)$ be a measure on $(0,\infty)$.
For a measurable function $f$, write $\langle
f(L_{0},L_{1})\rangle_{\gamma}:=\int
f(\mathcal{L}^{\gamma}_{\phi}(\partial_{0}\mathcal{C}_{\tau}),\mathcal{L}^{\gamma}_{\phi}(\partial_{1}\mathcal{C}_{\tau}))\mathrm{LF}_{\tau}(\mathrm{d}\phi)m(\mathrm{d}\tau)$.
Then $m(\mathrm{d}\tau)$ satisfies

(1.6)
$\displaystyle\int_{0}^{\infty}e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}m(\mathrm{d}\tau)=\frac{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}{\pi\gamma x\Gamma(1+ix)}\langle
L_{1}e^{-L_{1}}L_{0}^{ix}\rangle_{\gamma}\quad\textrm{for }x\in\mathbb{R}.$

Given Theorem 1.6, our proofs of Theorems 1.3 and 1.4 follow the same two-step
strategy:

  1. (1)

Show that $\mathrm{BA}$ and $\mathrm{QA}^{\gamma}$ can be written as
$\mathrm{LF}_{\tau}(\mathrm{d}\phi)m(\mathrm{d}\tau)$ for some measure
$m(\mathrm{d}\tau)$. We achieve this by proving a characterization of
$\mathrm{LF}_{\tau}$ in Section 2 and verifying this property for
$\mathrm{BA}$ and $\mathrm{QA}^{\gamma}$.

  2. (2)

The integral $\langle L_{1}e^{-L_{1}}L_{0}^{ix}\rangle_{\gamma}$ for
$\mathrm{BA}$ and $\mathrm{QA}^{\gamma}$ can be easily extracted from the
existing literature. Then computing $m(\mathrm{d}\tau)$ from (1.6) is
accomplished via an inverse Laplace transform.

We view Theorem 1.6 as a KPZ relation in the following sense. For many natural
random surfaces such as $\mathrm{BA}$ and $\mathrm{QA}^{\gamma}$, $\langle
L_{1}e^{-L_{1}}L_{0}^{ix}\rangle_{\gamma}$ is easy to compute from the
perspective that LQG describes the scaling limit of random planar maps. From
this we can compute $m(\mathrm{d}\tau)$ by (1.6). This reduces the computation
of a difficult quantity depending on the conformal structure to an easier
quantity in LQG. This is the essence of the derivation of the scaling
exponents/dimensions of random fractals using the traditional KPZ relation
[DS11, GHM15, BGRV16, GP19]. For example, for $\mathrm{QA}^{\gamma}$, the
$\tau\to\infty$ asymptotic of $m(d\tau)$ given in (1.4) encodes the dimension
of the $\mathrm{CLE}$ carpet [SSW09]. Therefore, using (1.6) to determine the
asymptotic of $m(d\tau)$ can be viewed as applying the traditional KPZ
relation to determine the $\mathrm{CLE}$ carpet dimension. In this sense
Theorem 1.6 upgrades the KPZ relation from the exponents/dimensions level to
the annular partition function level.

In principle our KPZ relation (1.6) allows one to obtain the annulus partition
$Z_{\mathrm{M}}(\tau)$ of any conformal matter $M$ from its quantum
counterpart. In physics $Z_{\mathrm{M}}(\tau)$ is usually obtained as
$\lim_{\delta\to 0}Z^{\delta}_{\mathrm{M}}(\tau)$ where
$Z^{\delta}_{\mathrm{M}}(\tau)$ is the partition function of $M$ on a lattice
of mesh size $\delta$ that approximates $\mathcal{C}_{\tau}$. When the scaling
limit of $M$ is described by SLE/CLE, the function $Z_{\mathrm{M}}(\tau)$ is
supposed to describe a natural SLE/CLE quantity that depends on the modular
parameter. For example, for $\mathrm{QA}^{\gamma}$, the corresponding
$Z_{\mathrm{M}}(\tau)$ describes the law of the modulus of the annulus bounded
by an outermost loop of a $\mathrm{CLE}_{\gamma^{2}}$ on a disk and the disk
boundary; see Theorem 1.7. However, in many cases the convergence of $M$ to
SLE/CLE is an outstanding problem. Theorem 1.6 allows us to bypass this
difficulty and obtain $Z_{\mathrm{M}}(\tau)$ from SLE/CLE using quantum
gravity .

In Section 1.4 we summarize our results for simple CLE based on the KPZ
approach. In Section 1.5 we apply the same approach to the case when $M$ is
the self avoiding loop.

###  1.4. Random Moduli for simple CLE loops

$\mathrm{CLE}_{\kappa}$ with $\kappa\in(8/3,8)$ was first defined in [She09]
as a canonical conformally invariant probability measure on infinite
collections of non-crossing loops, where each loop looks like an
$\mathrm{SLE}_{\kappa}$ curve. For $\kappa\in(8/3,4]$, the loops are simple
and $\mathrm{CLE}_{\kappa}$ can be characterized by a domain Markov property
and conformal invariance [SW12]. CLE has been proved or conjectured to be the
scaling limit of many important 2D loop models such as percolation, the Ising
model, the $O(n)$-loop model, and the random cluster model; see e.g. [CN08,
BH19, KS16]. LQG coupled with $\mathrm{CLE}$ [MSW20, MSW21] are proved or
conjectured to describe the scaling limit of random planar maps decorated with
these models. Recently, this coupling was used in [AS21, Mil21, MS22] to
derive results for CLE itself. As we recall in Section 4, the quantum annulus
$\mathrm{QA}^{\gamma}$ is defined by a symmetric modification of the law of
the annulus on a quantum disk bounded by an outermost loop and the disk
boundary. In fact choosing any loop in a CLE gives a natural LQG surface of
annular topology. Using the KPZ relation (1.6) we get an analog of Theorem 1.4
for these surfaces. This yields exact formulae for the moduli of the annuli
bounded by CLE loops, as we summarize below.

Let $\mathbb{D}$ be the unit disk. For a simple loop $\eta\subset\mathbb{D}$,
let $A_{\eta}$ be the annulus bounded by $\eta$ and $\partial\mathbb{D}$ and
let $\mathrm{Mod}(\eta)$ be the modulus of $A_{\eta}$. For
$\kappa\in(\frac{8}{3},4]$, let $\Gamma$ be a $\mathrm{CLE}_{\kappa}$ on
$\mathbb{D}$. Let $\\{\eta_{j}\\}_{j\geq 1}$ be the sequence of loops in
$\Gamma$ that surrounds the origin, ordered such that $\eta_{j+1}$ is
surrounded by $\eta_{j}$; see Figure 2 (middle). Our first result is for the
moment generating function of $\mathrm{Mod}(\eta_{j})$.

######  Theorem 1.7.

For $j\geq 1$ and $\lambda>\frac{3\kappa}{32}+\frac{2}{\kappa}-1$, we have

(1.7)
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{j})}]=\frac{(\frac{4}{\kappa}-1)\cos^{j}(\pi(\frac{4}{\kappa}-1))}{\sin(\pi(1-\frac{\kappa}{4}))}\times\frac{\sin(\frac{\kappa}{4}\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}{\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}}\cos^{j}(\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}.$

Figure 2.  Left: The cylinder $\mathcal{C}_{\tau}$ is obtained by identifying
the top and bottom edges of the rectangle $[0,\tau]\times[0,1]$. Middle: With
$\eta_{j}$ the $j^{\text{th}}$ nested loop around the origin, Theorem 1.7
identifies the law of the modulus of the annulus bounded by
$\partial\mathbb{D}$ and $\eta_{j}$. Right: For $\tau>0$, Corollary 1.10 gives
the law of the number $\mathcal{N}$ of non-contractible loops for
$\mathrm{CLE}$ in $\mathbb{A}_{\tau}$.

Let $\mathrm{CR}(\eta_{j})$ be the conformal radius of $\eta_{j}$ viewed from
0. Namely, $\mathrm{CR}(\eta_{j})=|\psi^{\prime}(0)|$ where $\psi$ is a
conformal map from the region surrounded by $\eta_{j}$ to $\mathbb{D}$ that
fixes the origin. The law of $\mathrm{CR}(\eta_{j})$ is obtained in [SSW09]:

(1.8)
$\mathbb{E}[\mathrm{CR}(\eta_{j})^{\lambda}]=\left(\frac{-\cos(4\pi\kappa)}{\cos(\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}\right)^{j}\quad\textrm{
for }\lambda>\frac{3\kappa}{32}+\frac{2}{\kappa}-1.$

Our next theorem gives the joint law of $\mathrm{Mod}(\eta_{j})$ and
$\mathrm{CR}(\eta_{j})$ for $\kappa\in(\frac{8}{3},4]$ and $j\geq 1$.

######  Theorem 1.8.

For $j\geq 1$, $\mathrm{Mod}(\eta_{j})$ and
$e^{2\pi\mathrm{Mod}(\eta_{j})}\mathrm{CR}(\eta_{j})$ are independent.

The case $\kappa=4,j=1$ for Theorems 1.7 and 1.8 was obtained in [ALS20] via
the level set coupling of $\mathrm{CLE}_{4}$ and the Gaussian free field. As
shown in [ALS20, SSW09], the law of $\mathrm{Mod}(\eta_{1})$ for $\kappa=4$
and $\mathrm{CR}(\eta_{1})$ for general $\kappa$ can both be described in
terms of hitting times of Brownian motion. This is also possible for
$\mathrm{Mod}(\eta_{j})$ for general $\kappa$ and $m$; see Remark 5.6. The
independence between $\mathrm{Mod}(\eta_{j})$ and
$e^{2\pi\mathrm{Mod}(\eta_{j})}\mathrm{CR}(\eta_{j})$ does not appear to be a
simple consequence of the domain Markov property of $\mathrm{CLE}$. Our proof
relies on the method in [AHS21, ARS21, AS21] which solves the law of conformal
radii of SLE/CLE loops using LCFT.

For a fixed $n>0$, the O($n$) loop model on a graph is a statistical physics
model where a configuration $\mathcal{L}$ is a collection of disjoint loops
(i.e. simple cycles). Given the temperature $T>0$, the weight of each
configuration is $T^{-\mathcal{V}(\mathcal{L})}n^{\mathcal{N}(\mathcal{L})}$,
where $\mathcal{V}(\mathcal{L})$ is the total number of vertices occupied by
all loops in $\mathcal{L}$ and $\mathcal{N}(\mathcal{L})$ is the total number
of loops in $\mathcal{L}$. When $n\in(0,2]$, for the O($n$) loop model on a
planar lattice, there exists a critical temperature $T_{c}>0$ such that at
$T_{c}$ the scaling limit is conjectured to be $\mathrm{CLE}_{\kappa}$ for
$\kappa\in(\frac{8}{3},4]$; see [She09] for more background. The parameters
$\kappa$ and $n$ related as follows:

(1.9)
$g=\frac{4}{\kappa}\in[1,\frac{3}{2}),\quad\chi=(1-g)\pi\in(-\frac{\pi}{2},0],\quad\textrm{and}\quad
n=2\cos\chi\in(0,2].$

The central charge of the CFT describing the scaling limit is given by:

(1.10)
$c=1-\frac{6(1-g)^{2}}{g}=1-6(\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2})^{2}\in(0,1].$

Consider the $O(n)$ loop model on a lattice approximation of the cylinder
$\mathcal{C}_{\tau}$. Let

(1.11) $q=e^{-\pi/\tau},\quad\tilde{q}=e^{-2\pi\tau},\quad\textrm{and}\quad
n^{\prime}=2\cos\chi^{\prime}\textrm{ for }\chi^{\prime}\in\mathbb{R}.$

Now suppose each contractible loop is assigned weight $n$ while each non-
contractible loop is assigned weight $n^{\prime}$ instead. Then as predicted
in [SB89, Car06], in the continuum limit at $T_{c}$ the partition function is

(1.12) $\displaystyle Z(\tau,\kappa,\chi^{\prime})$
$\displaystyle=q^{\frac{-c}{24}}\prod_{r=1}^{\infty}(1-q^{r})^{-1}\sum_{p\in\mathbb{Z}}\frac{\sin(p+1)\chi^{\prime}}{\sin\chi^{\prime}}q^{\frac{gp^{2}}{4}-\frac{(1-g)p}{2}}$
(1.13)
$\displaystyle=(\frac{2}{g})^{1/2}\tilde{q}^{-\frac{c}{12}}\prod_{r=1}^{\infty}(1-\tilde{q}^{2r})^{-1}\sum_{m\in\mathbb{Z}}\frac{\sin((\chi^{\prime}+2m\pi)/g)}{\sin\chi^{\prime}}\tilde{q}^{\frac{(\chi^{\prime}+2\pi
m)^{2}}{2\pi^{2}g}-\frac{(1-g)^{2}}{2g}}.$

Equations (1.12) and (1.13) were derived in [SB89] via the quantum group
method, and derived in [Car06] via the Coulomb gas method. (See also the
lecture notes [Jac09, Section 14.7.2]). As pointed out in [Car06], it would be
interesting to derive $Z(\tau,\kappa,\chi^{\prime})$ rigorously in the SLE
framework. As a consequence of Theorem 1.8, we solve this problem.

######  Theorem 1.9.

For any non-negative measurable function $f$ on $(0,\infty)$, we have

(1.14)
$\mathbb{E}\left[\sum_{j=1}^{\infty}\left(\frac{n^{\prime}}{n}\right)^{j-1}f(\mathrm{Mod}(\eta_{j}))\right]=\frac{\sqrt{\kappa}(\frac{4}{\kappa}-1)\cos(\pi(\frac{4}{\kappa}-1))}{\sqrt{2}\sin(\pi(1-\frac{\kappa}{4}))}\int_{0}^{\infty}f(\tau)e^{\frac{\kappa\pi}{4}(\frac{4}{\kappa}-1)^{2}\tau}Z(\tau,\kappa,\chi^{\prime})\eta(2i\tau)d\tau,$

where $\eta(i\tau)=e^{-\frac{\pi\tau}{12}}\prod_{k=1}^{\infty}(1-e^{-2\pi
k\tau})$ is the Dedekind eta function.

A more transparent way to see the geometric meaning of
$Z(\tau,\kappa,\chi^{\prime})$ is to consider CLE on the annulus. Given
$\Gamma$, we choose $\eta$ from the counting measure of $\\{\eta_{j}\\}_{j\geq
1}$ and let $A_{\eta}$ be the annulus bounded by $\eta$ and
$\partial\mathbb{D}$. Let $\mathrm{CLE}^{\tau}_{\kappa}$ be the conditional
law of the loops of $\Gamma$ inside $A_{\eta}$ conditioning on
$\mathrm{Mod(\eta)=\tau}$. Then we use conformal invariance to extend the
definition of $\mathrm{CLE}^{\tau}_{\kappa}$ to any annulus of modulus $\tau$.
This gives one natural definition of $\mathrm{CLE}_{\kappa}$ on an annulus
that allows non-contractible loops; see Figure 2 (right). In fact, if
$\eta=\eta_{j}$, then the number of non-contractible loops is $j-1$. Therefore
Theorem 1.9 has the following corollary.

######  Corollary 1.10.

For $\tau>0$, let $\mathcal{N}$ be the number of non-contractible loops of a
$\mathrm{CLE}^{\tau}_{\kappa}$. Then

(1.15)
$\mathbb{E}\left[\left(\frac{n^{\prime}}{n}\right)^{\mathcal{N}}\right]=\frac{Z(\tau,\kappa,\chi^{\prime})}{Z(\tau,\kappa,\chi)}$

where $\chi=(1-\frac{\kappa}{4})\pi$, $n=2\cos\chi$, and
$n^{\prime}=2\cos\chi^{\prime}$ for $\chi^{\prime}\in\mathbb{R}$.

Annulus partition functions are fundamental in the CFT framework of 2D
statistical physics. The annulus partition function of a CFT in term of the
$q$ variable is of the form
$q^{-c/24}\sum_{s\in\mathcal{S}_{\mathrm{bdy}}}c(s)K_{s}(q)$ as in (1.12),
where $c$ is the central charge and $\mathcal{S}_{\mathrm{bdy}}$ is the so
called boundary spectrum, and $K_{s}(q)$ is a universal $s$-dependent function
called the Virasoro character. This summation is called the conformal
bootstrap in the open string channel. When $\mathcal{S}_{\mathrm{bdy}}$ is
finite, the CFT is called rational. For example, the Ising model partition
function equals $Z(\tau,\kappa,\chi)$ with $n=1$. It can be written as
$Z_{++}+Z_{+-}$, where $Z_{++}$ (resp., $Z_{+-}$) is the annulus partition
function with $++$ (resp., $+-$) boundary condition. Both $Z_{++}$ and
$Z_{+-}$ are examples of Virasoro characters. In terms of the $\tilde{q}$
variable, annulus partition functions are of the form
$\tilde{q}^{-c/12}\sum_{s\in\mathcal{S}_{\mathrm{bul}}}|c(s)|^{2}K_{s}(\tilde{q}^{2})$.
The set $\mathcal{S}_{\mathrm{bul}}$ is called the bulk spectrum and the
summation is called the conformal bootstrap in the closed string channel. The
equality between (1.12) and (1.13) is called modular invariance or open-closed
duality. Conformal bootstrap formulas were also derived for torus partition
functions of various models; see e.g. [DFSZ87]. The spectrum is given by
$\mathcal{S}_{\mathrm{bul}}$ and a similar modular invariance holds. As a
cornerstone for 2D CFT, modular invariance imposes strong constraints on these
theories, which is crucial to the classification of an important class of
rational CFTs called the minimal models. The annulus and torus partition
functions can often be derived for lattice models by methods such as algebraic
CFT principle, Coulomb gas techniques, and transfer matrix, and along the way
one obtains $c$, $\mathcal{S}_{\mathrm{bdy}}$ and
$\mathcal{S}_{\mathrm{bul}}$. See [ISZ88, DFMS97] for more background on CFT
and 2D statistical physics.

The spectra $\mathcal{S}_{\mathrm{bdy}}$ and $\mathcal{S}_{\mathrm{bul}}$
determine the allowed boundary and bulk scaling dimensions for the
corresponding model. Many scaling dimensions were later rigorously derived in
the SLE framework as the exponents of certain arm events, or as the dimensions
of certain fractals. Our paper provides a method to derive the full annulus
partition function, strengthening the connection between the SLE and the CFT
approaches to 2D statistical physics. Although the connection between CFT and
SLE has been studied intensively from the martingale observable approach (e.g.
[FW03, BB03, KM21]), it appears challenging to derive results for CLE using
this approach. In the next subsection we demonstrate how our method applies to
another model: the self avoiding loop.

###  1.5. Annulus partition function of the SLE8/3 loop

Werner [Wer08] constructed an infinite conformally invariant measure on simple
loops on each Riemann surface which we call the _$\mathrm{SLE}_{8/3}$ loop
measure_. Modulo a multiplicative constant, the $\mathrm{SLE}_{8/3}$ loop
measure is uniquely characterized by the conformal restriction property. It
describes the conjectural scaling limit of self avoiding polygons [Wer08,
Conjecture 1]. A planar map version of this conjecture was proved in [AHS22]
based on [GM21a]. The construction of the $\mathrm{SLE}_{8/3}$ loop measure in
[Wer08] is based on the Brownian loop measure. Let $\mu_{\mathbb{C}}$ be an
$\mathrm{SLE}_{8/3}$ loop measure on the complex plane $\mathbb{C}$. In
Section 7 we will recall a construction of $\mu_{\mathbb{C}}$ due to Zhan
[Zha21].

Given a domain $D\subset\mathbb{C}$, let $\mu_{D}$ be the restriction of
$\mu_{\mathbb{C}}$ to the set of loops that are contained in $D$. Let $\psi$
be a conformal map between two domains $D$ and $D^{\prime}$. Then the
conformal restriction property of the $\mathrm{SLE}_{8/3}$ loop measure
asserts that $\mu_{D}$ is the pushforward of $\mu_{D^{\prime}}$ under $\psi$;
see Theorem 7.3. Given an annulus $A\subset\mathbb{C}$ of modulus $\tau$, let
$Z_{8/3}(\tau)$ be the $\mu_{\mathbb{C}}$-mass of loops in $A$ that are not
contractible. Then the conformal restriction property implies that
$Z_{8/3}(\tau)$ depends on $A$ through its modulus $\tau$. We call
$Z_{8/3}(\tau)$ the annulus partition function of the SLE8/3 loop. It is
supposed to be the scaling limit of $\sum\nu_{c}^{-\textrm{loop length}}$
where the summation runs over non-contractible loops of a lattice
approximation of the cylinder $\mathcal{C}_{\tau}$, and $\nu_{c}$ is the so-
called connectivity constant of the lattice. (For the honeycomb lattice
$\nu_{c}=\sqrt{2+\sqrt{2}}$ [DCS12].)

The sharp asymptotic of $Z_{8/3}(\tau)$ was obtained in [Wer08, Section 7],
where deriving an exact formula for $Z_{8/3}(\tau)$ was left as an open
question. A formula was conjectured by Cardy [Car06, Eq (5)] based on the
perspective that self-avoiding loop can be viewed as $O(n)$ loop model with
$n=0$. We rigorously prove Cardy’s formula for $Z_{8/3}(\tau)$ as the next
theorem.

######  Theorem 1.11.

There exists a constant $C>0$ such that
$Z_{8/3}(\tau)=CZ_{\mathrm{Cardy}}(\tau)$ where

(1.16)
$Z_{\mathrm{Cardy}}(\tau)=\prod_{r=1}^{\infty}(1-q^{r})^{-1}\sum_{k\in\mathbb{Z}}k(-1)^{k-1}q^{\frac{3k^{2}}{2}-k+\frac{1}{8}}\quad\textrm{for
}\tau>0\textrm{ and }q=e^{-\pi/\tau}.$

Since $\mu_{\mathbb{C}}$ is characterized up to a multiplicative constant, the
constant $C$ in Theorem 1.11 is not canonical. We prove Theorem 1.11 in
Section 7. The starting point of our proof is the following fact proved in
[AHS22]: gluing together two Brownian disks with equal boundary lengths gives
a Brownian sphere decorated with an $\mathrm{SLE}_{8/3}$ loop. From this we
can construct the $\mathrm{SLE}_{8/3}$-loop-decorated quantum annulus by
gluing together two Brownian annuli. Then the two-step strategy outlined below
(1.6) gives the law of its modulus. Finally, the conformal restriction
property of the $\mathrm{SLE}_{8/3}$ loop allows us to derive $Z_{8/3}(\tau)$
from the law of the modulus.

###  1.6. Perspectives and outlook

We discuss a few directions that a subset of the authors and their
collaborators are planning to investigate in subsequent works, and some open
questions.

  1. (1)

Our KPZ method for proving Theorems 1.4 and 1.11 can extend to other models,
such as uniform spanning tree and percolation. In a subsequent work we will
carry this out for $\mathrm{CLE}_{\kappa}\in(4,8)$, which will extend results
in Section 1.4 to $\kappa\in(4,8)$ and confirm conjectures in this regime from
[Car06]. We will also prove Cardy’s crossing formula for percolation on the
annulus from [Car02]; see also [Dub04].

  2. (2)

We can consider variants of (1.6) where we have one boundary marked point on
the annulus. Combining with the conformal bootstrap results for Liouville CFT
in [Wu22], this will give interesting relations between the 1-point torus
conformal block [GRSS20] and SLE/CLE observables.

  3. (3)

We can also consider variants of (1.6) where the underlying domain is a disk
with four boundary marked points or with one bulk and two boundary points.
This will give access to SLE observables such as crossing probabilities and
the one-point Green’s function.

  4. (4)

We will consider the variant of (1.6) in the pair of pants case, where we
expect to obtain the analogous results for the Brownian pair of pants and for
CLE on a disk with two holes. As a limiting case, we expect to obtain an exact
formula for certain two-point correlation functions for CLE on the disk.

  5. (5)

At the high level, our two-step strategy below (1.6) can extend to closed
surfaces such as the torus. But in the second step, we would need to find
other observables than the boundary length to obtain the law of the modulus.
This could still be tractable for the torus, but we consider it as a major
open question to extend our results to arbitrary Riemann surfaces. Essential
new ideas are required to understand the much more complicated moduli space
and the integrability of Liouville CFT on these surfaces. See [GRV19, Section
5] for precise conjectures in the closed surface case.

  6. (6)

Determining the full CFT content of CLE and related discrete models is an
active topic in both mathematics and physics. See [JPS20, Section 4.2] for a
summary of the state of the art in physics for the $Q$-Potts model. In
mathematics, the so-called imaginary DOZZ formula conjectured in [IJS16] was
proved in [AS21], and the annulus partition function is proved in this paper.
It would be a breakthrough to carry out a functional analytic conformal
bootstrap program for CLE similar to the Liouville case [GKRV20, GKRV21]. It
would already be very interesting to give a purely CLE-based proof of results
in our Section 1.4, as for the $\kappa=4$ case in [ALS20]. Such a proof might
be extended to prove the torus counterpart of Corollary 1.10 as predicted by
[DFSZ87, Pin94]. If knowledge on the conformal matter can be obtained without
LQG, then it can be used as a substitute for the boundary length observable in
(1.6) to determine the law of the modulus.

  7. (7)

It would be interesting to understand the coupling of the ghost field with the
matter CFT and Liouville CFT at the probabilistic level, which goes beyond the
factorization of partition functions. See [LZ91, BMP92] for related work at
the representation theoretic level.

### Organization of the paper.

In Section 2 we give a characterization of $\mathrm{LF}_{\tau}$. In Section 3
we prove Theorems 1.5 and 1.6. In Section 4 we carry out the two-step strategy
outlined below (1.6) to prove Theorem 1.4. In Section 5 we prove results for
CLE from Section 1.4. In Section 6 we prove Theorem 1.3 using the same two-
step strategy, which implies Theorem 1.2. In Section 7 we prove Theorem 1.11.

Acknowledgements. We thank Baojun Wu for communicating with us his concurrent
work [Wu22]. We thank Jesper Jacobsen and Hubert Saleur for very helpful
feedback on an earlier version of this paper. M.A. was partially supported by
NSF grant DMS-1712862. G.R. was supported by an NSF mathematical sciences
postdoctoral research fellowship, NSF Grant DMS-1902804. X.S. was supported by
the NSF grant DMS-2027986 and the NSF Career award 2046514.

##  2\. Liouville fields and their domain Markov property

In this section we introduce Liouville fields on the annulus and the disk, and
prove their domain Markov property (Propositions 2.4 and 2.9) that we will use
throughout the paper. We then give a characterization of the Liouville field
on the annulus (Proposition 2.13). In the rest of the paper, we let
$\mathbb{H}=\\{z\in\mathbb{C}:\Im z>0\\}$ be the upper half plane. For $x<y$,
let $\mathcal{C}(x,y)=[x,y]\times[0,1]/{\sim}$ be a finite horizontal cylinder
so that $\mathcal{C}_{\tau}=\mathcal{C}(0,\tau)$. Let
$\partial_{0}\mathcal{C}(x,y)=\\{x\\}\times[0,1]/{\sim}$ and
$\partial_{1}\mathcal{C}(x,y)=\\{y\\}\times[0,1]/{\sim}$. For infinite
measures, we will still adopt the probabilistic terminologies such as random
variable and law.

###  2.1. Gaussian free field

We first gather some facts about two dimensional Gaussian free field (GFF).
See e.g. [She07, Dub09] for more background. For concreteness, let $D$ be a
planar domain that is conformally equivalent of a disk or an annulus.
Throughout this section we assume that

(2.1) $\textrm{$\rho(\mathrm{d}x)$ is a compactly supported probability
measure such that
}\iint-\log|x-y|\,\rho(\mathrm{d}x)\rho(\mathrm{d}y)<\infty.$

Our discussion could extend to a broader class of measures but (2.1) is
sufficient for us. For a measure $\rho$ as in (2.1) supported on $D$, let
$H(D;\rho)$ be the Hilbert closure of $\\{f\in
C^{\infty}(D):\int_{D}f(x)\rho(\mathrm{d}x)=0\\}$ under the inner product
$(f,g)_{\nabla}=(2\pi)^{-1}\int_{D}\left(\nabla f\cdot\nabla
g\right)\mathrm{d}x$. Let $(f_{n})_{n\geq 1}$ be an orthonormal basis for
$H(D;\rho)$ and $(\alpha_{n})_{n\geq 1}$ be a sequence of independent standard
normal random variables. Then the random series
$\sum_{n=1}^{\infty}\alpha_{n}f_{n}$ converges almost surely as a random
generalized function. We call a random generalized function on $D$ with the
law of $\sum_{n=1}^{\infty}\alpha_{n}f_{n}$ a _free boundary Gaussian free
field_ on $D$ normalized to have average zero over $\rho(\mathrm{d}x)$.

Now suppose $A$ is conformally equivalent to an annulus, with the two boundary
components denoted by $\partial_{1}A$ and $\partial_{2}A$. Let $H(A)$ be the
Hilbert closure of $\\{f\in
C^{\infty}(A):\mathrm{supp}(f)\cap\partial_{1}A=\emptyset\\}$ under the inner
product $(f,g)_{\nabla}=(2\pi)^{-1}\int_{A}\left(\nabla f\cdot\nabla
g\right)\mathrm{d}x$. Here $\mathrm{supp}(f)$ means the support of $f$. Let
$(g_{n})_{n\geq 1}$ be an orthonormal basis for $H(A)$. Let
$(\alpha_{n})_{n\geq 1}$ be a sequence of independent standard normal random
variables. Then $\sum_{n=1}^{\infty}\alpha_{n}g_{n}$ converges almost surely
as a random generalized function. We call a random generalized function on $A$
with the law of $\sum_{n=1}^{\infty}\alpha_{n}g_{n}$ a Gaussian free field on
$A$ with zero-boundary condition on $\partial_{1}A$ and free boundary
condition on $\partial_{2}A$.

We will need an instance of the domain Markov property for the GFF. Let $D$ be
a domain conformally equivalent to an annulus or disk and such that $\partial
D$ is smooth. Let $\eta$ be a Jordan curve (namely simple closed curve) inside
$D$ such that we can find a connected component $A$ of $D\setminus\eta$ that
is conformally equivalent to an annulus. We let $\partial_{1}A$ be the
boundary component $\eta$ and $\partial_{2}A$ be the other boundary component
of $A$. Let $\rho(\mathrm{d}x)$ be a probability measure on $D\setminus A$
satisfying (2.1). Let $\mathrm{Harm}(D;\rho;A)$ be the space of functions in
$H(D;\rho)$ that are harmonic on $A$ and have normal derivative zero on
$\partial_{2}A$. Then we have the orthogonal decomposition

(2.2) $H(D,\rho)=H(A)\oplus\mathrm{Harm}(D;\rho;A),$

which yields the following domain Markov property of the free boundary GFF.

######  Lemma 2.1.

With $D,A,\rho$ defined in the paragraph right above, let $h$ be a free
boundary GFF on $D$ normalized by $\int h(x)\rho(\mathrm{d}x)=0$. Let
$h^{\mathrm{har}}$ be the harmonic extension of $h|_{D\setminus A}$ onto $A$
with zero normal derivative on $\partial_{2}A$. Let
$h_{A}=h-h^{\mathrm{har}}$. Then $h_{A}$ is a GFF on $A$ with zero boundary
condition on $\partial_{1}A$ (namely $\eta$), and free boundary condition on
$\partial_{2}A$. Moreover, $h_{A}$ is independent of $h^{\mathrm{har}}$.

###### Proof.

Since $h^{\mathrm{har}}$ is the projection of $h$ onto
$\mathrm{Harm}(D;\rho;A)$, we obtain Lemma 2.1 from (2.2) via the standard
argument from [She07]. ∎

###  2.2. Liouville fields on the annulus and the disk

We first define Liouville fields on annuli. For convenience we focus on the
horizontal cylinder
$\mathcal{C}_{\tau}=\mathcal{C}(0,\tau)=[0,\tau]\times[0,1]/{\sim}$.

######  Definition 2.2.

For $\tau>0$, let $\rho$ be a probability measure on $\mathcal{C}_{\tau}$
satisfying (2.1) and $\mathbb{P}_{\tau,\rho}$ be the law of the free boundary
Gaussian free field on $\mathcal{C}_{\tau}$ with
$\int_{\mathcal{C}_{\tau}}h(x)\rho(\mathrm{d}x)=0$. Sample $(h,\mathbf{c})$
from $\mathbb{P}_{\tau,\rho}\times dc$ and set $\phi=h+\mathbf{c}$. We write
$\mathrm{LF}_{\tau}$ as the law of $\phi$, and call $\phi$ a _Liouville field
on $\mathcal{C}_{\tau}$_.

######  Lemma 2.3.

The measure $\mathrm{LF}_{\tau}$ does not depend on the choice of the measure
$\rho$.

###### Proof.

This follows from the translation invariance of the Lebesgue measure $dc$. ∎

Lemma 2.1 and 2.3 yield a domain Markov property for $\mathrm{LF}_{\tau}$. To
state it, we recall that
$\partial_{0}\mathcal{C}_{\tau}=\\{0\\}\times[0,1]/{\sim}$ and
$\partial_{1}\mathcal{C}_{\tau}=\\{\tau\\}\times[0,1]/{\sim}$ are the two
boundary components of $\mathcal{C}_{\tau}$.

######  Proposition 2.4.

Let $\eta\subset\mathcal{C}_{\tau}$ be a non-contractible Jordan curve so that
both components of $\mathcal{C}_{\tau}\setminus\eta$ are of annular topology.
Let $A_{\eta}$ be the component bounded by $\eta$ and
$\partial_{0}\mathcal{C}_{\tau}$. Let $\phi$ be a Liouville field on
$\mathcal{C}_{\tau}$. Let $\phi^{\mathrm{har}}$ be the harmonic extension of
$\phi|_{\mathcal{C}_{\tau}\setminus A_{\eta}}$ onto $A_{\eta}$ with zero
normal derivative on $\partial_{0}\mathcal{C}_{\tau}$. Let
$\phi_{A_{\eta}}=\phi-\phi^{\mathrm{har}}$. Then conditioning on
$\phi|_{\mathcal{C}_{\tau}\setminus A_{\eta}}$, the conditional law of
$\phi_{A_{\eta}}$ is a GFF on $A_{\eta}$ with zero boundary condition on
$\eta$ and free boundary condition on $\partial_{0}\mathcal{C}_{\tau}$. If
$A_{\eta}$ is bounded by $\eta$ and $\partial_{1}\mathcal{C}_{\tau}$ instead,
the same holds with $\partial_{1}\mathcal{C}_{\tau}$ in place of
$\partial_{0}\mathcal{C}_{\tau}$.

Since $\mathrm{LF}_{\tau}$ is an infinite measure, the word “conditioning”
above requires a proper interpretation.

######  Definition 2.5.

Suppose $(\Omega,\mathcal{F})$ and $(\Omega^{\prime},\mathcal{F}^{\prime})$
are two measurable spaces. We say
$\Lambda:\Omega\times\mathcal{F}^{\prime}\to[0,1]$ is a Markov kernel if
$\Lambda(\omega,\cdot)$ is a probability measure on
$(\Omega^{\prime},\mathcal{F}^{\prime})$ for each $\omega\in\Omega$ and
$\Lambda(\cdot,A)$ is $\mathcal{F}$-measurable for each
$A\in\mathcal{F}^{\prime}$. If $(X,Y)\in\Omega\times\Omega^{\prime}$ is a
sample from $\Lambda(\omega,d\omega^{\prime})\mu(d\omega)$ for some measure
$\mu$ on $(\Omega,\mathcal{F})$, we say that the conditional law of $Y$ given
$X$ is $\Lambda(X,\cdot)$.

###### Proof of Proposition 2.4.

We focus on the case where $A_{\eta}$ is bounded by $\eta$ and
$\partial_{0}\mathcal{C}_{\tau}$ as the other case follows by symmetry. Let
$\rho$ be a probability measure on the connected component of
$\mathcal{C}_{\tau}\setminus\eta$ other than $A_{\eta}$ satisfying (2.1). Let
$\mathbb{P}_{\tau,\rho}$ be the law of the free boundary GFF on
$\mathcal{C}_{\tau}$ with $\int_{\mathcal{C}_{\tau}}h(x)\rho(\mathrm{d}x)=0$.
Sample $(h,\mathbf{c})$ from $\mathbb{P}_{\tau,\rho}\times dc$ and set
$\phi=h+\mathbf{c}$. Then $\phi$ is a Liouville field on $\mathcal{C}_{\tau}$
by definition.

Let $h^{\mathrm{har}}$ be the harmonic extension of
$h|_{\mathcal{C}_{\tau}\setminus A_{\eta}}$ onto $A_{\eta}$ with zero normal
derivative on $\partial_{0}\mathcal{C}_{\tau}$. Let
$h_{A_{\eta}}=h-h^{\mathrm{har}}$. Then
$\phi^{\mathrm{har}}=h^{\mathrm{har}}+\mathbf{c}$ and
$\phi_{A_{\eta}}=h_{A_{\eta}}$. By Lemma 2.1, conditioning on
$(h|_{\mathcal{C}_{\tau}\setminus A_{\eta}},\mathbf{c})$, the conditional law
of $h_{A_{\eta}}$ is a GFF on $A_{\eta}$ with zero boundary condition on
$\eta$ and free boundary condition on $\partial_{0}\mathcal{C}_{\tau}$. This
gives the desired conditional law of $\phi_{A_{\eta}}$ given
$\phi|_{\mathcal{C}_{\tau}\setminus A_{\eta}}$. ∎

We now recall the Liouville field on the disk. We focus on the upper half
plane $\mathbb{H}$.

######  Definition 2.6.

Fix $\gamma\in(0,2)$ and $Q=\frac{\gamma}{2}+\frac{2}{\gamma}$. Let
$\mathbb{P}_{\mathbb{H}}$ be the law of the free boundary Gaussian free field
on $\mathbb{H}$ with average zero over the uniform measure on the semi-circle
$\\{z\in\mathbb{H}:|z|=1\\}$. Sample $(h,\mathbf{c})$ from
$\mathbb{P}_{\mathbb{H}}\times e^{-Qc}dc$ and set
$\phi(z)=h(z)+\mathbf{c}-2Q\log|z|_{+}$, where $|z|_{+}=\max\\{|z|,1\\}$. We
write $\mathrm{LF}_{\mathbb{H}}$ as the law of $\phi$, and call $\phi$ a
_Liouville field on $\mathbb{H}$_.

Here we omit the dependence of $\mathrm{LF}_{\mathbb{H}}$ in $\gamma$ because
it will be a globally fixed parameter whenever we consider
$\mathrm{LF}_{\mathbb{H}}$. We recall Liouville fields on $\mathbb{H}$ with
one bulk insertion, following [ARS21].

######  Definition 2.7.

Fix $\alpha\in\mathbb{R}$ and $z_{0}\in\mathbb{H}$. Sample $(h,\mathbf{c})$
from
$(2\mathrm{Im}z_{0})^{-\alpha^{2}/2}|z_{0}|_{+}^{2\alpha(Q-\alpha)}\mathbb{P}_{\mathbb{H}}\times
e^{(\alpha-Q)c}dc$. Set $\phi(z)=h(z)+\mathbf{c}-2Q\log|z|_{+}+\alpha
G_{\mathbb{H}}(z,z_{0})$ where
$G_{\mathbb{H}}(z,w)=-\log(|z-w||z-\bar{w}|)+2\log|z|_{+}+2\log|w|_{+}$. We
write $\mathrm{LF}^{(\alpha,z_{0})}_{\mathbb{H}}$ as the law of $\phi$, and
call $\phi$ a _Liouville field on $\mathbb{H}$ with an $\alpha$-insertion at
$z_{0}$_.

Unlike the case of the annulus, the choice of normalization of the GFF $h$
affects the multiplicative constant of the Liouville field on the disk, a
phenomenon called the _Weyl anomaly_ [DKRV16, HRV18]. The following
computation was carried out in [HRV18, Section 3.3]; we rephrase it here and
give a quick proof.

######  Proposition 2.8.

Let $\rho$ be a probability measure on $\mathbb{H}$ satisfying (2.1). Set

$Z_{\rho}:=\exp(\frac{Q^{2}}{2}\iint(-\log|z-w|-\log|z-\overline{w}|)\rho(\mathrm{d}z)\rho(\mathrm{d}w)).$

Let $P_{\rho}$ denote the law of the GFF $h$ on $\mathbb{H}$ normalized so
$(h,\rho)=0$. Sample $(h,\mathbf{c})$ from
$P_{\rho}\times[Z_{\rho}e^{-Qc}\,dc]$. Then the law of
$h+\mathbf{c}-QG_{\rho}(\cdot,\infty)$ is $\mathrm{LF}_{\mathbb{H}}$, where
$G_{\rho}(\cdot,\cdot)$ is the covariance kernel for $P_{\rho}$.

###### Proof.

In Definition 2.6, the covariance kernel for $P_{\mathbb{H}}$ is
$G_{\mathbb{H}}$ and $G_{\mathbb{H}}(\cdot,\infty)=2\log|z|_{+}$. Therefore

$\displaystyle G_{\rho}(z,w)$ $\displaystyle=G_{\mathbb{H}}(z,w)-\int
G_{\mathbb{H}}(z,x)\rho(dx)-\int G_{\mathbb{H}}(y,w)\rho(\mathrm{d}y)+\iint
G_{\mathbb{H}}(x,y)\rho(\mathrm{d}x)\rho(\mathrm{d}y);$ (2.3) $\displaystyle
G_{\rho}(z,\infty)$ $\displaystyle=G_{\mathbb{H}}(z,\infty)-\int
G_{\mathbb{H}}(z,x)\rho(dx)-\int
G_{\mathbb{H}}(y,\infty)\rho(\mathrm{d}y)+\iint
G_{\mathbb{H}}(x,y)\rho(\mathrm{d}x)\rho(\mathrm{d}y).$

Sample $(h,\mathbf{c})$ from $P_{\mathbb{H}}(dh)e^{-Qc}\,dc$. Let
$\phi=h+\mathbf{c}-QG_{\mathbb{H}}(\cdot,\infty)$ so that the law of $\phi$ is
$\mathrm{LF}_{\mathbb{H}}$. Let $h_{0}=h-Q\int
G_{\mathbb{H}}(\cdot,x)\rho(\mathrm{d}x)$ and
$\mathbf{c}^{\prime}=(h,\rho)-Q\int
G_{\mathbb{H}}(y,\infty)\rho(\mathrm{d}y)+\mathbf{c}$. Then
$\phi=h-(h,\rho)+\mathbf{c}^{\prime}-QG_{\mathbb{H}}(\cdot,\infty)+Q\int
G_{\mathbb{H}}(y,\infty)\rho(\mathrm{d}y)$ hence by (2.3) we have
$\phi=h_{0}-(h_{0},\rho)+\mathbf{c}^{\prime}-QG_{\rho}(\cdot,\infty)$. By
Girsanov’s theorem the law of $(h_{0},\mathbf{c}^{\prime})$ is
$Z_{\rho}P_{\mathbb{H}}\times e^{-Qc^{\prime}}dc^{\prime}$, where we used that
$Z_{\rho}$ defined above equals
$\mathbb{E}_{P_{\mathbb{H}}}[e^{Q(h,\rho)}]e^{-Q^{2}\int
G_{\mathbb{H}}(y,\infty)\rho(\mathrm{d}y)}$. Since the law of
$(h_{0}-(h_{0},\rho),\mathbf{c}^{\prime})$ is
$P_{\rho}\times[Z_{\rho}e^{-Qc^{\prime}}\,dc^{\prime}]$ , we are done. ∎

We have the following analog of Proposition 2.4.

######  Proposition 2.9.

Let $\eta\subset\mathbb{H}$ be a Jordan curve. Let $A_{\eta}$ be the component
of $\mathbb{H}\setminus\eta$ bounded by $\eta$ and
$\partial\mathbb{H}=\mathbb{R}$. Let $\phi$ be a Liouville field on
$\mathbb{H}$. Let $\phi^{\mathrm{har}}$ be the harmonic extension of
$\phi|_{\mathbb{H}\setminus A_{\eta}}$ onto $A_{\eta}$ with zero normal
derivative on $\partial\mathbb{H}$ and with
$\lim_{|z|\to\infty}\phi^{\mathrm{har}}(z)/\log|z|=-2Q$. Let
$\phi_{A_{\eta}}=\phi-\phi^{\mathrm{har}}$. Then conditioning on
$\phi|_{\mathbb{H}\setminus A_{\eta}}$, the conditional law of
$\phi_{A_{\eta}}$ is a GFF on $A_{\eta}$ with zero boundary condition on
$\eta$ and free boundary condition on $\partial\mathbb{H}$.

###### Proof.

The same argument for the proof of Proposition 2.4 works here. Let $\rho$ be a
probability measure supported in $\mathbb{H}\backslash A_{\eta}$ satisfying
(2.1). Write $\phi=h+\mathbf{c}-QG_{\rho}(\cdot,\infty)$ as in Proposition
2.8. Let $h^{\mathrm{har}}$ be the harmonic extension of
$h|_{\mathbb{H}\setminus A_{\eta}}$ onto $A_{\eta}$ with zero normal
derivative on $\partial_{0}\mathbb{H}$. Let $h_{A_{\eta}}=h-h^{\mathrm{har}}$.
By (2.3), $G_{\mathbb{H}}(z,\infty)-G_{\rho}(z,\infty)=o(\log|z|)$ as
$z\to\infty$. Since $G_{\mathbb{H}}(z,\infty)=2\log|z|_{+}$, we have
$\lim_{z\to\infty}\frac{G_{\rho}(z,\infty)}{\log|z|}=2$. Therefore
$\phi^{\mathrm{har}}=h^{\mathrm{har}}+\mathbf{c}-QG_{\rho}(\cdot,\infty)$ and
$\phi_{A_{\eta}}=h_{A_{\eta}}$. By Lemma 2.1, conditioning on
$(h|_{\mathbb{H}\setminus A_{\eta}},\mathbf{c})$, the conditional law of
$h_{A_{\eta}}$ is a GFF on $A_{\eta}$ with zero boundary condition on $\eta$
and free boundary condition on $\partial\mathbb{H}$. This gives the desired
conditional law of $\phi_{A_{\eta}}$ given $\phi|_{\mathbb{H}\setminus
A_{\eta}}$. ∎

######  Lemma 2.10.

Fix $\alpha\in\mathbb{R}$ and $z_{0}\in\mathbb{H}$. Let
$\eta\subset\mathbb{H}$ be a Jordan curve surrounding $z_{0}$. Then the
statement of Proposition 2.9 holds with
$\mathrm{LF}^{(\alpha,z_{0})}_{\mathbb{H}}$ in place of
$\mathrm{LF}_{\mathbb{H}}$.

###### Proof.

By [ARS21, Lemma 2.2], we have
$\mathrm{LF}^{(\alpha,z_{0})}_{\mathbb{H}}(\mathrm{d}\phi)=\lim_{\epsilon\to
0}\epsilon^{\alpha^{2}/2}e^{\alpha\phi_{\epsilon}(z)}\mathrm{LF}_{\mathbb{H}}(d\phi)$
where $\phi_{\epsilon}$ is the average of $\phi$ over
$\\{z\in\mathbb{H}:|z-z_{0}|=\epsilon\\}$. Since $\eta$ surrounds
$\\{z\in\mathbb{H}:|z-z_{0}|=\epsilon\\}$ for small enough $\epsilon$, Lemma
2.10 follows from Proposition 2.9. ∎

We need the domain Markov property for
$\mathrm{LF}^{(\alpha,z_{0})}_{\mathbb{H}}$ to some local sets in the sense of
[SS13].We restrict ourselves to two concrete cases needed later; see [SS13,
Section 3] for the general background.

######  Lemma 2.11.

Fix $\alpha\in\mathbb{R}$ and $z_{0}\in\mathbb{H}$. Suppose $(\phi,\eta)$
satisfies one of the following conditions:

  * •

The law of $(\phi,\eta)$ is a product measure
$\mathrm{LF}^{(\alpha,z_{0})}_{\mathbb{H}}\times\mathbb{P}$ where $\mathbb{P}$
is a probability measure on Jordan curves in $\mathbb{H}$ surrounding $z_{0}$.

  * •

The law of $\phi$ is $\mathrm{LF}^{(\alpha,z_{0})}_{\mathbb{H}}$, and
$\eta\subset\mathbb{H}$ is a Jordan curve surrounding $z_{0}$ such that for
each deterministic open set $U\subset\mathbb{H}$, the event $\\{D\subset U\\}$
is measurable w.r.t. $\phi|_{U}$, where $D$ is the bounded simply-connected
component of $\mathbb{H}\backslash\eta$.

Then with $\phi_{A_{\eta}}$ defined as in Proposition 2.9, conditioning on
$(\phi|_{\mathbb{H}\setminus A_{\eta}},\eta)$, the conditional law of
$\phi_{A_{\eta}}$ is the GFF on $A_{\eta}$ with zero boundary condition on
$\eta$ and free boundary condition on $\partial\mathbb{H}$.

###### Proof.

Both of the two cases are examples of local sets as defined in [SS13, Section
3]. Using the standard argument from [SS13, Lemma 3.9] that extends the Markov
property to the strong Markov property, we obtain Lemma 2.11 from Lemma 2.10.
∎

###  2.3. A resampling characterization of $\mathrm{LF}_{\tau}$

Let $H^{1}(\mathcal{C}_{\tau})$ be the Sobolev space on $\mathcal{C}_{\tau}$
with inner product $\int_{\mathcal{C}_{\tau}}(fg+\nabla f\cdot\nabla
g)\mathrm{d}x$. Let $H^{-1}(\mathcal{C}_{\tau})$ be the dual of
$H^{1}(\mathcal{C}_{\tau})$. Then both the GFF on $\mathcal{C}_{\tau}$ and
$\mathrm{LF}_{\tau}$ can be viewed as measures on $H^{-1}(\mathcal{C}_{\tau})$
endowed with the Borel $\sigma$-algebra. Let $\phi^{\mathrm{har}}$ be the
harmonic extension of $\phi|_{\mathcal{C}(\frac{2\tau}{3},\tau)}$ onto
$\mathcal{C}(0,\frac{2\tau}{3})$ with zero normal derivative on
$\partial_{0}\mathcal{C}_{\tau}$. Let $\Lambda(\phi,\cdot)$ be the law of
$h+\phi^{\mathrm{har}}$ where $h$ is a GFF on $\mathcal{C}(0,\frac{2\tau}{3})$
with zero boundary condition on
$\\{\frac{2\tau}{3}\\}\times[0,1]/{\sim_{\gamma}}$ and free boundary condition
on $\partial_{0}\mathcal{C}_{\tau}$. By Proposition 2.4 $\mathrm{LF}_{\tau}$
is an invariant measure of the Markov kernel $\Lambda$, namely
$\mathrm{LF}_{\tau}=\int\Lambda(\phi,\cdot)\mathrm{LF}_{\tau}(d\phi)$.

Let $\bar{\Lambda}$ be the Markov kernel defined in the same way as $\Lambda$
in the flipped direction. More precisely, let $\phi^{\mathrm{har}}$ be the
harmonic extension of $\phi|_{\mathcal{C}(0,\frac{\tau}{3})}$ onto
$\mathcal{C}(\frac{\tau}{3},\tau)$ with zero normal derivative on
$\partial_{1}\mathcal{C}_{\tau}$. Let $\bar{\Lambda}(\phi,\cdot)$ be the law
of $h+\phi^{\mathrm{har}}$ where $h$ is a GFF on
$\mathcal{C}(\frac{\tau}{3},\tau)$ with zero boundary condition on
$\\{\frac{\tau}{3}\\}\times[0,1]/{\sim_{\gamma}}$ and free boundary condition
on $\partial_{1}\mathcal{C}_{\tau}$. Then by symmetry $\mathrm{LF}_{\tau}$ is
an invariant measure of $\bar{\Lambda}$ as well. We now explain that modulo a
multiplicative constant $\mathrm{LF}_{\tau}$ is the unique measure that is
invariant under both $\Lambda$ and $\bar{\Lambda}$. For both its proof and
applications, it will be convenient for us to truncate $\mathrm{LF}_{\tau}$ by
the quantum length.

We first recall the notion of _quantum length_. Let $h$ be a free boundary GFF
on a domain $D$ which contains a straight line segment $L$. Fix
$\gamma\in(0,2)$. Then the $\gamma$-LQG length measure on $L$ is defined by
$\mathcal{L}^{\gamma}_{h}=\lim_{\epsilon\to
0}\epsilon^{\frac{\gamma^{2}}{4}}e^{\frac{\gamma}{2}h_{\epsilon}(z)}dz$, where
$dz$ is the Lebesgue measure on $L$, and $h_{\epsilon}(z)$ is the average of
$h$ over $\\{w\in D:|w-z|=\epsilon\\}$. This is an example of Gaussian
multiplicative chaos (GMC); see e.g. [RV14], the limit exists in probability
in the weak topology of measures on $L$. Suppose $\phi$ is a Liouville field
on $C_{\tau}$, we can define $\mathcal{L}^{\gamma}_{\phi}=\lim_{\epsilon\to
0}\epsilon^{\frac{\gamma^{2}}{4}}e^{\frac{\gamma}{2}\phi_{\epsilon}(z)}dz$ as
well. Let $I$ be a finite interval in $(0,\infty)$. Let $\Omega_{I}\subset
H^{-1}(\mathcal{C}_{\tau})$ be the event that
$\mathcal{L}^{\gamma}_{\phi}(\partial_{0}\mathcal{C}_{\tau})\subset I$ and
$\mathcal{L}^{\gamma}_{\phi}(\partial_{1}\mathcal{C}_{\tau})\subset I$. By the
following lemma, $\mathrm{LF}_{\tau}(\Omega_{I})<\infty$.

######  Lemma 2.12.

Fix $\gamma\in(0,2)$ and $\tau>0$. Suppose we are in the setting of Definition
2.2 where $h$ is a free boundary GFF on the horizontal cylinder
$\mathcal{C}_{\tau}$ with a certain normalization. Then for non-negative
measurable functions $f$ and $g$ on $(0,\infty)$, we have

(2.4)
$\mathrm{LF}_{\tau}[f(\mathcal{L}^{\gamma}_{\phi}(\partial_{0}\mathcal{C}_{\tau}))g(\mathcal{L}^{\gamma}_{\phi}(\partial_{1}\mathcal{C}_{\tau}))]=\frac{2}{\gamma}\int_{0}^{\infty}\ell^{-1}f(\ell)\mathbb{E}\left[g\left(\ell\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}\right)\right]\mathrm{d}\ell.$

Note that the ratio
$\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}$
does not depend on normalization of $h$.

###### Proof.

By definition
$\mathrm{LF}_{\tau}[f(\mathcal{L}^{\gamma}_{\phi}(\partial_{0}\mathcal{C}_{\tau}))g(\mathcal{L}^{\gamma}_{\phi}(\partial_{1}\mathcal{C}_{\tau}))]=\int_{\mathbb{R}}\mathbb{E}[f(e^{\frac{\gamma
c}{2}}\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau}))g(e^{\frac{\gamma
c}{2}}\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau}))]dc.$ Setting
$\ell=e^{\frac{\gamma
c}{2}}\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})$ and use
$dc=\frac{2}{\gamma}\ell^{-1}\mathrm{d}\ell$ we conclude the proof. ∎

######  Proposition 2.13.

Let $I\subset(0,\infty)$ be a compact interval. Let $\mathrm{LF}_{\tau,I}$ be
the probability measure proportional to the restriction of
$\mathrm{LF}_{\tau}$ to $\Omega_{I}$. For $\phi\in\Omega$ and
$A\subset\Omega$, let
$\Lambda_{I}(\phi,A)=\frac{\Lambda(\phi,A)}{\Lambda(\phi,\Omega_{I})}$ and
$\bar{\Lambda}_{I}(\phi,A)=\frac{\bar{\Lambda}(\phi,A)}{\bar{\Lambda}(\phi,\Omega_{I})}$.
Then $\mathrm{LF}_{\tau,I}$ is the unique probability measure on $\Omega_{I}$
that is invariant under both $\Lambda_{I}$ and $\overline{\Lambda}_{I}$.

###### Proof.

We say that a Markov kernel $K$ is irreducible if there exists a positive
measure $\rho$ on the state space such that the following holds. For every $x$
and $\rho(A)>0$ there exists a positive integer $n$ (which may depend on $x$
and $A$) such that $K^{n}(x,A)>0$. It is known (from e.g. [MT09, Propositions
4.2.1 and 10.1.1, and Theorem 10.0.1]) that if an irreducible Markov kernel
has an invariant probability measure, then it has only one invariant
probability measure.

By Proposition 2.4, we see that $\mathrm{LF}_{I,\tau}$ is an invariant measure
of both $\Lambda_{I}$ and $\overline{\Lambda}_{I}$. Now we claim that
$K=\Lambda_{I}\cdot\overline{\Lambda}_{I}$ is irreducible, which will implies
the uniqueness of the invariant measure. It is well known that if $h$ is a GFF
in $\mathcal{C}(\frac{\tau}{3},\tau)$ with zero boundary conditions on
$\\{\frac{\tau}{3}\\}\times[0,1]/{\sim}$ and free boundary conditions on
$\\{\tau\\}\times[0,1]/{\sim_{\gamma}}$ and $g$ is a smooth function in
$\mathcal{C}(\frac{\tau}{3},\tau)$ with support bounded away from
$\\{\frac{\tau}{3}\\}\times[0,1]/{\sim}$ and with normal derivative zero on
$\\{\tau\\}\times[0,1]/{\sim}$, then the laws of
$h|_{\mathcal{C}(\frac{2\tau}{3},\tau)}$ and
$(h+g)|_{\mathcal{C}(\frac{2\tau}{3},\tau)}$ are mutually absolutely
continuous, see e.g. the proof of [MS17, Proposition 2.9]. Therefore
$K(x,d\phi)$ and $\mathrm{LF}_{\tau,I}(d\phi)$ are mutually absolutely
continuous. We can take $\rho=\mathrm{LF}_{\tau,I}$ to conclude that $K$ is
irreducible as desired. ∎

##  3\. The annulus boundary partition function for Liouville CFT

In this section we prove Theorem 1.5 for the free boundary GFF on
$\mathcal{C}_{\tau}$, and the KPZ relation in Theorem 1.6. We first recall the
result we need from [Wu22] for annulus partition function of Liouville CFT.

######  Theorem 3.1 ([Wu22]).

Fix $\tau>0$. For $\gamma\in(0,2)$, $\mu_{0}>0,\mu_{1}>0$ ,let
$L_{0}=\mathcal{L}^{\gamma}_{\phi}(\partial_{0}\mathcal{C}_{\tau})$ and
$L_{1}=\mathcal{L}^{\gamma}_{\phi}(\partial_{1}\mathcal{C}_{\tau})$. Then
there is a positive constant $C=C(\tau)$ such that

(3.1)
$\displaystyle\mathrm{LF}_{\tau}[L_{0}L_{1}e^{-\mu_{0}L_{0}-\mu_{1}L_{1}}]=\frac{C(\tau)}{2\pi}\int_{\mathbb{R}}\partial_{\mu_{0}}U(Q+iP,\mu_{0})\partial_{\mu_{1}}U(Q-iP,\mu_{1})e^{-\pi\tau
P^{2}}\,\mathrm{d}P,$

where
$U(\alpha,\mu_{i})=\frac{2}{\gamma}\Gamma(\frac{2(\alpha-Q)}{\gamma})\mu_{i}^{\frac{2(\alpha-Q)}{\gamma}}\Gamma(\frac{\gamma\alpha}{2}-\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma^{2}}{4})^{\frac{2(Q-\alpha)}{\gamma}}$.

Theorem 3.1 is a reformulation of [Wu22, Theorem 1.3] in our notation. Wu
[Wu22] also gave an annulus bootstrap formula for the joint law of quantum
length and area but we do not need it. The expression for $U(\alpha,\mu_{i})$
is for the one-point bulk structure constant for Liouville CFT on the disk
with zero bulk cosmological constant, which was derived in [Rem20]. The
unspecified constant $C(\tau)$ was due to some determinant of Laplacian
operator and some free field partition function that were not given explicitly
in [Wu22]. However, our proof of Theorem 1.5 below will in fact yield that
$C(\tau)=1$ for all $\tau>0$.

###### Proof of Theorem 1.5.

Starting from (3.1), by using the above formula for $U(\alpha,\mu_{i})$, the
identity $\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}$, and setting
$\mu_{1}=1$ we have:

$\displaystyle\mathrm{LF}_{\tau}[L_{1}L_{0}e^{-L_{1}-\mu_{0}L_{0}}]$
$\displaystyle=-\frac{2\pi
C(\tau)}{\mu_{0}\gamma^{2}}\int_{\mathbb{R}}\frac{\mu_{0}^{-\frac{2iP}{\gamma}}}{\sin(\frac{2\pi
iP}{\gamma})\sin(\frac{\gamma\pi iP}{2})}e^{-\pi\tau P^{2}}P^{2}\,\mathrm{d}P$
$\displaystyle=-\frac{\gamma\pi
C(\tau)}{4\mu_{0}}\int_{\mathbb{R}}\frac{\mu_{0}^{iu}}{\sin(u\pi
i)\sin(\frac{\gamma^{2}}{4}u\pi i)}e^{-\frac{\pi\gamma^{2}}{4}\tau
u^{2}}u^{2}\,\mathrm{d}u.$

with $C(\tau)$ as in Theorem 3.1. By Lemma 2.12:

(3.2)
$\displaystyle\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}L_{0}^{ix}]=\frac{2}{\gamma}\int_{0}^{\infty}\ell^{ix}e^{-\ell}\mathrm{d}\ell\mathbb{E}\left[\left(\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}\right)^{ix}\right]=\frac{2}{\gamma}\Gamma(1+ix)\mathbb{E}\left[\left(\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}\right)^{ix}\right].$

Using furthermore
$\Gamma(1-ix)L_{0}^{ix-1}=\int_{0}^{\infty}\lambda^{-ix}e^{-\lambda
L_{0}}\,\mathrm{d}\lambda=\int_{\mathbb{R}}e^{(1-ix)\omega}e^{-e^{\omega}L_{0}}\,\mathrm{d}\omega$,
we obtain:

$\displaystyle\mathbb{E}\left[\left(\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}\right)^{ix}\right]$
$\displaystyle=\frac{\gamma}{2\Gamma(1+ix)\Gamma(1-ix)}\int_{-\infty}^{\infty}e^{(1-ix)\omega}\mathrm{LF}_{\tau}[L_{0}L_{1}e^{-L_{1}}e^{-e^{\omega}L_{0}}]\,\mathrm{d}\omega$
$\displaystyle=-\frac{\gamma^{2}\sin(\pi
ix)C(\tau)}{8ix}\iint_{0}^{\infty}\frac{e^{i\omega u}}{\sin(u\pi
i)\sin(\frac{\gamma^{2}}{4}u\pi i)}e^{-\frac{\pi\gamma^{2}}{4}\tau
u^{2}}u^{2}\,\mathrm{d}u\,e^{-ix\omega}\,\mathrm{d}\omega.$

Using the Fourier inversion formula
$f(x)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-ix\omega}\int_{\mathbb{R}}e^{i\omega
u}f(u)\,\mathrm{d}u\,\mathrm{d}\omega$, we obtain:

(3.3)
$\displaystyle\mathbb{E}\left[\left(\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}\right)^{ix}\right]=-\frac{\gamma^{2}\sin(\pi
ix)C(\tau)}{8ix}\times\frac{2\pi e^{-\frac{\pi\gamma^{2}}{4}\tau
x^{2}}x^{2}}{\sin(x\pi i)\sin(\frac{\gamma^{2}}{4}x\pi
i)}=C(\tau)\frac{\pi\gamma^{2}xe^{-\frac{\pi\tau\gamma^{2}x^{2}}{4}}}{4\sinh(\frac{\gamma^{2}\pi
x}{4})}.$

Setting $x=0$ we get $C(\tau)=1$ for all $\tau>0$ as desired. ∎

######  Remark 3.2.

We can decompose $h$ into $h=R+Y$ where $R$ is the radial part of the field
given by $R(x)=\int_{0}^{1}h(x+iy)dy$, and $Y=h-R$ has zero average along the
$y$-axis for every $x$. It is well known that $Y$ and $R$ are independent.
From here it is straightforward to decompose our functional of interest as

$\displaystyle\mathbb{E}[\left(\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})\right)^{ix}(\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau}))^{-ix}]$
$\displaystyle=\mathbb{E}[e^{\frac{i\gamma
t}{2}(R(\tau)-R(0))}]\mathbb{E}[\left(\mathcal{L}^{\gamma}_{Y}(\partial_{1}\mathcal{C}_{\tau})\right)^{ix}(\mathcal{L}^{\gamma}_{Y}(\partial_{0}\mathcal{C}_{\tau}))^{-ix}],$

where here $\mathcal{L}^{\gamma}_{Y}(\partial_{0}\mathcal{C}_{\tau})$ has the
same definition as $\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})$
with $h$ replaced by $Y$, and similarly for
$\mathcal{L}^{\gamma}_{Y}(\partial_{1}\mathcal{C}_{\tau})$. By direct
computation on the covariance of $R$, one can obtain that
$\mathbb{E}[e^{\frac{i\gamma
x}{2}(R(\tau)-R(0))}]=e^{-\frac{\gamma^{2}x^{2}\pi\tau}{4}}$. By Theorem 1.5,
we have
$\mathbb{E}[\left(\mathcal{L}^{\gamma}_{Y}(\partial_{1}\mathcal{C}_{\tau})\right)^{ix}(\mathcal{L}^{\gamma}_{Y}(\partial_{0}\mathcal{C}_{\tau}))^{-ix}]=\frac{\pi\gamma^{2}x}{4\sinh(\frac{\gamma^{2}}{4}\pi
x)}$. Therefore, the law of
$\mathcal{L}^{\gamma}_{Y}(\partial_{1}\mathcal{C}_{\tau})/\mathcal{L}^{\gamma}_{Y}(\partial_{0}\mathcal{C}_{\tau})$
is a log-logistic distribution. We observe that interestingly the law of this
ratio does not depend on $\tau$. Moreover, as $\tau\to\infty$,
$\mathcal{L}^{\gamma}_{Y}(\partial_{1}\mathcal{C}_{\tau})$ and
$\mathcal{L}^{\gamma}_{Y}(\partial_{0}\mathcal{C}_{\tau})$ become independent
and both of them converge in law to the total mass of the Gaussian
multiplicative chaos on the circle explicitly solved in [Rem20]. This provides
an alternative route to proving Theorem 1.5 without relying on Theorem 3.1 but
we do not pursue it in this paper.

###### Proof of Theorem 1.6.

By (3.2) and Theorem 1.5, we have

(3.4)
$\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}L_{0}^{ix}]=\frac{2}{\gamma}\int_{0}^{\infty}\ell^{ix}e^{-\ell}\mathrm{d}\ell\mathbb{E}\left[\left(\frac{\mathcal{L}^{\gamma}_{h}(\partial_{1}\mathcal{C}_{\tau})}{\mathcal{L}^{\gamma}_{h}(\partial_{0}\mathcal{C}_{\tau})}\right)^{ix}\right]=\frac{\pi\gamma
xe^{-\frac{\pi\gamma^{2}\tau x^{2}}{4}}}{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}\Gamma(1+ix).$

Since $\langle
L_{1}e^{-L_{1}}L_{0}^{ix}\rangle_{\gamma}=\int_{0}^{\infty}\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}L_{0}^{ix}]m(\mathrm{d}\tau)$
by definition, we get (1.6). ∎

##  4\. Quantum annulus: Liouville field description and the random modulus

The quantum annulus is a quantum surface introduced in [AS21] in the context
of the quantum disk coupled with CLE. In this section we clarify its relation
with Liouiville CFT and determine the law of its modulus.

We first recall the definition of quantum surfaces. Fix $\gamma\in(0,2)$ and
$Q=\frac{\gamma}{2}+\frac{2}{\gamma}$. Consider pairs of the form $(D,h)$
where $D$ is a planar domain and $h$ is a generalized function on $D$. We say
that $(D,h)\sim_{\gamma}(\tilde{D},\tilde{h})$ if there exists a conformal map
$\psi:\tilde{D}\to D$ such that $\tilde{h}=h\circ\psi+Q\log|\psi^{\prime}|$. A
quantum surface in $\gamma$-LQG is an equivalence class under $\sim_{\gamma}$.
We write $(D,h)/{\sim_{\gamma}}$ as the quantum surface corresponding to
$(D,h)$. An embedding of a quantum surface is a choice of its representative.
We extend $\sim_{\gamma}$ to quantum surfaces with decorations/markings. For
example, for triples $(D,h,z)$ where $z\in D$, we further require
$z=\psi(\tilde{z})$ when defining
$(D,h,z)\sim_{\gamma}(\tilde{D},\tilde{h},\tilde{z})$ and call
$(D,h,z)/{\sim_{\gamma}}$ a quantum surface with an interior marked point. We
can similarly define quantum surfaces decorated by curves.

The notion of quantum length is intrinsic to quantum surfaces. Suppose
$(D,h)\sim_{\gamma}(\tilde{D},\tilde{h})$ through the conformal map
$\psi:\tilde{D}\to D$. Recall the GMC definition of quantum length above Lemma
2.12. If both $D$ and $\tilde{D}$ contain a line segments which are related by
$\psi$, then $\mathcal{L}^{\gamma}_{\tilde{h}}$ is the pushforward of
$\mathcal{L}^{\gamma}_{h}$ under $\psi$. Therefore, we can use this coordinate
change to consistently define $\mathcal{L}^{\gamma}_{h}$ on $\partial D$ even
if it does not contain a line segment. We now recall the Liouville field
description of the quantum disk with an $\alpha$ bulk insertion defined in
[ARS21].

######  Definition 4.1.

For $\gamma\in(0,2)$ and $\alpha>\frac{\gamma}{2}$, let
$\mathcal{M}^{\mathrm{disk}}_{1}(\alpha)$ be the law of the quantum surface
$(\mathbb{H},\phi,i)/{\sim_{\gamma}}$ where $\phi$ is sampled from
$\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}$. Let
$\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)$ be the disintegration
$\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}=\int_{0}^{\infty}\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)\mathrm{d}\ell$
such that samples from $\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)$ have
boundary length $\ell$. Let $\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)$ be
the law of $(\mathbb{H},\phi,i)/{\sim_{\gamma}}$ where $\phi$ is sampled from
$\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)$.

In this section we only use $\mathcal{M}^{\mathrm{disk}}_{1}(\gamma;\ell)$ but
we will use $\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)$ in the next one. By
[ARS21, Theorem 3.4],
$\frac{\gamma}{2\pi(Q-\gamma)^{2}}\mathcal{M}^{\mathrm{disk}}_{1}(\gamma;\ell)$
is the law of a quantum disk with boundary length $\ell$ and one interior
marked point as defined in [DMS21]. The reason we require
$\alpha>\frac{\gamma}{2}$ is that the total mass
$|\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)|$ of
$\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)$ (hence also
$\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)$) is finite in this case. The
precise value of $|\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)|$ was computed
in [Rem20] as explained in [ARS21, Lemma 2.7 and Proposition 2.8]:

(4.1)
$|\mathrm{LF}^{(\alpha,i)}_{\mathbb{H}}(\ell)|=\frac{2}{\gamma}2^{\frac{\alpha^{2}}{2}-\alpha
Q}\cdot\left(\frac{2\pi}{\Gamma(1-\frac{\gamma^{2}}{4})}\right)^{\frac{2}{\gamma}(Q-\alpha)}\Gamma(\frac{\gamma\alpha}{2}-\frac{\gamma^{2}}{4})\times\ell^{\frac{2}{\gamma}(\alpha-Q)-1}\quad\textrm{for
}\alpha>\frac{\gamma}{2}.$

Let $\kappa\in(\frac{8}{3},4)$ and $\gamma=\sqrt{\kappa}$. Let $\Gamma$ be a
$\mathrm{CLE}_{\kappa}$ on $\mathbb{H}$ and $h$ be a free boundary GFF on
$\mathbb{H}$ independent of $\Gamma$. Let $\eta$ be a loop in $\Gamma$. Let
$A_{\eta}$ and $D_{\eta}$ be the two connected components
$\mathbb{H}\setminus\eta$ where $D_{\eta}$ is simply connected and $A_{\eta}$
is the annulus. By the theory of quantum zipper [She16], the quantum length
measures of the quantum surfaces $(A_{\eta},h)/{\sim_{\gamma}}$ and
$(D_{\eta},h)/{\sim_{\gamma}}$ agree on $\eta$. This defines a quantum length
measure on $\eta$. If $h$ is replaced by a Liouville field, the quantum length
measure on $\eta$ can also be defined. The following theorem is proved in
Sections 2 and 3 of [AS21].

######  Theorem 4.2.

For $a>0$, let $(\phi,\Gamma)$ be a sample from
$\mathrm{LF}_{\mathbb{H}}^{\gamma,i}(a)\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}$
where $\mathrm{CLE}^{\mathbb{H}}_{\kappa}$ is the law of a
$\mathrm{CLE}_{\kappa}$ on $\mathbb{H}$. Let $\eta$ be the outermost loop of
$\Gamma$ surrounding $i$. Then the law of the quantum boundary length
$\mathcal{L}^{\gamma}_{\phi}(\eta)$ of $\eta$ is given by

(4.2)
$\mathrm{LF}_{\mathbb{H}}^{\gamma,i}(a)\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}[\mathcal{L}^{\gamma}_{\phi}(\eta)\in
I]=\int_{I}\frac{\cos(\pi(\frac{4}{\gamma^{2}}-1))}{\pi}\cdot\frac{b|\mathrm{LF}_{\mathbb{H}}^{\gamma,i}(b)|}{\sqrt{ab}(a+b)}\,\mathrm{d}b\quad\textrm{for
}I\subset(0,\infty).$

Moreover, conditioning on $(A_{\eta},h)/{\sim_{\gamma}}$, the condition law of
$(D_{\eta},\phi)/{\sim_{\gamma}}$ is
$\mathcal{M}^{\mathrm{disk}}_{1}(\gamma;\mathcal{L}^{\gamma}_{\phi}(\eta))$.

We now recall the definition of the quantum annulus.

######  Definition 4.3.

Consider the setting of Theorem 4.2. For $b>0$, let
$\mathrm{QA}^{\gamma}(a,b)^{\\#}$ be the conditional law of the quantum
surface $(A_{\eta},\phi)/{\sim_{\gamma}}$ conditioning on
$\mathcal{L}^{\gamma}_{\phi}(\eta)=b$, which is a probability measure. Let

(4.3)
$\mathrm{QA}^{\gamma}(a,b)=\frac{\cos(\pi(\frac{4}{\gamma^{2}}-1))}{\pi}\cdot\frac{\mathrm{QA}^{\gamma}(a,b)^{\\#}}{\sqrt{ab}(a+b)}.$

Let
$\mathrm{QA}^{\gamma}=\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{QA}^{\gamma}(a,b)\,\mathrm{d}a\mathrm{d}b$.
We call a sample of $\mathrm{QA}^{\gamma}$ a _quantum annulus_.

Here the conditioning $\\{\mathcal{L}^{\gamma}_{\phi}(\eta)=b\\}$ is
understood in the sense of disintegration, which is uniquely defined for
almost every $b\in(0,\infty)$. This ambiguity does not affect the definition
of $\mathrm{QA}^{\gamma}$ and the proof of Theorem 1.4 below. After we prove
Theorem 1.4 $\mathrm{QA}^{\gamma}(a,b)$ can be canonically defined for every
$(a,b)$. Therefore we omit this temporary ambiguity. The definition of
$\mathrm{QA}^{\gamma}$ is such that by Theorem 4.2, the joint law of
$(A_{\eta},\phi)/{\sim_{\gamma}}$ and $(D_{\eta},\phi)/{\sim_{\gamma}}$ is

(4.4)
$\int_{0}^{\infty}b\mathrm{QA}^{\gamma}(a,b)\times\mathcal{M}^{\mathrm{disk}}_{1}(\gamma;b)\mathrm{d}b.$

The intuitive meaning of the factor $b$ in (4.4) is that the number of ways of
gluing together $\mathrm{QA}^{\gamma}(a,b)$ and
$\mathcal{M}^{\mathrm{disk}}_{1}(\gamma;b)$ is proportional to the boundary
length $b$ of their interface; see [AS21, Sections 2.4].

Our main goal of this section is to prove Theorem 1.4 which describes the
matter-Liouville-ghost decomposition of $\mathrm{QA}^{\gamma}$. Namely, the
law of the quantum surface $(\mathcal{C}_{\tau},\phi)/{\sim_{\gamma}}$ is
$\mathrm{QA}^{\gamma}$ if we sample $(\tau,\phi)$ from

(4.5)
$1_{\tau>0}\cdot{\cos(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{\gamma}{2\pi}\theta_{1}(\frac{\gamma^{2}}{8},\frac{\gamma^{2}}{4}i\tau)\mathrm{LF}_{\tau}(\mathrm{d}\phi)\mathrm{d}\tau.$

In Section 4.1 we prove that $\mathrm{QA}^{\gamma}$ can be written in the form
of $\mathrm{LF}_{\tau}(d\phi)m(\mathrm{d}\tau)$ for some $m(\mathrm{d}\tau)$ .
This is based on the domain Markov property of Liouville fields from Section
2. Then in Section 4.2 we identify $m(\mathrm{d}\tau)$ as given in Theorem 1.4
using the KPZ relation Theorem 1.6. In Section 5, we will prove the CLE
results stated in Section 1.4 using Theorems 1.4 and 4.2.

###  4.1. Identification of the Liouville field via the domain Markov property

######  Proposition 4.4.

There exists a measure $m(\mathrm{d}\tau)$ on $(0,\infty)$ such that if we
sample $(\tau,\phi)$ from $\mathrm{LF}_{\tau}(d\phi)m(\mathrm{d}\tau)$, then
the law of the quantum surface $(\mathcal{C}_{\tau},\phi)/{\sim_{\gamma}}$ is
$\mathrm{QA}^{\gamma}$.

Our proof of Proposition 4.4 based on the domain Markov property will be used
multiple times to prove similar statements in later sections. We recall the
notions in Section 2. Let $\mathcal{C}(x,y)$ be the cylinder
$(x,y)\times[0,1]/{\sim}$ where $\sim$ is the identification of
$[x,y]\times\\{0\\}$ and $[x,y]\times\\{1\\}$. We will write the circle
$\\{z\\}\times[0,1]/{\sim}$ as $\\{z\\}\times[0,1]$ for simplicity. Let
$\mathcal{C}_{\tau}=\mathcal{C}(0,\tau)$ for each $\tau>0$. We prove the
needed domain Markov property through the following two lemmas.

######  Lemma 4.5.

Let $(\phi,\Gamma)$ be a sample from
$\mathrm{LF}^{\gamma,i}_{\mathbb{H}}\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}$.
Let $\eta$ be the outermost loop of $\Gamma$ surrounding $i$ and $A_{\eta}$ be
the component of $\mathbb{H}\setminus\eta$ bounded by $\eta$ and
$\partial\mathbb{H}$. Let $\hat{\tau}$ be such that $A_{\eta}$ is conformally
equivalent to $\mathcal{C}_{\hat{\tau}}$. Fix $x\in\\{0\\}\times[0,1]$ and let
$\psi:\mathcal{C}_{\hat{\tau}}\to A_{\eta}$ be the conformal map such that
$\psi(x)=0$. Let $\hat{\phi}(z)=\phi\circ\psi(z)+Q\log|\psi(z)^{\prime}|$ for
$z\in\mathcal{C}_{\hat{\tau}}$. Let $\hat{\phi}^{\mathrm{har}}$ be the
harmonic extension of
$\hat{\phi}|_{\mathcal{C}(\frac{2\hat{\tau}}{3},\hat{\tau})}$ onto
$\mathcal{C}(0,\frac{2\hat{\tau}}{3})$ with zero normal derivative on
$\\{0\\}\times[0,1]$. Then conditioning on $\hat{\tau}$ and
$\hat{\phi}|_{\mathcal{C}(\frac{2\hat{\tau}}{3},\hat{\tau})}$, the conditional
law of $\hat{\phi}-\hat{\phi}^{\mathrm{har}}$ is a GFF on
$\mathcal{C}(0,\frac{2\hat{\tau}}{3})$ with zero boundary condition on
$\\{\frac{2\hat{\tau}}{3}\\}\times[0,1]$ and free boundary condition on
$\\{0\\}\times[0,1]$.

###### Proof.

Let $\hat{\eta}\subset\mathbb{H}$ be the image of
$\\{\frac{2\hat{\tau}}{3}\\}\times[0,1]$ under $\psi$. Then the law of
$(\phi,\hat{\eta})$ is a product measure hence satisfies the first condition
in Lemma 2.11. Let $D_{\hat{\eta}}$ and $A_{\hat{\eta}}$ be the disk and
annulus component of $\mathbb{H}\setminus\hat{\eta}$, respectively. Let
$\phi^{\mathrm{har}}$ be the harmonic extension of $\phi|_{D_{\hat{\eta}}}$
onto $A_{\hat{\eta}}$ with zero normal derivative on $\partial\mathbb{H}$ and
satisfying $\lim_{|z|\to\infty}\phi^{\mathrm{har}}(z)/\log|z|=-2Q$. Let
$\phi_{A_{\hat{\eta}}}=\phi-\phi^{\mathrm{har}}$. Then by Lemma 2.11,
conditioning on $\hat{\eta}$ and $\phi|_{D_{\hat{\eta}}}$, the conditional law
of $\phi_{A_{\hat{\eta}}}$ is a GFF on $A_{\hat{\eta}}$ with zero boundary
condition on ${\hat{\eta}}$ and free boundary condition on
$\partial\mathbb{H}$. Since
$(\hat{\tau},\hat{\phi}|_{\mathcal{C}(\frac{2\hat{\tau}}{3},\hat{\tau})})$ is
determined by $(\hat{\eta},\phi|_{D_{\hat{\eta}}})$, by the conformal
invariance of GFF, conditioning on
$(\hat{\tau},\phi|_{\mathcal{C}(\frac{2\hat{\tau}}{3},\hat{\tau})})$ the
conditional law of $\phi_{A_{\hat{\eta}}}\circ\psi$ is a GFF on
$\mathcal{C}(0,\frac{2\hat{\tau}}{3})$ with zero boundary condition on
$\\{\frac{2\hat{\tau}}{3}\\}\times[0,1]$ and free boundary condition on
$\\{0\\}\times[0,1]$.

Note that
$\hat{\phi}-(\phi^{\mathrm{har}}\circ\psi+Q\log|\psi^{\prime}|)=\phi\circ\psi-\phi^{\mathrm{har}}\circ\psi=\phi_{A_{\hat{\eta}}}\circ\psi$,
which is zero on $\mathcal{C}(\frac{2\hat{\tau}}{3},\hat{\tau})$. We just need
to show that
$\hat{\phi}^{\mathrm{har}}=\phi^{\mathrm{har}}\circ\psi+Q\log|\psi^{\prime}|$.
To see this, note that $\phi^{\mathrm{har}}\circ\psi+Q\log|\psi^{\prime}|$ is
harmonic inside $\mathcal{C}(0,\frac{2\hat{\tau}}{3})$ since $\psi$ is
conformal. It remains to show that
$\phi^{\mathrm{har}}\circ\psi+Q\log|\psi^{\prime}|$ has the desired boundary
condition on $\\{0\\}\times[0,1]$. Let $\psi_{1}(z)=\frac{z-1}{i(z+1)}$ which
maps the unit disk $\mathbb{D}$ to $\mathbb{H}$. Then the normal derivative of
$\phi^{\mathrm{har}}_{1}:=\phi^{\mathrm{har}}\circ\psi_{1}+Q\log|\psi^{\prime}_{1}|$
on $\partial\mathbb{D}$ is zero everywhere. For
$w\in\mathcal{C}_{\hat{\tau}}$, let $\psi_{2}(w)=\psi_{1}^{-1}(\psi(w))$ so
that $\psi=\psi_{1}\circ\psi_{2}$. Then
$\phi^{\mathrm{har}}\circ\psi+Q\log|\psi^{\prime}|=\phi_{1}^{\mathrm{har}}\circ\psi_{2}+Q\log|\psi^{\prime}_{2}|$.
Since $\psi_{2}$ maps $\partial\mathbb{D}$ to $\\{0\\}\times[0,1]$, the normal
derivative of $\phi^{\mathrm{har}}\circ\psi+Q\log|\psi^{\prime}|$ is zero
everywhere on $\\{0\\}\times[0,1]$ as desired. ∎

Figure 3.  Illustration of objects in the proof of Lemma 4.5.

######  Lemma 4.6.

For a sample $(\phi,\Gamma)$ from
$\mathrm{LF}^{\gamma,i}_{\mathbb{H}}\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}$,
let $\mathcal{L}^{\gamma}_{\phi}(\eta)$ be the quantum length of the outermost
loop $\eta$ surrounding $i$. Let
$f(b)=b|\mathrm{LF}^{\gamma,i}_{\mathbb{H}}(b)|$ and consider the reweighted
measure
$M=\frac{1}{f(\mathcal{L}^{\gamma}_{\phi}(\eta))}\cdot\mathrm{LF}^{\gamma,i}_{\mathbb{H}}\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}$.
Let $A_{\eta}$ be the component of $\mathbb{H}\setminus\eta$ bounded by $\eta$
and $\partial\mathbb{H}$. Then the $M$-law of
$(A_{\eta},\phi)/{\sim_{\gamma}}$ is $\mathrm{QA}^{\gamma}$. Moreover, the
statement in Lemma 4.5 on the conditional law of
$\hat{\phi}-\hat{\phi}^{\mathrm{har}}$ given
$(\hat{\tau},\hat{\phi}|_{\mathcal{C}(\frac{2\hat{\tau}}{3},\hat{\tau})})$
still holds under $M$.

###### Proof.

By (4.4), the $M$-law of $(A_{\eta},\phi)/{\sim_{\gamma}}$ is
$\mathrm{QA}^{\gamma}$. For the assertion on the conditional law of
$\hat{\phi}-\hat{\phi}^{\mathrm{har}}$, we note that the most crucial point in
the proof of Lemma 4.5 is the following. Conditioning on $\hat{\eta}$ and
$\phi|_{D_{\hat{\eta}}}$, the conditional law of $\phi_{A_{\hat{\eta}}}$ is a
mixed boundary GFF on $A_{\hat{\eta}}$. Since $\eta$ and $\hat{\eta}$
determine each other, the quantum length $\mathcal{L}^{\gamma}_{\phi}(\eta)$
of $\eta$ is determined by $(\hat{\eta},\phi|_{D_{\hat{\eta}}})$. Therefore
under the reweighted measure $M$, the conditional law of
$\phi_{A_{\hat{\eta}}}$ given $(\hat{\eta},\phi|_{D_{\hat{\eta}}})$ does not
change. The rest of the proof of Lemma 4.5 now gives the last assertion in
Lemma 4.6. ∎

We recall the symmetry the quantum annulus proved as [AS21, Remark 3.12]. This
will allow us to flip the sides when applying the domain Markov property to
prove Proposition 4.4.

######  Lemma 4.7 ([AS21]).

For each $a,b>0$, we have
$\mathrm{QA}^{\gamma}(a,b)=\mathrm{QA}^{\gamma}(b,a)$.

###### Proof of Proposition 4.4.

Throughout this proof, we let $(\mathring{\phi},\mathring{\tau})$ be an
embedding of $\mathrm{QA}^{\gamma}$ such that the law of $\mathring{\phi}$ is
invariant under rotations against the axis of the cylinder
$\mathcal{C}(0,\mathring{\tau})$. Since rotations are the only conformal
automorphisms of a finite cylinder, the law of
$(\mathring{\phi},\mathring{\tau})$ is unique. We call
$(\mathring{\phi},\mathring{\tau})$ the rotational invariant embedding of
$\mathrm{QA}^{\gamma}$. We will show that the law of
$(\mathring{\phi},\mathring{\tau})$ is
$\mathrm{LF}(\mathrm{d}\phi)m(\mathrm{d}\tau)$ for some $m(\mathrm{d}\tau)$,
which will prove Proposition 4.4.

Let $I$ be a finite interval on $(0,\infty)$. Given a quantum surface of
annular topology, let $E_{I}$ be the event that both boundary lengths are in
$I$. Since $|\mathrm{QA}^{\gamma}(a,b)|$ is proportional to
$\frac{1}{\sqrt{ab}(a+b)}$, we have $\mathrm{QA}^{\gamma}(E_{I})=\int_{I\times
I}|\mathrm{QA}^{\gamma}(a,b)|\mathrm{d}a\mathrm{d}b<\infty$. Let
$\mathbb{P}_{I}$ be the probability measure describing the law of
$(\mathring{\phi},\mathring{\tau})$ conditioning on $E_{I}$. For $\tau>0$, let
$P_{I,\tau}$ be the regular conditional probability of $\mathring{\phi}$ under
$\mathbb{P}_{I}$ given $\\{\mathring{\tau}=\tau\\}$.

Note that the domain Markov property in Lemmas 4.5 and 4.6 hold for each fixed
$x\in\\{0\\}\times[0,1]$ in the definition of the conformal map $\psi$ from
$\mathcal{C}_{\hat{\tau}}$ to $A_{\eta}$. Therefore Lemmas 4.5 and 4.6 still
hold if we choose $x$ uniformly at random on $\\{0\\}\times[0,1]$. This way,
under the measure $M$ in Lemma 4.6 $(\hat{\phi},\hat{\tau})$ defined there is
a rotational invariant embedding of $\mathrm{QA}^{\gamma}$, hence equal in law
to $(\mathring{\phi},\mathring{\tau})$. Therefore by the domain Markov
property in Lemma 4.6, $P_{I,\tau}$ is an invariant probability measure of the
kernel $\Lambda_{I}$ in Proposition 2.13.

By the symmetry in Lemma 4.7, $P_{I,\tau}$ is also an invariant probability
measure of the kernel $\bar{\Lambda}_{I}$. Hence the uniqueness of invariant
measure in Proposition 2.13 gives $P_{I,\tau}=\mathrm{LF}_{\tau,I}$. More
precisely, for each $I$, $\mathrm{LF}_{\tau,I}$ is a version of the regular
conditional probability of $\mathring{\phi}$ given
$\\{\mathring{\tau}=\tau\\}$. We now write the assertion of Proposition 4.4 as
$\mathrm{QA}^{\gamma}=\int\mathrm{LF}_{\tau}m(d\tau)$ for simplicity. Then
$P_{I,\tau}=\mathrm{LF}_{\tau,I}$ implies that restricted to $E_{I}$, we have
$\mathrm{QA}^{\gamma}=\int\mathrm{LF}_{\tau}m_{I}(d\tau)$ for some measure
$m_{I}(d\tau)$ on $(0,\infty)$. Setting $m(d\tau)=m_{I}(d\tau)$ on each
interval $I$ we get a well-defined measure $m(d\tau)$ on $(0,\infty)$. Then
$\mathrm{QA}^{\gamma}=\int\mathrm{LF}_{\tau}m(d\tau)$. ∎

###  4.2. Identification of the law of the modulus by comparing the boundary
lengths

By Theorem 1.6 and Proposition 4.4, the measure $m(\mathrm{d}\tau)$ satisfies
the following: for $x\in\mathbb{R}$

(4.6)
$\displaystyle\int_{0}^{\infty}e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}m(\mathrm{d}\tau)=\frac{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}{\pi\gamma x\Gamma(1+ix)}\mathrm{QA}^{\gamma}[L_{1}e^{-L_{1}}L_{0}^{ix}].$

Here we abuse notion and use $L_{0}$ and $L_{1}$ to represent the quantum
lengths of the two boundaries of a sample from $\mathrm{QA}^{\gamma}$. We now
identify the measure $m$ hence proving Theorem 1.4.

######  Lemma 4.8.

For $\gamma\in(\sqrt{\frac{8}{3}},2)$, we have

(4.7)
$\mathrm{QA}^{\gamma}[L_{1}e^{-L_{1}}L_{0}^{ix}]={\cos(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{\Gamma(1+ix)}{\cosh(\pi
x)}\quad\textrm{for }x\in\mathbb{R}.$

###### Proof.

By the definition of $\mathrm{QA}^{\gamma}$, the total mass
$|\mathrm{QA}^{\gamma}(a,b)|$ of $\mathrm{QA}^{\gamma}(a,b)$ is given by

(4.8)
$|\mathrm{QA}^{\gamma}(a,b)|=\frac{\cos(\pi(\frac{4}{\gamma^{2}}-1))}{\pi}\cdot\frac{1}{\sqrt{ab}(a+b)}\quad\textrm{for
}a>0\textrm{ and }b>0.$

Therefore

$\displaystyle\mathrm{QA}^{\gamma}[L_{i}e^{-L_{i}}L_{o}^{ix}]=\iint_{0}^{\infty}ae^{-a}b^{ix}|\mathrm{QA}^{\gamma}(a,b)|dadb=\frac{\cos(\pi(\frac{4}{\gamma^{2}}-1))}{\pi}\iint_{0}^{\infty}\frac{ae^{-a}b^{ix}}{\sqrt{ab}(a+b)}dadb.$

Setting $b=at$ and using $\int_{0}^{\infty}a^{ix}e^{-a}da=\Gamma(1+ix)$ and
$\int_{0}^{\infty}\frac{t^{ix-\frac{1}{2}}}{1+t}dt=\frac{\pi}{\cosh(\pi x)}$,
we get

(4.9)
$\displaystyle\iint_{0}^{\infty}\frac{ae^{-a}b^{ix}}{\sqrt{ab}(a+b)}dadb=\iint_{0}^{\infty}\frac{ae^{-a}(at)^{ix}}{a\sqrt{t}(1+t)}dadt=\int_{0}^{\infty}e^{-a}a^{ix}da\int_{0}^{\infty}\frac{t^{ix-\frac{1}{2}}}{1+t}dt=\frac{\pi\Gamma(1+ix)}{\cosh(\pi
x)}.$

This gives (4.7). ∎

###### Proof of Theorem 1.4.

By (4.6) and Lemma 4.8 we have

(4.10)
$\int_{0}^{\infty}e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}m(\mathrm{d}\tau)={\cos(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}{\pi\gamma x\cosh(\pi x)}\quad\textrm{for }x\in\mathbb{R}.$

By (A.3) in Appendix A, we have

(4.11)
$\int_{0}^{\infty}\theta_{1}(\frac{\gamma^{2}}{8},\frac{\gamma^{2}}{4}i\tau)e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}d\tau=\frac{4\sinh(\frac{\gamma^{2}}{4}\pi
x)}{\gamma^{2}x\cosh(\pi x)}\quad\textrm{for }x\in\mathbb{R}.$

Taking the inverse Laplace transform, we get
$m(\mathrm{d}\tau)={\cos(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{\gamma}{2\pi}\theta_{1}(\frac{\gamma^{2}}{8},\frac{\gamma^{2}}{4}i\tau)\,\mathrm{d}\tau$
on $(0,\infty)$. ∎

##  5\. The random moduli for CLE loops

In this section we prove the three results on $\mathrm{CLE}_{\kappa}$
presented in Section 1.4. We will work with $(\mathbb{H},i)$ instead of
$(\mathbb{D},0)$. Namely, let $\kappa\in(8/3,4]$. We assume that $\Gamma$ is a
$\mathrm{CLE}_{\kappa}$ on the upper half plane $\mathbb{H}$. Given $\Gamma$,
we let $\\{\eta_{j}\\}_{j\geq 1}$ be the sequence of loops surrounding the
point $i$ ordered such that $\eta_{j+1}$ is surrounded by $\eta_{j}$.
Moreover, let $D_{j}$ be the Jordan domain bounded by $\eta_{j}$, and $A_{j}$
be the annulus bounded by $\eta_{j}$ and $\partial\mathbb{H}$. Finally, let
$\mathrm{Mod}(\eta_{j})$ be the modulus of $A_{j}$.

###  5.1. Random modulus of a single loop: proof of Theorem 1.7

We first prove the $j=1$ case of Theorem 1.7 in the case
$\kappa\in(\frac{8}{3},4)$.

######  Proposition 5.1.

For $\kappa\in(\frac{8}{3},4)$ and $\gamma=\sqrt{\kappa}$, let
$m(\mathrm{d}\tau)=1_{\tau>0}\cdot{\cos(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{\gamma}{2\pi}\theta_{1}(\frac{\gamma^{2}}{8},\frac{\gamma^{2}}{4}i\tau)\mathrm{d}\tau$.
Then the law of $\mathrm{Mod}(\eta_{1})$ has a density proportional to
$e^{\frac{\pi\kappa}{4}(\frac{4}{\kappa}-1)^{2}\tau}m(\mathrm{d}\tau)$.
Moreover,

(5.1)
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{1})}]=\frac{(\frac{4}{\kappa}-1)\cos(\pi(\frac{4}{\kappa}-1))}{\sin(\pi(1-\frac{\kappa}{4}))}\times\frac{\sin(\frac{\kappa}{4}\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}{\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}}\cos(\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}$

for $\lambda>\frac{3\kappa}{32}+\frac{2}{\kappa}-1$.

###### Proof.

For $\tau>0$, define $\mathrm{LF}_{\tau}(a,b)$ by the disintegration

(5.2)
$\mathrm{LF}_{\tau}=\iint_{0}^{\infty}\mathrm{LF}_{\tau}(a,b)\mathrm{d}a\mathrm{d}b$

where $\mathrm{LF}_{\tau}(a,b)$ is supported on the event that the quantum
boundary lengths of $\\{0\\}\times[0,1]$ and $\\{\tau\\}\times[0,1]$ are $a$
and $b$, respectively. Let $C_{\gamma}\in(0,\infty)$ be a $\gamma$-dependent
constant that can vary from line to line. Let
$f(\ell)=\ell|\mathrm{LF}_{\mathbb{H}}^{(\gamma,i)}(\ell)|$, which equals
$C_{\gamma}\ell^{1-\frac{4}{\gamma^{2}}}$ for some $C_{\gamma}$. By (4.4) and
Theorem 1.4, for a non-negative measurable function $\rho$ on $(0,\infty)$ we
have

$\displaystyle\mathrm{LF}_{\mathbb{H}}^{(\gamma,i)}(a)\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}[\rho(\mathrm{Mod}(\eta_{1}))]=\int_{0}^{\infty}\int_{0}^{\infty}\rho(\tau)|\mathrm{LF}_{\tau}(a,b)|f(b)\,\mathrm{d}b\,m(\mathrm{d}\tau).$

For a sample from $\mathrm{LF}_{\mathbb{H}}^{(\gamma,i)}$ let $L$ be the
quantum boundary length of $\partial\mathbb{H}$. Integrating over
$ae^{-a}\mathrm{d}a$, we have

(5.3)
$\mathrm{LF}_{\mathbb{H}}^{(\gamma,i)}[Le^{-L}]\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}[\rho(\mathrm{Mod}(\eta_{1}))]=\int_{0}^{\infty}\rho(\tau)\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}f(L_{0})]\,m(\mathrm{d}\tau).$

Recall from (3.4) that
$\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}L_{0}^{ix}]=\frac{\pi\gamma
x\Gamma(1+ix)}{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}$ for $x\in\mathbb{R}$. This gives

$\frac{\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}f(L_{0})]}{\mathrm{LF}_{\mathbb{H}}^{(\gamma,i)}[Le^{-L}]}=C_{\gamma}e^{\frac{\pi\gamma^{2}\tau}{4}\cdot(1-\frac{4}{\gamma^{2}})^{2}}=C_{\gamma}e^{\frac{\pi\kappa}{4}\cdot(\frac{4}{\kappa}-1)^{2}\tau}.$

Therefore the law of $\mathrm{Mod}(\eta_{1})$ is
$C_{\gamma}e^{\frac{\pi\kappa}{4}\cdot(\frac{4}{\kappa}-1)^{2}\tau}m(\mathrm{d}\tau)$
as desired. Consequently,

(5.4)
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{1})}]=C_{\gamma}\int_{0}^{\infty}e^{-2\pi\lambda\tau}e^{\frac{\pi\kappa}{4}(\frac{4}{\kappa}-1)^{2}\tau}m(\mathrm{d}\tau).$

By (A.2) the function
$\tau\mapsto\theta_{1}(\frac{\gamma^{2}}{8},\frac{\gamma^{2}}{4}i\tau)$ decays
as $O(e^{-\frac{\pi\gamma^{2}}{16}\tau})$ as $\tau\to\infty$. Therefore
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{1})}]$ is finite and analytic in
$\lambda$ on a complex neighborhood of
$(\frac{3\kappa}{32}+\frac{2}{\kappa}-1,\infty)$. By (5.4) and (A.3), for
$\lambda$ sufficiently large, we can evaluate the integral on the right side
of (5.5) to get

(5.5)
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{1})}]=\frac{C_{\gamma}\sin(\frac{\kappa}{4}\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}{\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}}\cos(\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}.$

By the analyticity in $\lambda$ (5.5) holds for all
$\lambda\in(\frac{3\kappa}{32}+\frac{2}{\kappa}-1,\infty)$. Finally, setting
$\lambda=0$ we see that the constant $C_{\gamma}$ in (5.4) equals
$\frac{(\frac{4}{\kappa}-1)\cos(\pi(\frac{4}{\kappa}-1))}{\sin(\pi(1-\frac{\kappa}{4}))}$,
completing the proof. ∎

Our derivation for $\eta_{1}$ works exactly the same way for $\eta_{j}$ with
$j\geq 2$. We first introduce the analog of $\mathrm{QA}^{\gamma}$ following
Section 4.

######  Definition 5.2.

Fix $j\geq 2$. For $a>0$, let $(\phi,\Gamma$) be a sample from
$\mathrm{LF}_{\mathbb{H}}^{\gamma,i}(a)\times\mathrm{CLE}_{\kappa}^{\mathbb{H}}$
and let $\eta_{j}$ be the $j$-th outermost loop of $\Gamma$ surrounding $i$.
For $b>0$, let $\mathrm{QA}_{j}^{\gamma}(a,b)^{\\#}$ be the conditional law of
the quantum surface $(A_{j},\phi)/{\sim_{\gamma}}$ conditioning on
$\mathcal{L}^{\gamma}_{\phi}(\eta)=b$.

######  Proposition 5.3.

Let $\mathrm{QA}^{\gamma}_{1}=\mathrm{QA}^{\gamma}$. For $j\geq 2$, let
$\mathrm{QA}^{\gamma}_{j}(a,b)=|\mathrm{QA}^{\gamma}_{j}(a,b)|\mathrm{QA}^{\gamma}_{j}(a,b)^{\\#}$
where the total mass $|\mathrm{QA}_{j}^{\gamma}(a,b)|$ of
$\mathrm{QA}_{j}^{\gamma}(a,b)$ is given by

(5.6)
$|\mathrm{QA}_{j}^{\gamma}(a,b)|=\int_{0}^{\infty}\ell|\mathrm{QA}^{\gamma}(a,\ell)||\mathrm{QA}^{\gamma}_{j-1}(\ell,b)|\mathrm{d}\ell.$

Then the joint law of $(A_{j},h)/{\sim_{\gamma}}$ and
$(D_{j},\phi,0)/{\sim_{\gamma}}$ is

(5.7)
$\int_{0}^{\infty}b\mathrm{QA}^{\gamma}_{j}(a,b)\times\mathcal{M}^{\mathrm{disk}}_{1}(\gamma;b)\mathrm{d}b\quad\textrm{for
}j\geq 1.$

###### Proof.

By the Markov property of $\mathrm{CLE}_{\kappa}$, conditioning on $\eta_{j}$,
the law of the loops of inside $D_{j}$ is a $\mathrm{CLE}_{\kappa}$ on
$D_{j}$. Iteratively applying (4.4) we get (5.7). ∎

We now extend Theorem 1.4 to $\mathrm{QA}^{\gamma}_{j}$.

######  Proposition 5.4.

Fix $j\geq 2$. Let
$\mathrm{QA}^{\gamma}_{j}=\iint_{0}^{\infty}\mathrm{QA}^{\gamma}_{j}(a,b)\mathrm{d}a\mathrm{d}b$.
Let $m_{j}(\mathrm{d}\tau)$ be the unique measure on $(0,\infty)$ such that

(5.8)
$\int_{0}^{\infty}e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}m_{j}(\mathrm{d}\tau)={\cos^{j}(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}{\pi\gamma x\cosh^{j}(\pi x)}\quad\textrm{for }x\in\mathbb{R}.$

Sample $(\tau,\phi)$ from
$\mathrm{LF}_{\tau}(\mathrm{d}\phi)m_{j}(\mathrm{d}\tau)$. Then the law of the
quantum surface $(\mathcal{C}_{\tau},\phi)/{\sim_{\gamma}}$ is
$\mathrm{QA}^{\gamma}_{j}$.

To prove Proposition 5.4, we first prove the analog of Lemma 4.8.

######  Lemma 5.5.

For $j\geq 2$, we have

(5.9)
$\displaystyle\mathrm{QA}^{\gamma}_{j}[L_{1}e^{-L_{1}}L_{0}^{ix}]=\frac{\cos(\pi(\frac{4}{\gamma^{2}}-1))}{\cosh(\pi
x)}\mathrm{QA}^{\gamma}_{j-1}[L_{1}e^{-L_{1}}L_{0}^{ix}]\quad\textrm{for
}x\in\mathbb{R}.$

###### Proof.

By definition,
$\mathrm{QA}^{\gamma}_{j}[L_{1}e^{-L_{1}}L_{0}^{ix}]=\iiint_{0}^{\infty}ae^{-a}b^{ix}\ell|\mathrm{QA}^{\gamma}_{j-1}(a,\ell)||\mathrm{QA}(\ell,b)|\mathrm{d}\ell\mathrm{d}a\mathrm{d}b$,
which equals

$\displaystyle\frac{\cos(\pi(\frac{4}{\gamma^{2}}-1))}{\pi}\iiint_{0}^{\infty}\frac{ae^{-a}b^{ix}\ell|\mathrm{QA}^{\gamma}_{j-1}(a,\ell)|}{\sqrt{\ell
b}(\ell+b)}\mathrm{d}\ell\mathrm{d}a\mathrm{d}b.$

Setting $b=\ell t$ and using
$\int_{0}^{\infty}\frac{t^{ix}}{\sqrt{t}(1+t)}dt=\frac{\pi}{\cosh(\pi x)}$, we
get

$\displaystyle\iiint_{0}^{\infty}\frac{ae^{-a}b^{ix}\ell|\mathrm{QA}^{\gamma}_{j-1}(a,\ell)|}{\sqrt{\ell
b}(\ell+b)}\mathrm{d}\ell\mathrm{d}a\mathrm{d}b=\iiint_{0}^{\infty}\frac{ae^{-a}(\ell
t)^{ix}|\mathrm{QA}^{\gamma}_{j-1}(a,\ell)|}{\sqrt{t}(1+t)}\mathrm{d}\ell\mathrm{d}a\mathrm{d}t$
$\displaystyle=$ $\displaystyle\frac{\pi}{\cosh(\pi
x)}\iint_{0}^{\infty}ae^{-a}\ell^{ix}|\mathrm{QA}^{\gamma}_{j-1}(a,\ell)|\mathrm{d}\ell\mathrm{d}a=\frac{\pi}{\cosh(\pi
x)}\mathrm{QA}^{\gamma}_{j-1}[L_{1}e^{-L_{1}}L_{0}^{ix}].\qed$

###### Proof of Proposition 5.4.

By the identical argument as for Proposition 4.4, there exists a measure
$m_{j}(\mathrm{d}\tau)$ on $(0,\infty)$ such that if we sample $(\tau,\phi)$
from $\mathrm{LF}_{\tau}(d\phi)m_{j}(\mathrm{d}\tau)$, then the law of the
quantum surface $(\mathbb{A}_{\tau},\phi)/{\sim_{\gamma}}$ is
$\mathrm{QA}^{\gamma}_{j}$. Similar as for the $j=1$ case in (4.6), by Theorem
1.6 we have

(5.10)
$\displaystyle\int_{0}^{\infty}e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}m_{j}(\mathrm{d}\tau)=\frac{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}{\pi\gamma
x\Gamma(1+ix)}\mathrm{QA}^{\gamma}_{j}[L_{1}e^{-L_{1}}L_{0}^{ix}]\quad\textrm{for
}x\in\mathbb{R}.$

By Lemmas 4.8 and 5.5, we conclude the proof. ∎

###### Proof of Theorem 1.7.

First fix $\kappa=\gamma^{2}\in(\frac{8}{3},4)$. By the same calculation as
for (5.4) we have

(5.11)
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{j})}]=C_{\gamma}\int_{0}^{\infty}e^{-2\pi\lambda\tau}e^{\frac{\pi\kappa}{4}(\frac{4}{\kappa}-1)^{2}\tau}m_{j}(\mathrm{d}\tau).$

for some $\gamma$-dependent constant $C_{\gamma}$. By Proposition 5.4 and the
argument for (5.4), we have

(5.12)
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{j})}]=\frac{C_{\kappa}\sin(\frac{\kappa}{4}\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}{\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}}\cos^{j}(\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}\quad\text{
for }\lambda>\frac{3\kappa}{32}+\frac{2}{\kappa}-1$

for some constant $C_{\kappa}$. Setting $\lambda=0$ we get
$C_{\kappa}=\frac{(\frac{4}{\kappa}-1)\cos^{j}(\pi(\frac{4}{\kappa}-1))}{\sin(\pi(1-\frac{\kappa}{4}))}$
as desired. This proves Theorem 1.7 for $\kappa\in(\frac{8}{3},4)$. The
$\kappa=4$ case follows from sending $\kappa\to 4$. See [AS21, Appendix A] for
the needed continuity in $\kappa$. ∎

######  Remark 5.6 (Relation to Brownian motion).

For $a\geq 0$, let $B^{a}$ be a Brownian motion starting from $a$ and
$Y_{a}=\inf\\{t:|B^{a}_{t}|=1\\}$ and $T_{0}=\sup\\{t<Y_{0}:B^{0}_{t}=0\\}$.
Then as explained in e.g. [BPY01], we have

$\mathbb{E}[e^{-\frac{\theta^{2}}{2}Y_{a}}]=\frac{\cosh(a\theta)}{\cosh\theta}\quad\textrm{and}\quad\mathbb{E}[e^{-\frac{\theta^{2}}{2}T_{0}}]=\frac{\tanh\theta}{\theta}\quad\textrm{for
}\theta\in\mathbb{R}.$

For $j\geq 1$ let $m_{j}(\mathrm{d}\tau)$ be the measure such that
$\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{j})}]=C_{\gamma}\int_{0}^{\infty}e^{-2\pi\lambda\tau}e^{\frac{\pi\kappa}{4}(\frac{4}{\kappa}-1)^{2}\tau}m_{j}(\mathrm{d}\tau)$
as in (5.11). Let $M_{j}$ is sampled from the probability measure proportional
to $m_{j}(\mathrm{d}\tau)$. Then by (4.10) we have the equality $2\pi
M_{1}\overset{d}{=}\frac{\kappa\pi^{2}}{4}T_{0}+\frac{4\pi^{2}}{\kappa}Y_{\frac{\kappa}{4}}$
in law. For $\kappa=4$, since $Y_{1}=0$ we have
$M_{1}\overset{d}{=}\frac{\pi}{2}T_{1}$ as shown in [ALS20]. Since
$\mathbb{E}[e^{-\frac{\theta^{2}}{2}Y_{0}}]=\frac{1}{\cosh\theta}$, by (5.11)
we have $2\pi M_{j}\overset{d}{=}2\pi M_{1}+\sum_{m=1}^{j-1}X_{m}$ where
$X_{m}$’s are independent copies of $Y_{0}$ that are independent with $M_{1}$.

###  5.2. Joint law with the conformal radius: proof of Theorem 1.8

Let $\mathrm{Loop}_{\kappa}$ be the law of $\eta_{1}$, which is the outermost
loop surrounding $i$ of a $\mathrm{CLE}_{\kappa}$ on $\mathbb{H}$ . For a
sample $\eta$ from $\mathrm{Loop}_{\kappa}$, let $\psi_{\eta}:\mathbb{H}\to
D_{\eta}(i)$ be the unique conformal map with $\psi(i)=i$ and
$\psi^{\prime}(i)>0$. For $\alpha\in\mathbb{R}$, let
$\mathrm{Loop}_{\kappa}^{\alpha}$ be the measure defined by reweighting
$\mathrm{Loop}_{\kappa}$ as follows:

(5.13)
$\frac{d\mathrm{Loop}_{\kappa}^{\alpha}}{d\mathrm{Loop}_{\kappa}}(\eta)=|\psi_{\eta}^{\prime}(i)|^{2\Delta_{\alpha}-2}\quad\textrm{where
}\Delta(\alpha)=\frac{\alpha}{2}(Q-\frac{\alpha}{2}).$

Recall $\mathcal{M}^{\mathrm{disk}}_{1}(\alpha)$ from Definition 4.1. We use
the following proposition to prove Theorem 1.8.

######  Proposition 5.7.

For $\kappa\in(\frac{8}{3},4)$, $\gamma=\sqrt{\kappa}$ and
$\alpha>\frac{\gamma}{2}$, let $(\phi,\eta_{1})$ be sampled from
$\mathrm{LF}^{\alpha,i}_{\mathbb{H}}\times\mathrm{Loop}_{\kappa}^{\alpha}$.
Let $A_{1}$ be the annulus bounded by $\eta_{1}$ and $\partial\mathbb{H}$, and
$D_{1}$ be the Jordan domain bounded by $\eta_{1}$. Then the joint law of
$(A_{1},\phi)/{\sim_{\gamma}}$ and $(D_{1},\phi,0)/{\sim_{\gamma}}$ is

(5.14)
$\int_{0}^{\infty}b\mathrm{QA}^{\gamma}(a,b)\times\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;b)db.$

###### Proof.

When $\alpha=\gamma$, this is simply the $j=1$ case of Proposition 5.3. The
general $\alpha$ case can be obtained from the $\alpha=\gamma$ case via a
reweighting argument as in [ARS21, Section 4.2]. There the loop touches
$\partial\mathbb{H}$ but the exact same argument works here. ∎

###### Proof of Theorem 1.8.

Let
$f_{\alpha}(\ell)=|\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)|\ell=|\mathrm{LF}_{\mathbb{H}}^{\alpha,i}(\ell)|\ell=C_{\alpha,\gamma}\ell^{\frac{2}{\gamma}(\alpha-Q)}$
for some $(\alpha,\gamma)$-dependent constant $C_{\alpha,\gamma}$. By
Proposition 5.7, the identity (5.3) holds with $\gamma$-insertion replaced by
a generic $\alpha$. Namely

(5.15)
$\mathrm{LF}_{\mathbb{H}}^{(\alpha,i)}[Le^{-L}]\times\mathrm{Loop}_{\kappa}^{\alpha}[\rho(\mathrm{Mod}(\eta_{1}))]=\int_{0}^{\infty}\rho(\tau)\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}f_{\alpha}(L_{0})]m(\mathrm{d}\tau).$

where
$m(\mathrm{d}\tau)=1_{\tau>0}\cdot{\cos(\pi(\frac{4}{\gamma^{2}}-1))}\cdot\frac{\gamma}{4\pi}\theta_{1}(\frac{\gamma^{2}}{8},\frac{\gamma^{2}}{4}i\tau)\mathrm{d}\tau$]
and $\rho$ is a non-negative measurable function.

Let $\mathbb{E}$ be the expectation for $\mathrm{CLE}_{\kappa}^{\mathbb{H}}$.
Let $X=e^{2\pi\mathrm{Mod}(\eta_{1})}|\psi_{\eta}^{\prime}(i)|$. Then for
$\alpha>\frac{\gamma}{2}$ we have

(5.16)
$\mathbb{E}[X^{2\Delta_{\alpha}-2}\rho(\mathrm{Mod}(\eta_{1}))]=\mathrm{Loop}_{\kappa}^{\alpha}[e^{2\pi\mathrm{Mod}(\eta_{1})(2\Delta_{\alpha}-2)}\rho(\mathrm{Mod}(\eta_{1}))]=\int_{0}^{\infty}\rho(\tau)g(\tau,\alpha)m(\mathrm{d}\tau)$

where
$g(\tau,\alpha)=\frac{e^{2\pi(2\Delta_{\alpha}-2)\tau}\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}f(L_{0})]}{\mathrm{LF}_{\mathbb{H}}^{(\alpha,i)}[Le^{-L}]}$.
Since

$\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}L_{0}^{ix}]=\frac{\pi\gamma
x\Gamma(1+ix)}{2\sinh(\frac{\gamma^{2}}{4}\pi
x)}e^{-\frac{\pi\gamma^{2}x^{2}\tau}{4}}\quad\textrm{and}\quad
e^{\frac{\pi\tau\gamma^{2}}{4}(\frac{2}{\gamma}(Q-\alpha))^{2}}e^{2\pi(2\Delta_{\alpha}-2)\tau}=e^{\pi\tau(Q^{2}-4)},$

for a fixed $\gamma$, the function $(\alpha,\tau)\mapsto
e^{2\pi(2\Delta_{\alpha}-2)\tau}\mathrm{LF}_{\tau}[L_{1}e^{-L_{1}}L_{0}^{\frac{2}{\gamma}(\alpha-Q)}]$
can be factorized as a function of $\alpha$ times a function of $\tau$.
Therefore $g(\alpha,\tau)$ can be factorized as a function of $\alpha$ times a
function of $\tau$ as well. Therefore $X$ is independent of
$\mathrm{Mod}(\eta_{1})$. Since
$X=e^{2\pi\mathrm{Mod}(\eta_{1})}|\psi_{\eta}^{\prime}(i)|$, by the definition
of conformal radius we obtain Theorem 1.8 with $j=1$ and $\kappa\in(8/3,4)$.
For $j\geq 2$, the identity (5.16) holds with $m(\mathrm{d}\tau)$ replaced by
$m_{j}(\mathrm{d}\tau)$ from Proposition 5.4. This gives Theorem 1.8 for
$\kappa\in(8/3,4)$. The $\kappa=4$ case follows by sending $\kappa\uparrow 4$
as in the proof of Theorem 1.7. ∎

###  5.3. Nesting statistics: proof of Theorem 1.9

By Theorem 1.7, we have for

(5.17) $\sum_{j\geq
1}\beta^{j-1}\mathbb{E}[e^{-2\pi\lambda\mathrm{Mod}(\eta_{j})}]=\frac{(\frac{4}{\kappa}-1)\sin(\frac{\kappa}{4}\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})}{\sin(\pi(1-\frac{\kappa}{4}))\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}}}\frac{\cos(\pi(\frac{4}{\kappa}-1))}{\cos(\pi\sqrt{(\frac{4}{\kappa}-1)^{2}-\frac{8\lambda}{\kappa}})-\beta\cos(\pi(\frac{4}{\kappa}-1))}.$

Suppose $t=2\pi\lambda-\frac{\kappa\pi}{4}(\frac{4}{\kappa}-1)^{2}>0$. Set
$C_{\kappa}=\frac{\sqrt{\kappa\pi}(\frac{4}{\kappa}-1)\cos(\pi(\frac{4}{\kappa}-1))}{2\sin(\pi(1-\frac{\kappa}{4}))}$.
By (5.17), we have

$\sum_{j\geq
1}\beta^{j-1}\mathbb{E}[e^{-t\mathrm{Mod}(\eta_{j})}e^{-\frac{\kappa\pi}{4}(\frac{4}{\kappa}-1)^{2}\mathrm{Mod}(\eta_{j})}]=\frac{\sinh(\frac{\kappa}{4}\pi\sqrt{\frac{4t}{\kappa\pi}})}{\sqrt{\frac{4t}{\kappa\pi}}}\frac{C_{\kappa}}{\cosh(\pi\sqrt{\frac{4t}{\kappa\pi}})-\beta\cos(\pi(\frac{4}{\kappa}-1))}$

Let $n=2\cos(\pi(1-\frac{4}{\kappa}))$. For $n^{\prime}\in[-2,2]$, let
$\beta=\frac{n^{\prime}}{n}$ so that
$\frac{n^{\prime}}{2}=\beta\cos(\pi(1-\frac{4}{\kappa}))$. Let
$\chi^{\prime}=-\arccos(\frac{n^{\prime}}{2})$. Then for
$f(\tau)=e^{-t\tau}e^{-\frac{\kappa\pi}{4}(\frac{4}{\kappa}-1)^{2}\tau}$, we
have

(5.18)
$\mathbb{E}\left[\sum_{j=1}^{\infty}\left(\frac{n^{\prime}}{n}\right)^{j-1}f(\mathrm{Mod}(\eta_{n}))\right]=\frac{C_{\kappa}\sinh(\sqrt{\kappa\pi
t/4})}{{\sqrt{t}}({\cosh\sqrt{4\pi t/\kappa}-\cos\chi^{\prime}})}.$

Now Theorem 1.9 is immediate from the following fact on the function
$Z(\tau,\kappa,\chi^{\prime})$ in (1.12).

######  Lemma 5.8.

Let $\mathcal{L}[f(\tau)](t)=\int_{0}^{\infty}e^{-t\tau}f(\tau)d\tau$ be the
Laplace transform. For $\kappa\in(\frac{8}{3},4]$ and
$\chi^{\prime}\in\mathbb{R}$,

(5.19)
$\displaystyle\mathcal{L}[Z(\tau,\kappa,\chi^{\prime})\eta(2i\tau)](t)=\frac{\sqrt{\pi}\sinh(\sqrt{\kappa\pi
t/4})}{{\sqrt{2t}}({\cosh\sqrt{4\pi
t/\kappa}-\cos\chi^{\prime}})}\quad\textrm{for }t>0.$

###### Proof.

By the modular property $\eta(i/\tau)=\sqrt{\tau}\eta(i\tau)$ of the eta
function, we have

(5.20)
$\eta(2i\tau)=(2\tau)^{-1/2}\eta(\frac{i}{2\tau})=(2\tau)^{-1/2}{q}^{\frac{1}{24}}\prod_{r=1}^{\infty}(1-q^{r})\quad\textrm{with
}q=e^{-\frac{\pi}{\tau}}.$

By the representation (1.13) of $Z(\tau,\kappa,\chi^{\prime})$, for
$g=\frac{4}{\kappa}$ we have

(5.21)
$Z(\tau,\kappa,\chi^{\prime})\eta(2i\tau)=\frac{1}{\sqrt{2\tau}}\sum_{p\in\mathbb{Z}}\frac{\sin(p+1)\chi^{\prime}}{\sin\chi^{\prime}}q^{\frac{g}{4}(p-\frac{1-g}{g})^{2}}$

Since
$\mathcal{L}[\frac{1}{\sqrt{\tau}}e^{-\frac{a}{4\tau}}](t)=\frac{\sqrt{\pi}e^{-\sqrt{at}}}{\sqrt{t}}$
for each $a>0$, we have
$\mathcal{L}[\frac{1}{\sqrt{\tau}}q^{\frac{g}{4}(p-\frac{1-g}{g})^{2}}](t)=\frac{\sqrt{\pi}e^{-|p+1-g^{-1}|\sqrt{g\pi
t}}}{\sqrt{t}}$ hence

(5.22)
$\mathcal{L}[Z(\tau,\kappa,\chi^{\prime})\eta(2i\tau)](t)=\sqrt{\frac{\pi}{2}}\sum_{p\in\mathbb{Z}}\frac{\sin(p+1)\chi^{\prime}}{\sin\chi^{\prime}}\frac{e^{-|p+1-g^{-1}|\sqrt{g\pi
t}}}{\sqrt{t}}.$

Shifting $p+1$ and dividing the sum according to $p>0$ or $p<0$, we get

$\displaystyle\sum_{p\in\mathbb{Z}}\frac{\sin(p+1)\chi^{\prime}}{\sin\chi^{\prime}}\frac{e^{-|p+1-g^{-1}|\sqrt{g\pi
t}}}{\sqrt{t}}=\sum_{p\geq
1}\frac{\sin(p\chi^{\prime})}{{\sqrt{t}}\sin\chi}(e^{-(p-g^{-1})\sqrt{g\pi
t}}-e^{-(p+g^{-1})\sqrt{g\pi t}})$ $\displaystyle=$ $\displaystyle\sum_{p\geq
1}\frac{2\sin(p\chi^{\prime})\sinh(\sqrt{\pi
t/g})}{{\sqrt{t}}\sin\chi^{\prime}}e^{-p\sqrt{g\pi t}}=\frac{\sinh(\sqrt{\pi
t/g})}{{\sqrt{t}}({\cosh\sqrt{g\pi t}-\cos\chi^{\prime}})},$

where the last step uses

$\sum_{p\geq 1}{2\sin(p\chi^{\prime})}e^{-p\sqrt{g\pi
t}}=\frac{\sin\chi^{\prime}}{\cosh\sqrt{g\pi t}-\cos\chi^{\prime}}.$

Now (5.19) follows from (5.22). ∎

###### Proof of Theorem 1.9.

Comparing (5.18) and (5.19), we get

(5.23)
$\mathbb{E}\left[\sum_{j=1}^{\infty}\left(\frac{n^{\prime}}{n}\right)^{j-1}f(\mathrm{Mod}(\eta_{j}))\right]=\frac{\sqrt{2}C_{\kappa}}{\sqrt{\pi}}\mathcal{L}[Z(\tau,\kappa,\chi^{\prime})\eta(2i\tau)](t)\quad\textrm{for
}t>0,$

where
$C_{\kappa}=\frac{\sqrt{\kappa\pi}(\frac{4}{\kappa}-1)\cos(\pi(\frac{4}{\kappa}-1))}{2\sin(\pi(1-\frac{\kappa}{4}))}$
and $f(\tau)=e^{-t\tau}e^{-\frac{\kappa\pi}{4}(\frac{4}{\kappa}-1)^{2}\tau}$.
This gives (1.14) for this choice of $f$. Varying $t$ we see that (1.14) holds
if $f$ is an arbitrary non-negative measurable function.∎

##  6\. The random modulus for the Brownian annulus

In this section we prove Theorem 1.2. In Section 6.1 we give a precise
construction of the conformally embedded Brownian disk and Brownian annulus as
quantum surfaces in $\sqrt{8/3}$-LQG, with more background on Brownian
surfaces provided in Appendix B. In Section 6.2 we use the strategy in Section
4 based on the KPZ relation (Theorem 1.6) to complete the proof.

###  6.1. Preliminaries on Brownian surfaces

The _Brownian sphere_ is a random metric-measure space obtained in [Le 13,
Mie13] as the Gromov-Hausdorff-Prokhorov scaling limit of uniform
quadrangulations, under the name _Brownian map_. We write
$\mathrm{BS}_{2}(1)^{\\#}$ as the law of the unit-area Brownian sphere with
two marked points. For $A,L>0$, the _Brownian disk_ with area $A$ and boundary
length $L$ is a random metric-measure space obtained in [BM17, Section 2.3] as
the scaling limit of random quadrangulations with boundary. We denote its law
by $\mathrm{BD}_{0,1}(L;A)^{\\#}$. We use the subscripts $\\{0,1\\}$ because a
sample from $\mathrm{BD}_{0,1}(L;A)^{\\#}$ has no interior marked points and
one boundary marked point. For $L>0$, define

(6.1) $\mathrm{BD}_{0,1}(L)^{\\#}=\int_{0}^{\infty}\frac{L^{3}}{\sqrt{2\pi
A^{5}}}e^{-\frac{L^{2}}{2A}}\mathrm{BD}_{0,1}(L;A)^{\\#}\,\mathrm{d}A.$

Then $\mathrm{BD}_{0,1}(L)^{\\#}$ is the probability measure for the law of
the Brownian disk with boundary length $L$ and free area. See Theorems B.1 and
B.2 in Appendix B for precise scaling limit results concerning
$\mathrm{BS}_{2}(1)^{\\#}$ and $\mathrm{BD}_{0,1}(L)^{\\#}$. Both the Brownian
sphere and the Brownian disk can be defined purely in the continuum using the
Brownian snake as done in [Le 13, Mie13, BM17], but this construction is not
explicitly needed for Sections 6 and 7. We now introduce a few variants of
$\mathrm{BS}_{2}(1)^{\\#}$ and $\mathrm{BD}_{0,1}(L)^{\\#}$ which are more
convenient for our purpose.

######  Definition 6.1.

Given a sample of $\mathrm{BS}_{2}(1)^{\\#}$, if its area measure is rescaled
by some $A>0$, we write $\mathrm{BS}_{2}(A)^{\\#}$ for the law of this new
metric-measure space of area $A$ with two marked points. Let

$\mathrm{BS}_{2}=\int_{0}^{\infty}A^{-3/2}\mathrm{BS}_{2}(A)^{\\#}\mathrm{d}A.$

We call a sample from $\mathrm{BS}_{2}$ a _free Brownian sphere_. For $L>0$
and $A>0$, let

$\mathrm{BD}_{0,1}(L)=L^{-\frac{5}{2}}\mathrm{BD}_{0,1}(L)^{\\#}\quad\textrm{and}\quad\mathrm{BD}_{0,1}(L;A)=\frac{L^{\frac{1}{2}}}{\sqrt{2\pi
A^{5}}}e^{-\frac{L^{2}}{2A}}\mathrm{BD}_{0,1}(L;A)^{\\#}.$

Given a sample from
$\int_{0}^{\infty}\frac{A}{L}\mathrm{BD}_{0,1}(L;A)\;\mathrm{d}A$, forget the
marked point on the boundary and add an interior marked point according to its
area measure. We denote the law of the resulting metric-measure space with one
interior marked point by $\mathrm{BD}_{1,0}(L)$, and the probability measure
proportional to $\mathrm{BD}_{1,0}(L)$ by $\mathrm{BD}_{1,0}(L)^{\\#}$.

The following Proposition 6.2 and Lemma 6.3 allow us to find Brownian disks
and annuli inside a Brownian sphere. Here whenever a subset of a metric space
is viewed as a new metric space, we will use the internal metric on the
subset.

######  Proposition 6.2 ([LG19, Theorem 3]).

Sample $(\mathcal{S},x,y)$ from $\mathrm{BS}_{2}$. On the event $d(x,y)>2$ let
$D_{y}$ be the connected component of $\mathcal{S}\setminus B(x,1)$ containing
$y$, where $B(x,1)$ is the metric ball of radius 1 around $x$. Conditioned on
$d(x,y)>2$ and on the boundary length $\mathcal{L}_{y}$ of $D_{y}$, the
conditional law of $(D_{y},y)$ is given by
$\mathrm{BD}_{1,0}(\mathcal{L}_{y})^{\\#}$ conditioned on the event that the
distance between the interior marked point and the disk boundary is greater
than 1.

In the setting of Proposition 6.2, we write
$\mathcal{B}^{\bullet}(x,1)=\mathcal{S}\setminus D_{y}$ and called it a filled
metric ball of radius 1 around $x$. We define $\mathcal{B}^{\bullet}(y,1)$ in
the same way with the role of $x$ and $y$ swapped.

######  Lemma 6.3.

Sample $(\mathcal{S},x,y)$ from $\mathrm{BS}_{2}$. Conditioned on $d(x,y)>2$
and on the boundary lengths $\mathcal{L}_{x}$ of $\mathcal{B}^{\bullet}(x,1)$
and $\mathcal{L}_{y}$ of $\mathcal{B}^{\bullet}(y,1)$, the conditional law of
$\mathcal{S}\backslash(\mathcal{B}^{\bullet}(x,1)\cup\mathcal{B}^{\bullet}(y,1))$
is $\mathrm{BA}(\mathcal{L}_{x},\mathcal{L}_{y})^{\\#}$.

###### Proof.

This follows from Definition 1.1 of $\mathrm{BA}(a,b)^{\\#}$and Proposition
6.2. ∎

Lemma 6.3 implies the desired symmetry of the Brownian annulus needed for our
proof of Theorem 1.2.

######  Lemma 6.4.

As measures on metric-measure spaces, we have
$\mathrm{BA}(a,b)^{\\#}=\mathrm{BA}(b,a)^{\\#}$ for $a,b>0$.

###### Proof.

For $(\mathcal{S},x,y)$ sampled from $\mathrm{BS}_{2}$, the law of
$(\mathcal{S},y,x)$ is also $\mathrm{BS}_{2}$. Now Lemma 6.4 follows from
Lemma 6.3. ∎

Miller and Sheffield [MS20, MS21a, MS21b] gave the $\sqrt{8/3}$-LQG
description of the conformally embedded Brownian sphere and the Brownian disk.
In this section we only need their disk result, which we recall now. The
sphere result will be recalled as Theorem 7.1 in Section 7. Fix
$\gamma=\sqrt{8/3}$. For $a>0$, recall that the measure
$\mathrm{LF}^{(\gamma,i)}_{\mathbb{H}}(a)$ is defined by the disintegration
$\mathrm{LF}^{(\gamma,i)}_{\mathbb{H}}=\int_{0}^{\infty}\mathrm{LF}^{(\gamma,i)}_{\mathbb{H}}(a)\mathrm{d}a$
such that samples from $\mathrm{LF}_{\mathbb{H}}^{(\gamma,i)}(a)$ have
boundary length $a$. By (4.1) one has
$|\mathrm{LF}^{(\gamma,i)}_{\mathbb{H}}(a)|<\infty$. Let
$\mathrm{LF}^{(\gamma,i)}_{\mathbb{H}}(a)^{\\#}$ be the probability measure
proportional to $\mathrm{LF}^{(\gamma,i)}_{\mathbb{H}}(a)$. Now sample $\phi$
from $\mathrm{LF}^{\gamma,i}_{\mathbb{H}}(a)^{\\#}$. Let
$\mu_{\phi}=e^{\gamma\phi(z)}dz$ be the quantum area measure [DS11] defined
via the Gaussian multiplicative chaos. By [DDDF20, GM21b], we can find a
smooth regularization $\phi_{\varepsilon}$ and a normalizing sequence
$z_{\varepsilon}$ such that as $\varepsilon\to 0$ the Riemannian metric on
$\mathbb{H}$ induced by metric tensor $z^{-1}_{\varepsilon}e^{\frac{1}{4}\cdot
2\gamma\phi_{\varepsilon}}(\mathrm{d}x^{2}+\mathrm{d}y^{2})$ converges in
probability to a metric $d_{\phi}$ on $\mathbb{H}$, which is called the
$\sqrt{8/3}$-LQG metric of $\phi$. Moreover, modulo a multiplicative constant,
$d_{\phi}$ agrees with the metric constructed earlier in [MS20, MS21a] via the
quantum Loewner evolution. Here the exponent $\frac{1}{4}$ is the Hausdorff
dimension of a Brownian surface. The following theorem summarizes the relation
between $\mathrm{LF}^{(\gamma,i)}_{\mathbb{H}}$ and the Brownian disk.

######  Theorem 6.5.

There exist constants $c_{1}$ and $c_{2}$ such that the following holds.333The
metric constant $c_{1}$ can be set to $1$ by choosing the normalizing sequence
in the definition of $d_{\phi}$ appropriately. The area constant $c_{2}$ is
canonically defined due to the different definitions of area for Brownian
surfaces and $\sqrt{8/3}$-LQG surfaces. As explained in [GHS21, Remark 3.12],
$c_{2}=\sqrt{3}$. Suppose $\phi$ is sampled from
$\mathrm{LF}^{\gamma,i}_{\mathbb{H}}(a)^{\\#}$ for some $a>0$. Let
$(\mathcal{D},p,d,\mu)$ be the (marked) metric-measure space given by
$(\mathbb{H},i,c_{1}d_{\phi},c_{2}\mu_{\phi})$. Then $(\mathcal{D},p,d,\mu)$
is a Brownian disk with boundary length $a$ and an interior marked point;
namely its law is $\mathrm{BD}_{1,0}(a)^{\\#}$. Moreover, the notion of
boundary length for the Brownian disk agrees with the $\sqrt{8/3}$-LQG length.
Finally, $(\mathcal{D},p,d,\mu)$ and the quantum surface
$(\mathbb{H},\phi,i)/{\sim_{\gamma}}$ are measurable with respect to each
other.

###### Proof.

By [ARS21, Theorem 3.4], $(\mathbb{H},\phi,i)/{\sim_{\gamma}}$ is a quantum
disk with boundary length $a$ and an interior marked point as defined in the
mating-of-trees framework [DMS21]. By [MS20, MS21a] together with [LG19,
Section 6], it is possible to require the two notions of boundary lengths
agree and find scaling constants $c_{1}$ and $c_{2}$ such that the law of
$(\mathcal{D},p,d,\mu)$ is $\mathrm{BD}_{1,0}(a)^{\\#}$. The measurability
result was first proved in [MS21b] and a more constructive proof was given in
[GMS20]. ∎

######  Remark 6.6.

By (4.1) with $\gamma=\sqrt{8/3}$, we have

$|\mathrm{LF}^{\gamma,i}_{\mathbb{H}}(a)|=Ca^{-\frac{4}{\gamma^{2}}}=Ca^{-\frac{3}{2}}=C|\mathrm{BD}_{1,0}(a)|\quad\textrm{for
some constant $C>0$}.$

Therefore Theorem 6.5 still holds with
$\mathrm{LF}^{\gamma,i}_{\mathbb{H}}(a)$ and $C\cdot\mathrm{BD}_{1,0}(a)$ in
place of $\mathrm{LF}^{\gamma,i}_{\mathbb{H}}(a)^{\\#}$ and
$\mathrm{BD}_{1,0}(a)^{\\#}$.

Let $d_{\mathbb{H}}=c_{1}d_{\phi}$ and $\mu_{\mathbb{H}}=c_{2}\mu_{\phi}$.
Then $(\mathbb{H},d_{\mathbb{H}},\mu_{\mathbb{H}})$ is a conformal embedding
of $(\mathcal{D},p,d,\mu)$ as discussed below Definition 1.1 of the Brownian
annulus. On the event that $d(p,\partial\mathcal{D})>1$, let $\mathcal{A}$ be
the (annular) connected component of $\mathcal{D}\setminus B_{d}(p,1)$ whose
boundary contains $\partial\mathcal{D}$. Let
$B^{\bullet}(p,1)=\mathcal{D}\setminus\mathcal{A}$, which we call the filled
metric ball of radius $1$ around $p$. Let
$B_{d_{\mathbb{H}}}(i,1)=\\{x\in\mathbb{H}:d_{\mathbb{H}}(i,x)\leq 1\\}$ and
$A$ the annular component of $\mathbb{H}\setminus B_{d_{\mathbb{H}}}(0,1)$.
Then $(A,d_{\mathbb{H}},\mu_{\mathbb{H}})$ is a conformal embedding of