# Tightness of approximations to the chemical distance
metric for simple conformal loop ensembles

Jason Miller

###### Abstract.

Suppose that $\Gamma$ is a conformal loop ensemble (${\rm CLE}_{\kappa}$) with
simple loops ($\kappa\in(8/3,4)$) in a simply connected domain
$D\subseteq\mathbf{C}$ whose boundary is itself a type of ${\rm CLE}_{\kappa}$
loop. Let $\Upsilon$ be the carpet of $\Gamma$, i.e., the set of points in $D$
not surrounded by a loop of $\Gamma$. We prove that certain approximations to
the chemical distance metric in $\Upsilon$ are tight. More precisely, for each
path $\omega\colon[0,1]\to\Upsilon$ and $\epsilon>0$ we let
${\mathfrak{N}}_{\epsilon}(\omega)$ be the Lebesgue measure of the
$\epsilon$-neighborhood of $\omega$. For $z,w\in\Upsilon$ we let
$\mathfrak{d}_{\epsilon}(z,w;\Gamma)=\inf_{\omega}{\mathfrak{N}}_{\epsilon}(\omega)$
where the infimum is over all paths $\omega\colon[0,1]\to\Upsilon$ with
$\omega(0)=z$, $\omega(1)=w$ and let ${\mathfrak{m}}_{\epsilon}$ be the median
of $\sup_{z,w\in\partial D}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$. We prove that
$(z,w)\mapsto{\mathfrak{m}}_{\epsilon}^{-1}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$
is tight and that any subsequential limit defines a geodesic metric on
$\Upsilon$ which is Hölder continuous with respect to the Euclidean metric. We
conjecture that the subsequential limit is unique, conformally covariant, and
describes the scaling limit of the chemical distance metric for discrete loop
models which converge to ${\rm CLE}_{\kappa}$ for $\kappa\in(8/3,4)$ such as
the critical Ising model.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Crossing estimates
  4. 4 Percolation exploration
  5. 5 Tightness on the boundary
  6. 6 Tightness in the interior
  7. 7 Positive definiteness and geodesics
  8. A Modulus of continuity of space-filling ${\rm SLE}$ with quantum parameterization
  9. B Density of quantum carpet typical points
  10. C Lévy process estimates

##  1\. Introduction

###  1.1. Overview

The conformal loop ensemble ${\rm CLE}_{\kappa}$ [40, 42] is the canonical
conformally invariant measure on loops in a simply connected domain
$D\subseteq\mathbf{C}$. It is the loop version of the Schramm-Loewner
evolution (${\rm SLE}_{\kappa}$) [39]. It consists of a countable collection
of loops in $D$ each of which looks locally like an ${\rm SLE}_{\kappa}$ curve
where $\kappa\in(8/3,8)$. Just like for ${\rm SLE}_{\kappa}$, the loops are
simple and do not intersect each other or the domain boundary for
$\kappa\in(8/3,4]$ while for $\kappa\in(4,8)$ the loops are self-intersecting
and intersect the domain boundary. Its importance is that it describes (or is
conjectured to describe) the scaling limit of all of the interfaces for a
number of discrete models from statistical mechanics in two dimensions. For
example, on deterministic planar lattices, ${\rm CLE}_{3}$ was shown to be the
scaling limit of the interfaces in the Ising model [7, 3], ${\rm CLE}_{16/3}$
the interfaces in the FK-Ising model [16, 44], ${\rm CLE}_{6}$ the interfaces
in the percolation model [43, 5], and ${\rm CLE}_{8}$ the Peano curve
associated with the uniform spanning tree (UST) [20].

The purpose of this article is to initiate the program of constructing a
conformally covariant metric associated with a ${\rm CLE}_{\kappa}$. We will
focus on the case that $\kappa\in(8/3,4)$ so that the ${\rm CLE}_{\kappa}$
loops are simple and do not intersect each other. Suppose that
$D\subseteq\mathbf{C}$ is a simply connected domain and let $\Gamma$ be a
${\rm CLE}_{\kappa}$ in $D$. Let $\Upsilon$ be the carpet of $\Gamma$, i.e.,
the set of $z\in D$ which are not surrounded by any loop of $\Gamma$. For each
path $\omega\colon[0,1]\to\Upsilon$ and $\epsilon>0$, we let
${\mathfrak{N}}_{\epsilon}(\omega)$ be the Lebesgue measure of the
$\epsilon$-neighborhood of $\omega$. For $z,w\in\Upsilon$, we let
$\mathfrak{d}_{\epsilon}(z,w;\Gamma)=\inf_{\omega}{\mathfrak{N}}_{\epsilon}(\omega)$
where the infimum is over all paths $\omega\colon[0,1]\to\Upsilon$ with
$\omega(0)=z$, $\omega(1)=w$. Finally, we let ${\mathfrak{m}}_{\epsilon}$ be
the median of the random variable $\sup_{z,w\in\partial
D}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$. Our aim is to show that
$(z,w)\mapsto{\mathfrak{m}}_{\epsilon}^{-1}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$
is tight and that any subsequential limit defines a geodesic metric on
$\Upsilon$ which is Hölder continuous with respect to the Euclidean metric.

We note that for $z,w\in\Upsilon$ which are away from $\partial D$, it is
natural to expect that the “shortest” path in $\Upsilon$ from $z$ to $w$ is a
fractal curve and that there exists $\alpha>1$ so that the median of
$\sup_{z,w\in\Upsilon}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$ is
$\epsilon^{2-\alpha+o(1)}$ as $\epsilon\to 0$. If $\partial D$ is smooth and
$z,w\in\partial D$, then one could take the path which follows $\partial D$
from $z$ to $w$ hence in this case
${\mathfrak{m}}_{\epsilon}=\epsilon^{1+o(1)}$. This means that if $\partial D$
is smooth then ${\mathfrak{m}}_{\epsilon}$ is the wrong normalization to get a
finite metric on $\Upsilon$. For this reason, we will state our main result
for a domain with rough boundary. In particular, we will take $\partial D$ to
be itself a ${\rm CLE}_{\kappa}$ loop.

###  1.2. Main result

The main result of this paper is the tightness of
${\mathfrak{m}}_{\epsilon}^{-1}\mathfrak{d}_{\epsilon}(\cdot,\cdotp;\Gamma)$,
which we note is a function defined on a random set. Let us now specify with
respect to which topology we will prove tightness.

We let $\mathbf{K}_{n}$ be the set of pairs consisting of a compact subset
$K\subseteq\mathbf{R}^{n}$ and a function $f\colon K\to\mathbf{R}$. We define
a metric on $\mathbf{K}_{n}$ as follows. Let $d_{\mathrm{H}}$ denote the
Hausdorff distance on compact subsets of $\mathbf{R}^{n}$. For
$(f,K),(g,L)\in\mathbf{K}_{n}$ we let

${\mathbf{d}}_{0,n}((f,K),(g,L))=\sup\\{|f(z)-g(w)|:z\in K,w\in L,\ |z-w|\leq
d_{\mathrm{H}}(K,L)\\}.$

We note that ${\mathbf{d}}_{0,n}$ defines a pseudometric on $\mathbf{K}_{n}$
but not a metric. Indeed, if $K$, $L$ are distinct compact subsets of
$\mathbf{R}^{n}$ and $f,g$ are the zero function then
${\mathbf{d}}_{0,n}((f,K),(g,L))=0$. To define a metric, we thus set

${\mathbf{d}}_{n}((f,K),(g,L))={\mathbf{d}}_{0,n}((f,K),(g,L))+d_{\mathrm{H}}(K,L).$

Fix $R,S>0$. Suppose that $\mathbf{F}$ is a family in $\mathbf{K}_{n}$ so
that:

  1. (i)

for every $\epsilon>0$ there exists $\delta>0$ so that for every
$(f,K)\in\mathbf{F}$, $z,w\in K$, and $|z-w|<\delta$ we have that
$|f(z)-f(w)|<\epsilon$ and

  2. (ii)

for every $(f,K)\in\mathbf{F}$ we have that $K\subseteq B(0,R)$ and
$\sup_{z\in K}|f(z)|\leq S$.

Then the closure of $\mathbf{F}$ is compact in
$(\mathbf{K}_{n},{\mathbf{d}}_{n})$. Indeed, suppose that $((f_{n},K_{n}))$ is
a sequence in $\mathbf{F}$. Then there exists a subsequence
$((f_{n_{j}},K_{n_{j}}))$ so that $(K_{n_{j}})$ converges with respect to
$d_{\mathrm{H}}$. The usual proof of the Arzela-Ascoli theorem then implies
that, by taking a further subsequence if necessary, there exists
$(f,K)\in\mathbf{F}$ so that
${\mathbf{d}}_{0,n}((f_{n_{j}},K_{n_{j}}),(f,K))\to 0$ as $j\to\infty$.

The main argument used in this paper works whenever $\partial D$ locally looks
like an ${\rm SLE}_{\kappa}$ curve, though we will state our main result with
a particular choice for the sake of concreteness (see Figure 1.1 for a
numerical simulation of the setup).

Figure 1.1.  Top left: A ${\rm CLE}_{3}$ on the unit disk. Top right: The loop
which surrounds the origin from the top left (red) with an independent ${\rm
CLE}_{3}$ in the domain surrounded by it. Bottom: The top right with the ${\rm
CLE}_{3}$ carpet colored according to the chemical distance from a uniformly
chosen point in the carpet; the closest points are red and the farthest points
are magenta.

######  Theorem 1.1.

Fix $\kappa\in(8/3,4)$. Suppose that $\Gamma_{\mathbf{D}}$ is a ${\rm
CLE}_{\kappa}$ in the unit disk $\mathbf{D}$ and let $D$ be the set of points
surrounded by the loop $\mathcal{L}\in\Gamma_{\mathbf{D}}$ which surrounds
$0$. Given $\mathcal{L}$, let $\Gamma$ be a ${\rm CLE}_{\kappa}$ in $D$. Let
${\mathfrak{m}}_{\epsilon}$ be the median of $\sup_{z,w\in\partial
D}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$. Then the law of the map
$(z,w)\mapsto{\mathfrak{m}}_{\epsilon}^{-1}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$
is tight in the space $(\mathbf{K}_{4},{\mathbf{d}}_{4})$. Any subsequential
limit is a.s. a geodesic metric on $\Upsilon$ which is Hölder continuous with
respect to the Euclidean metric.

We conjecture that the limit in Theorem 1.1 exists and is characterized by a
certain list of axioms, as in the case of the Liouville quantum gravity metric
[23, 13]. We plan to address this in future work. We also conjecture that the
limit describes the scaling limit of the chemical distance metric for discrete
models which converge to ${\rm CLE}$, such as the critical Ising model [3]
($\kappa=3$).

###  1.3. Outline and proof strategy

Fix $\kappa\in(8/3,4)$. In proving Theorem 1.1, we will often work in an
infinite volume setup which describes the local behavior of $\mathcal{L}$ near
a “typical” point in $\mathcal{L}$. That is, we will replace $\mathcal{L}$ by
a so-called two-sided whole-plane ${\rm SLE}_{\kappa}$ process $\eta$ and we
let $\Gamma_{+}$ be a ${\rm CLE}_{\kappa}$ in the component of
$\mathbf{C}\setminus\eta$ which is to the left of $\eta$. Rather than
normalize using the median and quantiles associated with the setup described
in Theorem 1.1, we will instead use the median and quantiles associated with
this infinite volume setup and then at the very end of the proof show that
they are comparable. In order to distinguish between these two situations, we
will use the superscript HP to denote a quantity associated with the infinite
volume (half-planar) setup and will not use a superscript when referring to a
quantity associated with the setup described in the statement of Theorem 1.1.

The basic strategy to prove Theorem 1.1 is the following. For each
$\epsilon>0$ and $p\in(0,1)$ we let
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$ denote the $p$th quantile of
$\sup_{z,w\in\eta([0,1])}\mathfrak{d}_{\epsilon}(z,w;\Gamma_{+})$ and let
${\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}={{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(1/2)$
be the median. Note that for any $M_{0}\geq 1$ there exists $M_{1}\geq 1$ so
that

(1.1)
$\frac{1}{M_{1}}{{\mathfrak{q}}}_{M_{0}\epsilon}^{\mathrm{HP}}(p)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)\leq
M_{1}{{\mathfrak{q}}}_{\epsilon/M_{0}}^{\mathrm{HP}}(p).$

That is, replacing $\epsilon$ with a fixed multiple of $\epsilon$ changes the
quantiles by at most a factor which does not depend on $\epsilon$. (See also
Lemma 5.4 for a more quantitative version of (1.1).)

The proof is divided into two main steps:

  * •

Establish the comparability of the quantiles
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$ uniformly in $\epsilon>0$ and
prove the tightness of
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(\cdot,\cdot;\Gamma)$
in $(\mathbf{K}_{4},{\mathbf{d}}_{4})$. We emphasize here that the tightness
result will be proved in the finite volume setting but using the normalization
from the median defined in the infinite volume setting.

  * •

Show that every subsequential limit is a.s. positive definite hence a metric.

The geodesic property of a subsequential limit and the comparability of the
infinite volume and finite volume quantiles will not be difficult to establish
once we have accomplished both of the steps above.

Let $\kappa^{\prime}=16/\kappa\in(4,6)$. In [31], it was shown how one can
view an ${\rm SLE}_{\kappa^{\prime}}(\kappa^{\prime}-6)$ process
$\eta^{\prime}$ as a _conformal percolation interface_ (CPI) inside of the
carpet $\Upsilon$ of $\Gamma$ which always keeps the loops of $\Gamma$ which
it hits on its right side. Moreover, the process obtained by following the
loops of $\Gamma$ which are hit by $\eta^{\prime}$ is an ${\rm
SLE}_{\kappa}(\kappa-6)$ process $\eta$. More generally, if we fix
$\rho^{\prime}\in[\kappa^{\prime}-6,0]$ then there exists a value of
$\beta\in[-1,1]$ so that we can couple an ${\rm
SLE}_{\kappa^{\prime}}(\rho^{\prime};\kappa-6-\rho^{\prime})$ process with
$\Upsilon$ so that it keeps each loop of $\Gamma$ that it hits on its left
(resp. right) side with probability $(1-\beta)/2$ (resp. $(1+\beta)/2$) and
the process obtained by following the loops which are hit by $\eta^{\prime}$
is an ${\rm SLE}_{\kappa}^{\beta}(\kappa-6)$ process $\eta$. We will often
abuse notation and refer to $\eta^{\prime}$ together with the loops of
$\Gamma$ that it hits (i.e., $\eta$) also as a CPI.

CPIs give us a way to “explore” the ${\rm CLE}_{\kappa}$ carpet in a Markovian
way so that the conditional law of the loops of $\Gamma$ which are in the
components of $D\setminus\eta([0,t])$ not surrounded by a loop of $\Gamma$ are
independent ${\rm CLE}_{\kappa}$’s. This means that we can explore further
into each such component using a CPI which can also start from any point on
the component boundary. This type of exploration will be the basis for a
spatial decomposition of the ${\rm CLE}_{\kappa}$ into “chunks”. Since the
_shape_ of a domain strongly affects the chemical distance metric (e.g., as we
explained earlier if the domain boundary is smooth then it is natural to
expect that points along the boundary will be close to each other), knowing
the shape of the successive chunks of the exploration will restrict the form
of the chemical distance metric in the unexplored region. We will thus want to
consider a type of exploration which avoids revealing the exact shape of the
explored region as much as possible. For this reason, we will explore the
${\rm CLE}_{\kappa}$ using CPIs in a “quantum” manner using the relationship
between ${\rm SLE}$ and Liouville quantum gravity (LQG) developed in [41, 9]
as well as the relationship between ${\rm CLE}$ and LQG developed in [33]. The
reason for this is that exploring a ${\rm CLE}$ in a quantum manner is
analogous to exploring a random planar map using a peeling process and in such
an exploration the regions which are cut out depend only on the exploration
through their boundary length. In particular, conditioning on chunks which
have been explored this way means that we do not have to condition on their
actual shape – only on how they are glued together.

We will call a chunk “good” if it is possible to traverse it with a path
$\omega$ with
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$
and “bad” if it is not. We will prove that by making the chunk size
sufficiently small (but with $p\in(0,1)$ fixed), we can make the probability
that a chunk is good as close to $1$ as we wish uniformly in $\epsilon\to 0$.
We will explain these estimates in Section 3. We emphasize that this is the
particular point in the proof where we make use of the normalization by
${\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}$ instead of making a choice such as
by the median length of a crossing $\omega$ which minimizes
${\mathfrak{N}}_{\epsilon}(\omega)$ among those which connect two disjoint
boundary arcs. The reason is that we will show that the probability that the
diameter of the set of points $z\in\Upsilon_{+}$, $\Upsilon_{+}$ the carpet of
$\Gamma_{+}$, with
$\mathfrak{d}_{\epsilon}(0,z;\Gamma_{+})\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$
is at least $\delta$ tends to $1$ as $\delta\to 0$ provided $\epsilon>0$ is
sufficiently small relative to $\delta$. The reason that this is the case is
that $\eta([0,1])$ enjoys certain self-similarity properties. In particular,
if there was a positive chance that the aforementioned set was arbitrarily
small with positive probability then that would have to be very likely to be
true somewhere along $\eta([0,1])$ which in turn would contradict the
definition of ${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.

We will show that the good vs bad chunks can be viewed as a type of strongly
supercritical percolation Section 4, though in a more general framework. In
particular, we will show that it is likely that two points on the domain
boundary are connected by a collection of such chunks. The initial estimate
that we will obtain here will not have sufficiently good concentration in
order to deduce the tightness of
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(\cdot,\cdot;\Gamma)$
because the exploration that we will consider has the possibility of hitting
obstacles such as large ${\rm CLE}_{\kappa}$ loops. In order to obtain
stronger concentration, we will consider an adaptive version which circumvents
these obstacles.

Using the definition of good vs bad chunks from Section 3 and the adaptive
percolation exploration from Section 4, we will show in Section 5 that it is
possible to string together the crossings between good chunks in order to
deduce that with high probability we can connect two boundary points by a path
$\omega$ with ${\mathfrak{N}}_{\epsilon}(\omega)$ at most a constant times
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$. The adaptive version of the
exploration will give sufficiently good concentration so that we can establish
the tightness of
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(\cdot,\cdot;\Gamma)$
on $\partial D$ and deduce the comparability of the quantiles
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.

In Section 6 we then extend the tightness of
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(\cdot,\cdot;\Gamma)$
from the boundary to the interior. The first step is to define collections of
points $P_{j}$ in $\Upsilon$ which consist of i.i.d. samples from the natural
LQG measure on $\Upsilon$ constructed in [33]. We will show in Appendix B that
there exists $\alpha_{\mathrm{NET}}>0$ so that if
$|P_{j}|=2^{\alpha_{\mathrm{NET}}j}$ then with overwhelming probability
$P_{j}$ forms a $2^{-j}$-net of $\Upsilon$ (with respect to the Euclidean
metric). We then aim to show that there exists $\beta>0$ so that if $z,w\in
P_{j}$ have Euclidean distance at most $2^{-j}$ then
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$
is very likely to be at most $2^{-\beta j}$. Proving this statement will
require a number of estimates. For example, we will have to rule out the
possibility that such points are separated from each other by a loop of
$\Gamma$ with diameter significantly larger than $2^{-j}$. Upon proving this,
the remainder of the proof of tightness follows along the proof of the
Kolmogorov-Centsov continuity criterion where the role of the dyadic
decomposition in the latter is played by the sets $P_{j}$ in the former.

In the statement of Theorem 1.1, it is important to note that the
normalization is by the median of
$\sup_{z,w\in\eta([0,1])}\mathfrak{d}_{\epsilon}(z,w;\Gamma_{+})$ as opposed
to the median length of a crossing which connects two disjoint boundary arcs.
In particular, due to this choice it is not obvious that a subsequential limit
is positive definite (the other properties of being a metric will at this
point be obvious due to how $\mathfrak{d}_{\epsilon}(\cdot,\cdot;\Gamma)$ is
defined). Establishing the positive definiteness will be the focus of Section
7. The first observation is that a subsequential limit
$\mathfrak{d}(\cdot,\cdot;\Gamma)$ cannot be completely degenerate (i.e.,
vanish) because this would contradict the comparability of the quantiles
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$ uniformly in $\epsilon>0$ which
we have discussed earlier. In particular, this implies that if we take two
disjoint arcs $I,J$ on $\partial D$ then it is a positive probability event
that the $\mathfrak{d}(I,J;\Gamma)>0$. On this event, let $K=\\{z\in
D:\mathfrak{d}(z,I;\Gamma)=0\\}$ and let $U$ be the component of $D\setminus
K^{*}$ with $J$ on its boundary. Then $\partial U\setminus\partial D$ has the
property that there cannot exist $z\in U$ and $w\in K$ with
$\mathfrak{d}(z,w;\Gamma)=0$ (for otherwise $z\in K$). Thus $D\cap\partial U$
can be thought of as a type of “shield” through which a path with
“$\mathfrak{d}(\cdot,\cdot;\Gamma)$-length zero” cannot pass. The thrust of
the argument in Section 7 is to use that such shields exist with positive
probability and independence to show that there a.s. exists a family of such
shields which separate $I$ from $J$ and therefore $\mathfrak{d}(I,J;\Gamma)>0$
a.s. Since $I$, $J$ were arbitrary boundary intervals, this will imply that
$\mathfrak{d}(\cdot,\cdot;\Gamma)$ is positive definite on $\partial D$.
Finally, using a CPI exploration, we will argue that if it is a positive
probability event that exist $z,w\in\Upsilon\setminus\partial D$ distinct with
$\mathfrak{d}(z,w;\Gamma)=0$ then it is also a positive probability event that
there exists $z,w\in\partial D$ distinct with $\mathfrak{d}(z,w;\Gamma)=0$.
Altogether, this gives the positive definiteness of
$\mathfrak{d}(\cdot,\cdot;\Gamma)$ everywhere. At the end of Section 7, we
will explain why the subsequential limit $\mathfrak{d}(\cdot,\cdot;\Gamma)$ is
a.s. geodesic and deduce the comparability of the quantiles defined in
infinite volume and finite volume settings.

In Appendix A, we will give an upper bound for the modulus of continuity of
the space-filling version of ${\rm SLE}$ from [26] when drawn on top of an
independent quantum disk and parameterized by quantum area. In Appendix B we
will collect some estimates for the natural measure on the carpet of a ${\rm
CLE}_{\kappa}$ which comes from LQG constructed in [33]. Finally, in Appendix
C, we will collect some estimates for Lévy processes.

###  1.4. Relationship with other work

Let us make some comments on the relationship between this work and some
recent works focused on the construction of the LQG metric. Recall that LQG
refers to the random two-dimensional Riemannian manifold whose metric tensor
is (formally) given by $e^{\gamma h(z)}(dx^{2}+dy^{2})$ where $h$ is an
instance of (some form of) the Gaussian free field (GFF) on a planar domain
$D$. Since $h$ is a distribution and not a function, this expression does not
make literal sense and requires interpretation. The rigorous construction of
the two-point distance function associated with LQG was first accomplished in
the case that $\gamma=\sqrt{8/3}$ in [28, 29, 30] and then subsequently (using
a different method) in [8, 13] for all $\gamma\in(0,2)$. In the program of
constructing a metric in the ${\rm CLE}_{\kappa}$ carpet for
$\kappa\in(8/3,4)$, the present work plays the role of [8] in the program of
constructing the LQG metric for $\gamma\in(0,2)$.

The argument that we will employ to establish tightness here will be rather
different to that given in [8], however. Let us briefly highlight a few of the
main differences. First of all, the heart of [8] is to consider a certain
mollification $h_{\epsilon}$ of the GFF $h$ and then to prove the (uniform in
$\epsilon$) comparability of quantiles for lengths of side-to-side crossings
in a box measured using the metric $e^{\xi h_{\epsilon}}$ where $\xi>0$ is
determined by $\gamma$. Establishing this comparability in [8] is based on
decomposing the larger box into smaller boxes and then using the Efron-Stein
inequality to get a recursive relationship in the quantiles for different
values of $\epsilon>0$. Once the comparability of side-to-side quantiles is
accomplished, a standard sort of Kolmogorov-Centsov argument makes it possible
to string together paths with length comparable to the median length crossing
and deduce the tightness of the diameter and indeed the whole metric.
Moreover, the positive definiteness of the subsequential limit is immediate
because if there were two points with zero distance in the limit then there
would have to be some box across which the quantiles are not all comparable.

In this work, we will instead make the choice of looking at quantiles for the
diameter of the whole boundary associated with our approximations. We will
then get comparability of the quantiles of the boundary diameter while at the
same time proving that the subsequential limit is continuous with respect to
the Euclidean topology. The argument that we will employ does not use the
Efron-Stein inequality or a recursion in order to compare the approximations
for different values of $\epsilon$. Moreover, stringing together short paths
in [8] follows the usual dyadic subdivision as in the proof of the Kolmogorov-
Centsov continuity criterion. In the present work, stringing together short
paths is rather delicate because of the presence of obstacles such as loops.
Finally, because we are normalizing our approximations by the median boundary
diameter, it is not immediate that the subsequential limit is positive
definite.

Next, we mention that the ${\rm CLE}_{8}$ metric was constructed in [14],
though the nature of the problem is rather different than for
$\kappa\in(8/3,8)$. Recall that ${\rm CLE}_{8}$ is the scaling limit of the
UST [20] so defining the metric in this case corresponds to properly assigning
a length to the continuous analog of the UST branches. These branches are
${\rm SLE}_{2}$-type curves [24, 26] and the appropriate notion of length to
consider is the so-called natural parameterization of ${\rm SLE}$ [21, 19, 2].

Finally, we mention the work [22] which studies random planar maps where the
face sizes have law which is in the domain of attraction of an $\alpha$-stable
distribution and it is shown that the resulting metric space properly
renormalized is tight in the Gromov-Hausdorff topology. The subsequentially
limiting metric constructed in [22] should correspond to a quantum version
(i.e., where the lengths of paths in the approximations are defined using a
quantity from LQG rather than a Euclidean quantity such as Lebesgue measure)
of our main result, but we will not focus on that in the present paper.

###  1.5. Notation

We will make use of the following notation in this article. We will write
$\mathbf{D}$ for the unit disk and $\mathbf{H}$ for the upper half-plane. If
$f,g$ are two functions, then we will write $f\lesssim g$ (resp. $f\gtrsim g$)
if there exists a constant $c>0$ so that $f\leq cg$ (resp. $f\geq cg$). For a
simply connected domain $D\subseteq\mathbf{C}$ and prime ends $a,b\in\partial
D$, we let $[a,b]_{\partial D}^{\bm{\circlearrowleft}}$ (resp.
$[a,b]_{\partial D}^{\bm{\circlearrowright}}$) denote the counterclockwise
(resp. clockwise) arc of prime ends in $\partial D$ from $a$ to $b$. For a set
$A$, we let $\mathrm{int}(A)$ (resp. $\mathrm{cl}(A)$) denote the interior
(resp. closure) of $A$.

### Acknowledgements

This research was supported by the ERC starting grant 804166 (SPRS). We thank
Wendelin Werner for helpful discussions related to this work. We also thank
Naotaka Kajino for helpful discussions on related work.

##  2\. Preliminaries

The purpose of this section is to collect a number of preliminaries. We will
start by reviewing some facts about ${\rm SLE}$ in Section 2.1. We will then
review ${\rm CLE}$ in Section 2.2. Next we will review some relevant results
from imaginary geometry [24, 26] in Section 2.3. Finally, we will review some
basics of Liouville quantum gravity (LQG) and its relationship to ${\rm SLE}$
in Section 2.4. We assume that the reader has some familiarity with these
objects and we will provide references for the various facts that we will use.

###  2.1. Schramm-Loewner evolution

The Schramm-Loewner evolution (${\rm SLE}_{\kappa}$) is a random fractal curve
in a simply connected domain and was introduced by Schramm in [39]. The
chordal version corresponds to a curve which connects two boundary points. In
the case that the domain is $\mathbf{H}$ and the curve goes from $0$ to
$\infty$, it is defined by solving the chordal Loewner equation

(2.1) $\partial_{t}g_{t}(z)=\frac{2}{g_{t}(z)-W_{t}},\quad g_{0}(z)=z$

where $W_{t}=\sqrt{\kappa}B_{t}$ and $B$ is a standard Brownian motion. Let
$\mathbf{H}_{t}$ be the domain of $g_{t}$ and
$K_{t}=\mathbf{H}\setminus\mathbf{H}_{t}$. In the case that $\kappa\neq 8$, it
was shown by Rohde-Schramm [38] that there a.s. exists a curve $\eta$ so that
$\mathbf{H}_{t}$ is equal to the unbounded component of
$\mathbf{H}\setminus\eta([0,t])$. The continuity in the case that $\kappa=8$
follows from the convergence of the uniform spanning tree Peano curve to ${\rm
SLE}_{8}$ established in [20]. If $D\subseteq\mathbf{C}$ is another simply
connected domain (i.e., not $\mathbf{H}$) and $x,y\in\partial D$ are distinct,
then an ${\rm SLE}_{\kappa}$ in $D$ from $x$ to $y$ is defined as
$\varphi(\eta)$ where $\eta$ is an ${\rm SLE}_{\kappa}$ in $\mathbf{H}$ from
$0$ to $\infty$ and $\varphi\colon\mathbf{H}\to\mathbf{D}$ is a conformal map
which takes $0$ (resp. $\infty$) to $x$ (resp. $y$).

The so-called ${\rm SLE}_{\kappa}(\underline{\rho})$ processes are an
important variant of ${\rm SLE}_{\kappa}$ in which one has extra marked
points, say at $x_{1},\ldots,x_{n}$, associated with weights
$\underline{\rho}=(\rho_{1},\ldots,\rho_{n})\in\mathbf{R}^{n}$. The $x_{j}$
for $1\leq j\leq n$ are sometimes called force points and the associated
weight $\rho_{j}$ determines how the curve interacts with $x_{j}$. In
particular, if $\rho_{j}$ is positive (resp. negative) then the ${\rm
SLE}_{\kappa}(\underline{\rho})$ is pushed away from (resp. pulled towards)
$x_{j}$. These processes were first introduced in [17] and are defined by
solving (2.1) where $W$ is taken to be the solution to the SDE

$\displaystyle
dW_{t}=\sqrt{\kappa}dB_{t}+\sum_{i=1}^{n}\frac{\rho_{i}}{W_{t}-V_{t}^{i}}dt,\quad
dV_{t}^{i}=\frac{2}{V_{t}^{i}-W_{t}}dt,\quad
V_{0}^{i}=x_{i}\quad\text{for}\quad i=1,\ldots,n.$

Up until the first time that $W$ collides with one of the $V^{i}$, the law of
an ${\rm SLE}_{\kappa}(\underline{\rho})$ is absolutely continuous with
respect to the law of an ${\rm SLE}_{\kappa}$ hence corresponds to a
continuous curve. More generally, the standard ${\rm
SLE}_{\kappa}(\underline{\rho})$ processes are defined up to the so-called
continuation threshold [24]: the first time $t$ that
$\sum_{i=1}^{n}\rho_{i}{\bf 1}_{W_{t}=V_{t}^{i}}\leq-2$. The continuity of the
${\rm SLE}_{\kappa}(\underline{\rho})$ processes up to the continuation
threshold was proved in [24]. One can also consider ${\rm SLE}_{\kappa}(\rho)$
processes with $-2-\kappa/2<\rho<-2$, though in this case their behavior is
rather than different. In particular, an ${\rm SLE}_{\kappa}(\rho)$ process
with $-2-\kappa/2<\rho<-2$ is self-intersecting for $\kappa\in(0,4]$ while
standard ${\rm SLE}_{\kappa}$ and ${\rm SLE}_{\kappa}(\rho)$ curves are
simple. The continuity in the case $-2-\kappa/2<\rho<-2$ was proved in [31,
27] in the presence of a single force point. We will sometimes use the
notation ${\rm SLE}_{\kappa}(\underline{\rho}_{L};\underline{\rho}_{R})$ to
indicate an ${\rm SLE}_{\kappa}(\underline{\rho})$ process where
$\underline{\rho}_{L}$ (resp. $\underline{\rho}_{R}$) indicates the weights of
the of the force points which are to the left (resp. right) of $0$. An ${\rm
SLE}_{\kappa}(\underline{\rho})$ connecting two boundary points in a simply
connected domain is defined as the conformal image of an ${\rm
SLE}_{\kappa}(\underline{\rho})$ in $\mathbf{H}$ from $0$ to $\infty$.

Let us mention some of the special ranges of $\rho$ values. Suppose that
$\eta$ is an ${\rm SLE}_{\kappa}(\underline{\rho}_{L};\underline{\rho}_{R})$
with force points located at $(\underline{x}_{L};\underline{x}_{R})$.

  * •

If $\sum_{i=1}^{k}\rho_{i,R}\geq\kappa/2-2$ then $\eta$ a.s. does not hit
$[x_{i,R},x_{i+1,R})$ (and likewise with $L$ in place of $R$).

  * •

If $\sum_{i=1}^{k}\rho_{i,R}\in(-2,\kappa/2-2)$ then $\eta$ can hit
$[x_{i,R},x_{i+1,R})$ (and likewise with $L$ in place of $R$).

  * •

In the case $\kappa^{\prime}>4$ and
$\sum_{i=1}^{k}\rho_{i,R}\in(-2,\kappa^{\prime}/2-4]$ then with positive
probability $\eta^{\prime}$ fills an interval of the form $[x_{i,R},b)$ for
$b\in(x_{i,R},x_{i+1,R})$ (and likewise with $L$ in place of $R$).

We can also consider whole-plane ${\rm SLE}_{\kappa}$ processes, which are
defined by solving the equation

$\partial_{t}g_{t}(z)=g_{t}(z)\frac{W_{t}+g_{t}(z)}{W_{t}-g_{t}(z)}.$

Here we take $W_{t}=e^{i\sqrt{\kappa}B_{t}}$ where $B_{t}$ is a two-sided
(i.e., defined for $t\in\mathbf{R}$) Brownian motion. For each time $t$,
$g_{t}$ is the unique conformal transformation from the unbounded component of
$\mathbf{C}\setminus\eta([0,t])$ to
$\mathbf{C}\setminus\mathrm{cl}(\mathbf{D})$ with positive derivative at
$\infty$. The whole-plane ${\rm SLE}_{\kappa}$ processes also correspond to a
continuous curve [38, 20] meaning there exists a continuous curve
$\eta\colon\mathbf{R}\to\mathbf{C}$ so that for each $t\in\mathbf{R}$ the
domain of $g_{t}$ is equal to the unbounded component of
$\mathbf{C}\setminus\eta((-\infty,t])$. There are also the whole-plane ${\rm
SLE}_{\kappa}(\rho)$ processes where $\rho>-2$ which are constructed by
replacing $W_{t}$ with the solution to the SDE (defined for all
$t\in\mathbf{R}$)

$\displaystyle dW_{t}$
$\displaystyle=\left(\frac{\rho}{2}\widetilde{\Psi}(O_{t},W_{t})-\frac{\kappa}{2}W_{t}\right)dt+i\sqrt{\kappa}W_{t}dB_{t},\quad
dO_{t}=\Psi(W_{t},O_{t})dt$

where

$\Psi(w,z)=-z\frac{z+w}{z-w}\quad\text{and}\quad\widetilde{\Psi}(z,w)=\frac{\Psi(z,w)+\Psi(1/\overline{z},w)}{2}.$

The continuity of the whole-plane ${\rm SLE}_{\kappa}(\rho)$ processes with
$\rho>-2$ was proved in [26].

###  2.2. Conformal loop ensembles

We are now going to give a brief review of the conformal loop ensembles (${\rm
CLE}_{\kappa}$) [40, 42]. We will focus on their construction using the so-
called boundary conformal loop ensembles ($\mathrm{BCLE}_{\kappa}$) which is
developed in [31] since this is the construction which will be most relevant
for this work. We will primarily focus on the case of the simple ${\rm
CLE}_{\kappa}$ (i.e., $\kappa\in(8/3,4)$).

Fix $\kappa\in(2,4)$ and $\rho\in(-2,\kappa-4)$. A
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\rho)$ (so that the loops
have a clockwise orientation) is defined as follows. Let $\eta$ be an ${\rm
SLE}_{\kappa}(\rho;\kappa-6-\rho)$ in $\mathbf{H}$ from $0$ to $\infty$ with
force points at $0^{-}$, $0^{+}$. Then the law of $\eta$ is _target
invariant_. This means that if $\varphi$ is a conformal transformation
$\mathbf{H}\to\mathbf{H}$ which fixes $0$ then $\varphi(\eta)$ and $\eta$,
viewed modulo time parameterization, have the same law up until first
disconnecting $\infty$ and $\varphi(\infty)$. Let $(x_{n})$ be a countable
dense set in $\mathbf{R}$ and, for each $n\in\mathbf{N}$, let $\eta_{n}$ be an
${\rm SLE}_{\kappa}(\rho;\kappa-6-\rho)$ in $\mathbf{H}$ from $0$ to $x$ with
force points at $0^{-}$, $0^{+}$. The target invariance implies that we can
couple the $\eta_{n}$ together so that any finite collection of them agree up
until their target points are disconnected and then afterwards evolve
independently. The family of paths $(\eta_{n})$ should be thought of as
branches of a tree which is rooted at $0$. As we will explain shortly, the
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\rho)$ in $\mathbf{H}$ is
defined from the $(\eta_{n})$ in a certain way;
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\rho)$ in domains other than
$\mathbf{H}$ are defined by applying a conformal transformation.

The way that the loops are defined from the family $(\eta_{n})$ is as follows.
Suppose that $\eta_{n}$ makes an excursion away from
$[0,x_{n}]_{\partial\mathbf{H}}^{\bm{\circlearrowleft}}$ in the time-interval
$[s,t]$. Then $\eta_{n}|_{[s,t]}$ corresponds to part of a loop. If
$x_{m}\in(\eta_{n}(s),\eta_{n}(t))$ then there exists $t^{\prime}>t$ so that
$\eta_{m}$ makes an excursion away from
$[0,x_{m}]_{\partial\mathbf{H}}^{\bm{\circlearrowleft}}$ in $[s,t^{\prime}]$
and $\eta_{n}|_{[s,t]}=\eta_{m}|_{[s,t]}$. Thus $\eta_{m}|_{[s,t^{\prime}]}$
corresponds to a larger portion of the same loop. The whole loop is obtained
by by taking a sequence $(x_{m_{j}})$ in
$[\eta_{n}(s),\eta_{n}(t)]_{\partial\mathbf{H}}^{\bm{\circlearrowleft}}$ which
decreases to $\eta_{n}(s)$ and considering the excursion of $\eta_{m_{j}}$
which contains $\eta_{n}|_{[s,t]}$. It was shown in [31] that the loops are
simple continuous curves and that the law of the collection of loops does not
depend on the root point (i.e., the origin). By considering the intervals
$[0,x_{n}]_{\partial\mathbf{H}}^{\bm{\circlearrowright}}$ in place of
$[0,x_{n}]_{\partial\mathbf{H}}^{\bm{\circlearrowleft}}$ one can define the
“false” loops of $\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\rho)$ and
note that these are surrounded counterclockwise.
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(\rho)$ is defined in the same
way except we take ${\rm SLE}_{\kappa}(\kappa-6-\rho;\rho)$ in place of ${\rm
SLE}_{\kappa}(\rho;\kappa-6-\rho)$ and the loops of a
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(\rho)$ correspond to the
complementary components which are surrounded counterclockwise.

There is a natural exploration path of a
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\rho)$ process $\Gamma$ on a
Jordan domain $D\subseteq\mathbf{C}$. Namely, let $\eta_{0}$ be a path which
traverse $\partial D$ counterclockwise. We then let $\eta$ be the path which
follows $\eta_{0}$ except whenever it hits a loop of $\Gamma$ for the first
time, it follows the loop in its entirety in the clockwise direction, and then
continues following $\eta_{0}$. The continuity of $\eta$ was proved in [31] as
a consequence of the continuity of space-filling ${\rm SLE}_{\kappa^{\prime}}$
established in [26]. If we take $\eta$ and target it at a fixed boundary point
(i.e., we parameterize $\eta$ by capacity as viewed from the given boundary
point), then we recover the branch of the ${\rm
SLE}_{\kappa}(\rho;\kappa-6-\rho)$ tree used to build the
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\rho)$. We can similarly
define an exploration path of a
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(\rho)$.

For $\kappa^{\prime}\in(4,8)$ and
$\rho\in(\kappa^{\prime}/2-4,\kappa^{\prime}/2-2)$ a
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}(\rho^{\prime})$ is
defined in an analogous manner to the case $\kappa\in(2,4)$. Moreover, one can
associate with a
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}(\rho^{\prime})$ an
exploration path which is defined in the same way as the case
$\kappa\in(2,4)$. When $\rho^{\prime}=0$ we will simply write
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}$. It turns out that
the loops of a $\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}$ have
the same law as the boundary touching loops of a ${\rm
CLE}_{\kappa^{\prime}}$.

Fix $\kappa\in(8/3,4)$ and let $\kappa^{\prime}=16/\kappa\in(4,6)$. The way
that the iterated $\mathrm{BCLE}$ construction works is as follows. Suppose
that we first sample a
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}$ process
$\Gamma^{\prime}$. Then in each of the complementary components which are
surrounded clockwise we sample a conditionally independent
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(-\kappa/2)$. This procedure is
then repeated in each of the components which are surrounded by a false (i.e.,
counterclockwise) loop of the original
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}$ or by a false
(i.e., clockwise) loop of one of the
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(-\kappa/2)$. The resulting
collection of $\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(-\kappa/2)$
loops (from all of the iterations) gives a ${\rm CLE}_{\kappa}$ process
$\Gamma$.

Let $\eta^{\prime}$ be the exploration path associated with the
$\Gamma^{\prime}$. Let $\eta$ be defined to be the path which follows
$\eta^{\prime}$ except whenever it hits a
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(-\kappa/2)$ loop for the first
time then it follows that loop counterclockwise. It is shown in [31, Theorem
7.4] that if we target $\eta$ at any fixed boundary point then we get an ${\rm
SLE}_{\kappa}(\kappa-6)$ process and recall that we know that $\eta^{\prime}$
targeted at any fixed boundary point is an ${\rm
SLE}_{\kappa^{\prime}}(\kappa^{\prime}-6)$ process. We call $\eta$ a
_conformal percolation interface_ (CPI) in $\Gamma$ and $\eta^{\prime}$ its
trunk. If we draw $\eta$ up to any stopping time in which it is not drawing a
loop of $\Gamma$, then the conditional law of the loops of $\Gamma$ in the
complementary components are conditionally independent ${\rm CLE}_{\kappa}$’s.
In order to emphasize that all of the loops that $\eta$ hits are to its right,
we will sometimes write $\eta\sim{\rm SLE}_{\kappa}^{1}(\kappa-6)$. By
replacing the $\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}$ and
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(-\kappa/2)$ with
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa^{\prime}}$ and
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(-\kappa/2)$ we can similarly
construct an ${\rm SLE}_{\kappa}^{-1}(\kappa-6)$ process meaning that the
${\rm CLE}_{\kappa}$ loops which it hits are always to its left.

What we have described above is a _totally asymmetric_ CPI because all of the
loops which are hit by $\eta$ lie to its right. We can also consider the
symmetric version which is constructed as follows. We first sample a
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}(\kappa^{\prime}/2-3)$
process $\Gamma^{\prime}$. In each complementary component which is surrounded
clockwise we sample a conditionally independent
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(\kappa/4-2)$ and in each
component which is surrounded counterclockwise we sample a conditionally
independent $\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\kappa/4-2)$.
This procedure is then iterated in each of the false loops of the
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(\kappa/4-2)$ and
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\kappa/4-2)$. The resulting
ensemble of loops is again a ${\rm CLE}_{\kappa}$ (see [31, Theorem 7.4,
Corollary 7.5]). Let $\eta^{\prime}$ be the exploration path associated with
$\Gamma^{\prime}$ and then let $\eta$ be the path which is obtained by
following $\eta^{\prime}$ and except it traverses each loop of a
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(\kappa/4-2)$ (resp.
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa}(\kappa/4-2)$) that it hits
counterclockwise (resp. clockwise). Then $\eta$ targeted at any fixed boundary
point has the law of an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process. Each time
that $\eta$ encounters a new loop of the ${\rm CLE}_{\kappa}$, it goes to its
left or right based on the toss of a fair coin flip.

We will not need to consider even more general versions of these explorations,
but we remark that one can get ${\rm SLE}_{\kappa}^{\beta}(\kappa-6)$ for
$\beta\in[-1,1]$ using other combinations of $\rho$ values in the above
construction properly (see [31, Theorem 7.4]). The formula which relates
$\beta$ and $\rho$ was computed in [33]. (The corresponding formula with the
roles of $\kappa$ and $\kappa^{\prime}$ reversed was computed in [34].)

###  2.3. Imaginary geometry

We are now going to collect some background from the theory of imaginary
geometry [24, 26] which will be relevant for the present work. Throughout, we
take the convention that $\kappa\in(0,4)$ and $\kappa^{\prime}=16/\kappa>4$.
We also let

$\chi=\frac{2}{\sqrt{\kappa}}-\frac{\sqrt{\kappa}}{2},\quad\lambda=\frac{\pi}{\sqrt{\kappa}},\quad\text{and}\quad\lambda^{\prime}=\frac{\pi}{\sqrt{\kappa^{\prime}}}.$

Suppose that we have weights $(\underline{\rho}_{L};\underline{\rho}_{R})$ and
force point locations $(\underline{x}_{L};\underline{x}_{R})$. Let $h$ be a
GFF on $\mathbf{H}$ with boundary conditions given by

$-\lambda\left(1+\sum_{i=1}^{k}\rho_{i,L}\right)\quad\text{in}\quad(x_{k+1,L},x_{k,L}]\quad\text{and}\quad\lambda\left(1+\sum_{i=1}^{k}\rho_{i,R}\right)\quad\text{in}\quad[x_{k,R},x_{k+1,R}).$

It is shown in [24] that an ${\rm
SLE}_{\kappa}(\underline{\rho}_{L};\underline{\rho}_{R})$ process with force
points located at $(\underline{x}_{L};\underline{x}_{R})$ can be coupled with
$h$ as a flow line of the formal vector field $e^{ih/\chi}$. This means that
if $(f_{t})$ is the centered Loewner flow for $\eta$ (i.e.,
$f_{t}=g_{t}-W_{t}$ where $(g_{t})$ is the Loewner flow) and $\tau$ is a
stopping time for $\eta$ then $h\circ
f_{\tau}^{-1}-\chi\arg(f_{\tau}^{-1})^{\prime}$ is a GFF on $\mathbf{H}$ with
boundary conditions given by

$-\lambda\left(1+\sum_{i=1}^{k}\rho_{i,L}\right)\quad\text{in}\quad(f_{\tau}(x_{k+1,L}),f_{\tau}(x_{k,L})]\quad\text{and}\quad\lambda\left(1+\sum_{i=1}^{k}\rho_{i,R}\right)\quad\text{in}\quad[f_{\tau}(x_{k,R}),f_{\tau}(x_{k+1,R})).$

Moreover, the boundary conditions on the image of the left (resp. right) side
of $\eta|_{[0,\tau]}$ are given by $-\lambda$ (resp. $\lambda$). Suppose that
$D\subseteq\mathbf{C}$ is a simply connected domain,
$\varphi\colon\mathbf{H}\to D$ is a conformal transformation, and
$\widetilde{h}=h\circ\varphi^{-1}-\chi\arg(\varphi^{-1})^{\prime}$. Then we
can similarly view $\varphi(\eta)$ as the flow line of
$e^{i\widetilde{h}/\chi}$. In the coupling of an ${\rm
SLE}_{\kappa}(\underline{\rho}_{L};\underline{\rho}_{R})$ as a flow line of a
GFF, we also have that the former is a.s. determined by the latter. Finally,
we can also define the flow line of a GFF $h$ with angle $\theta$ by
considering the flow line of $h+\theta\chi$.

We now turn to the case of ${\rm SLE}_{\kappa^{\prime}}$ processes. Suppose
that we have weights $(\underline{\rho}_{L};\underline{\rho}_{R})$ and force
point locations $(\underline{x}_{L};\underline{x}_{R})$. Let $h$ be a GFF on
$\mathbf{H}$ with boundary conditions

$\lambda^{\prime}\left(1+\sum_{i=1}^{k}\rho_{i,L}^{\prime}\right)\quad\text{in}\quad(x_{k+1,L},x_{k,L}]\quad\text{and}\quad-\lambda^{\prime}\left(1+\sum_{i=1}^{k}\rho_{i,R}^{\prime}\right)\quad\text{in}\quad[x_{k,R},x_{k+1,R}).$

Then we can view an ${\rm
SLE}_{\kappa^{\prime}}(\underline{\rho}_{L}^{\prime};\underline{\rho}_{R}^{\prime})$
process $\eta^{\prime}$ as a counterflow line of $h$ from $0$ to $\infty$.
This means that if $(f_{t})$ is the centered Loewner flow for $\eta^{\prime}$
and $\tau$ is a stopping time for $\eta^{\prime}$ then we have that $h\circ
f_{\tau}^{-1}-\chi\arg(f_{\tau}^{-1})^{\prime}$ is a GFF on $\mathbf{H}$ with
boundary conditions given by

$\lambda^{\prime}\left(1+\sum_{i=1}^{k}\rho_{i,L}^{\prime}\right)\quad\text{in}\quad(f_{\tau}(x_{k+1,L}),f_{\tau}(x_{k,L})]\quad\text{and}\quad-\lambda^{\prime}\left(1+\sum_{i=1}^{k}\rho_{i,R}^{\prime}\right)\quad\text{in}\quad[f_{\tau}(x_{k,R}),f_{\tau}(x_{k+1,R})).$

Moreover, the boundary conditions on the image of the left (resp. right) side
of $\eta|_{[0,\tau]}$ are given by $\lambda^{\prime}$ (resp.
$-\lambda^{\prime}$). As in the case of GFF flow lines, counterflow lines are
a.s. determined by the GFF and counterflow lines for GFFs defined on domains
other than $\mathbf{H}$ are defined by applying the same change of coordinates
formula as in the case of flow lines.

The discussion above implies that the iterated $\mathrm{BCLE}$ exploration is
naturally coupled with the GFF. We will explain this point in further detail
in Appendix A.

We will also have occasion to consider GFF flow lines which start from an
interior point, as developed in [26]. If we fix $\alpha>-\chi$ and we have a
whole-plane GFF with values modulo $2\pi(\chi+\alpha)$, then its flow line
starting from $0$ is a whole-plane ${\rm SLE}_{\kappa}(\rho)$ process with
$\rho=2-\kappa+2\pi\alpha/\lambda$. We can also consider flow lines of
different angles starting from the origin and talk about their conditional
laws.

Lastly, let us also mention the space-filling version of ${\rm SLE}$ which is
considered in [26]. It is constructed by fixing a countable dense set of
points, starting a flow line from each one, and then ordering the points
according to how the flow lines merge. It is shown in [26] that there is a
continuous path which visits these points in this order and this is space-
filling ${\rm SLE}$.

###  2.4. Liouville quantum gravity

A Liouville quantum gravity (LQG) surface is the random surface whose
Riemannian metric admits the formal expression:

$e^{\gamma h(z)}(dx^{2}+dy^{2})$

where $h$ is an instance of some form of the GFF and $\gamma\in(0,2)$ is a
parameter. This expression does not make literal sense because $h$ is a
distribution and not a function. The volume form was constructed in [10] (see
also the references therein). If $h$ is (some form of) the GFF with free
boundary conditions, then one can also make sense of a boundary length measure
by considering $e^{\gamma h(x)/2}dx$. The metric (i.e., two-point distance
function) for LQG was constructed first for $\gamma=\sqrt{8/3}$ in [28, 29,
30] and subsequently for all $\gamma\in(0,2)$ in [8, 13].

If $\varphi\colon D\to\widetilde{D}$ is a conformal transformation and we let

(2.2)
$\widetilde{h}=h\circ\varphi+Q\log|\varphi^{\prime}|\quad\text{where}\quad
Q=\frac{2}{\gamma}+\frac{\gamma}{2}$

then we have that $\mu_{\widetilde{h}}(\varphi(A))=\mu_{h}(A)$ for all Borel
sets $A\subseteq D$. We say that two fields $h$, $\widetilde{h}$ are
equivalent as quantum surfaces if they are related as in (2.2). A quantum
surface is defined as an equivalence class with respect to this equivalence
relation. More generally, we can consider a quantum surface with marked points
$x_{1},\ldots,x_{k}\in D$. Two quantum surfaces with marked points are
considered to be equivalent if the fields are related as in (2.2) where the
conformal map $\varphi$ takes the marked points for one surface to the marked
points for the other. We remark that it is not always obvious that two fields
define equivalent quantum surfaces.

We will always assume that the parameters are matched by
$\kappa=\gamma^{2}\in(0,4)$ and $\kappa^{\prime}=16/\kappa>4$. There is a way
to define the quantum length of an ${\rm SLE}_{\kappa}$ on a $\gamma$-LQG
surface which comes as a consequence of the main results of [41]. The main
results of [9] provide a way to define the quantum length (so-called quantum
natural time) of an ${\rm SLE}_{\kappa^{\prime}}$ process on a $\gamma$-LQG
surface.

There are a number of important types of LQG surfaces: quantum wedges, cones,
disks, and spheres. We will not give a precise definition of these objects
here because their exact definition will not matter for what follows, with the
exception of the technical estimates which are proved in Appendices B and A.
Briefly, a quantum wedge is an infinite volume surface which comes with two
marked points (the “origin” and “infinity”). We will denote the law of a
quantum wedge of weight $W>0$ as
$\mathsf{QWedge}_{\bm{\gamma}=\gamma}^{\mathbf{W}=W}$. Quantum wedges are
homeomorphic to $\mathbf{H}$ when $W\geq\tfrac{\gamma^{2}}{2}$ and in this
case we will use the notation $(\mathbf{H},h,0,\infty)$ to indicate a quantum
wedge which is parameterized by $\mathbf{H}$ with the origin point at $0$ and
the infinity point at $\infty$.

Following [33], we will refer to a sample from the law
$\mathsf{QWedge}_{\bm{\gamma}=\gamma}^{\mathbf{W}=2}$ as a _quantum half-
plane_. The reason for this terminology is that the quantum half-plane is the
LQG analog of the so-called Brownian half-plane [12]. It possesses the
following special property [41] which distinguishes it from other quantum
wedges. If $(\mathbf{H},h,0,\infty)$ is a quantum half-plane, $r>0$ is fixed,
and $x\in[0,\infty)$ is such that $\nu_{h}([0,x])=r$, then
$(\mathbf{H},h,x,\infty)$ is a quantum half-plane.

A quantum cone is an infinite volume surface which is homeomorphic to
$\mathbf{C}$ and also has two marked points (the “origin” and “infinity”). We
will denote the law of a quantum cone of weight $W>0$ as
$\mathsf{QCone}_{\bm{\gamma}=\gamma}^{\mathbf{W}=W}$. We will use the notation
$(\mathbf{C},h,0,\infty)$ to indicate a quantum cone parameterized by
$\mathbf{C}$ where the origin point is at $0$ and the infinity point is at
$\infty$. A quantum disk is a finite volume surface which is homeomorphic to
$\mathbf{D}$ and its definition naturally equips it with two marked boundary
points. We will also denote the law of a quantum disk with boundary length
$\ell$ using $\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$ and use the
notation $(D,h,x,y)$ to denote a quantum disk where the marked boundary points
are at $x,y$.

The following theorem is proved in [41] in the case of
$\mathsf{QWedge}_{\bm{\gamma}=\gamma}^{\mathbf{W}=4}$ and an ${\rm
SLE}_{\kappa}$ process and then generalized in [9] to the case of
$\mathsf{QWedge}_{\bm{\gamma}=\gamma}^{\mathbf{W}=W}$ for $W>0$ and ${\rm
SLE}_{\kappa}(\rho_{1};\rho_{2})$ for $\rho_{1},\rho_{2}>-2$.

######  Theorem 2.1.

Fix $W\geq\tfrac{\gamma^{2}}{2}$ and suppose that $W_{1},W_{2}>0$ are such
that $W=W_{1}+W_{2}$. Suppose that $\mathcal{W}=(\mathbf{H},h,0,\infty)$ has
law $\mathsf{QWedge}_{\bm{\gamma}=\gamma}^{\mathbf{W}=W}$. Let $\eta$ be an
independent ${\rm SLE}_{\kappa}(W_{1}-2;W_{2}-2)$ in $\mathbf{H}$ from $0$ to
$\infty$. Let $\mathcal{W}_{1}$ (resp. $\mathcal{W}_{2}$) be the quantum
surface parameterized by the component of $\mathbf{H}\setminus\eta$ which are
to the left (resp. right) of $\eta$. Then $\mathcal{W}_{1}$ (resp.
$\mathcal{W}_{2}$) has law
$\mathsf{QWedge}_{\bm{\gamma}=\gamma}^{\mathbf{W}=W_{1}}$ (resp.
$\mathsf{QWedge}_{\bm{\gamma}=\gamma}^{\mathbf{W}=W_{2}}$) and
$\mathcal{W}_{1},\mathcal{W}_{2}$ are independent.

This is proved in [9].

######  Theorem 2.2.

Fix $W>0$ and suppose that $\mathcal{C}=(\mathbf{C},h,0,\infty)$ has law
$\mathsf{QCone}_{\bm{\gamma}=\gamma}^{\mathbf{W}=W}$. Let $\eta$ be an
independent whole-plane ${\rm SLE}_{\kappa}(W-2)$ process in $\mathbf{C}$ from
$0$ to $\infty$. Then the quantum surface parameterized by
$\mathbf{C}\setminus\eta$ is a quantum wedge of weight $W$.

######  Remark 2.3.

As a consequence of Theorems 2.1, 2.2 we have the following fact. Suppose that
$\mathcal{C}=(\mathbf{C},h,0,\infty)$ has law
$\mathsf{QCone}_{\bm{\gamma}=\gamma}^{\mathbf{W}=4}$. Let $\eta_{-}$ be an
independent whole-plane ${\rm SLE}_{\kappa}(2)$ in $\mathbf{C}$ from $0$ to
$\infty$ and, given $\eta_{-}$, let $\eta_{+}$ be a chordal ${\rm
SLE}_{\kappa}$ in $\mathbf{C}\setminus\eta_{-}$ from $0$ to $\infty$. Let
$H_{+},H_{-}$ be the two components of
$\mathbf{C}\setminus(\eta_{-}\cup\eta_{+})$ and let
$\mathcal{H}_{\pm}=(H_{\pm},h,0,\infty)$ be the quantum surfaces parameterized
by $H_{\pm}$. Then $\mathcal{H}_{\pm}$ are independent quantum half-planes. We
note that if one considers the concatenation of the time-reversal of
$\eta_{-}$ with $\eta_{+}$ then one obtains a curve in the Riemann sphere from
$\infty$ to $\infty$ which in the literature is sometimes referred to as a
two-sided whole-plane ${\rm SLE}_{\kappa}$ from $\infty$ to $\infty$ through
$0$. One can also construct the law $\eta_{\pm}$ using the whole-plane GFF
using the machinery from [26]. Namely, let $h^{w}$ be a whole-plane GFF with
values modulo $2\pi(\chi+\sqrt{\kappa}/2)=4\pi/\sqrt{\kappa}$ and let
$h^{\mathrm{IG}}=h^{w}-\tfrac{\sqrt{\kappa}}{2}\arg(\cdot)$, also with values
defined modulo $4\pi/\sqrt{\kappa}$. Then the joint law of the pair
$(\eta_{-},\eta_{+})$ is equal to that of the flow lines of $h^{\mathrm{IG}}$
from $0$ to $\infty$ with angles $2\pi/(2-\kappa/2)$, $0$, respectively.

Using the invariance of the law of the quantum-plane under the operation of
shifting its origin point a given amount of quantum length to the left or
right, one also has the following. Suppose that $\eta_{+}$ is parameterized
according to quantum length and $r>0$. Then
$(H_{+},h|_{H_{+}},\eta_{+}(r),\infty)$,
$(H_{-},h|_{H_{-}},\eta_{+}(r),\infty)$ are independent quantum half-planes.
Therefore the quantum surface $(\mathbf{C},h,\eta_{+}(r),\infty)$ also has law
$\mathsf{QCone}_{\bm{\gamma}=\gamma}^{\mathbf{W}=4}$. Moreover, up to a
translation and rescaling the law of the paths $\eta_{+}|_{[r,\infty)}$ and
the concatenation of the time-reversal of $\eta_{+}|_{[0,r]}$ with $\eta_{-}$
is the same as the law of $(\eta_{-},\eta_{+})$.

The following is proved in [33].

######  Theorem 2.4.

Suppose that $\mathcal{H}=(\mathbf{H},h,0,\infty)$ is a quantum half-plane.
Fix $\beta\in[-1,1]$. Let $\eta$ be an independent ${\rm
SLE}_{\kappa}^{\beta}(\kappa-6)$ in $\mathbf{H}$ from $0$ to $\infty$ which is
subsequently parameterized by the quantum natural time of its trunk. Then for
each time $t$, the quantum surface $\mathcal{H}_{t}$ parameterized by the
unbounded component of $\mathbf{H}\setminus\eta([0,t])$ is a quantum half-
plane which is independent of the quantum surface ${\mathcal{K}}_{t}$
parameterized by its complement in $\mathbf{H}$ and decorated by
$\eta|_{[0,t]}$. This holds more generally if we replace $t$ by a stopping
time of the filtration $\mathcal{F}_{t}=\sigma({\mathcal{K}}_{s}:s\leq t)$.
Finally, let $L_{t}$ denote the quantum length of the part of
$\mathbf{H}\cap\partial{\mathcal{K}}_{t}$ which is to the left of $\eta(t)$
minus the quantum length of the part of
$\partial\mathbf{H}\cap\partial{\mathcal{K}}_{t}$ which is to the left of $0$.
Define $R_{t}$ analogously with right in place of left. Then the pair
$(L_{t},R_{t})$ evolve as independent $4/\kappa$-stable Lévy processes.

In the statement of Theorem 2.4, the positivity parameter (which determines
the relative intensity of upward vs downward jumps) and relative overall
intensity of jumps for $L_{t}$ and $R_{t}$ depend on $\beta$ and were
determined in [33]. We will have some further discussion of this in Appendix
C.

##  3\. Crossing estimates

The purpose of this section is to start to set up the percolation argument
that will be used in order to establish Theorem 1.1. In Section 3.1, we will
describe the setup that we will use throughout this section as well as give
the main two statements. The first main statement (Lemma 3.1) gives that it is
very likely that a quantum surface $\mathcal{N}$ cut out of a quantum half-
plane by an independent ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process run until
the first time it hits the boundary after time $\delta^{4/\kappa}$ contains
paths $\omega$ in the ${\rm CLE}_{\kappa}$ carpet with
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$
(as defined in Section 1.3; we will also recall the definition of
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$ just below) which connect the
bottom right corner of $\mathcal{N}$ to any sufficiently large interval on the
top of $\mathcal{N}$ which intersects the ${\rm CLE}_{\kappa}$ carpet and
otherwise stays away from $\partial\mathcal{N}$, provided $\delta>0$ is
sufficiently small. The second main statement (Lemma 3.2) is of a similar
flavor except with two chunks of quantum surface cut out by successive ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ processes. In order to read the rest of this
article, one only needs these two main statements, so the rest of the section
can be skipped on a first reading. In Section 3.2, we will show that a CPI in
a ${\rm CLE}_{\kappa}$ carpet $\Upsilon$ has positive conditional probability
given $\Upsilon$ of being close to any fixed path in $\Upsilon$. The focus of
Section 3.3 is to prove an intermediate result for the likelihood of such a
crossing $\omega$ in $\Upsilon$ as above but in a slightly different setting.
In Section 3.4 we will use this intermediate result in order to complete the
proofs of the two main statements.

###  3.1. Setup and main statements

Throughout this section, we assume that we have the following setup. We assume
that $\mathcal{C}=(\mathbf{C},h,0,\infty)$ has law
$\mathsf{QCone}_{\bm{\gamma}=\gamma}^{\mathbf{W}=4}$ with the circle average
embedding. We let $h^{w}$ be a whole-plane GFF with values modulo
$2\pi(\chi+\sqrt{\kappa}/2)=4\pi/\sqrt{\kappa}$ and set
$h^{\mathrm{IG}}=h^{w}-\tfrac{\sqrt{\kappa}}{2}\arg(\cdot)$, also with values
modulo $4\pi/\sqrt{\kappa}$. We assume that $h^{\mathrm{IG}}$ is independent
of $\mathcal{C}$. We let $\eta_{-}$, $\eta_{+}$ be the flow lines of
$h^{\mathrm{IG}}$ from $0$ to $\infty$ with angles $2\pi/(2-\kappa/2)$, $0$,
respectively. Then $\eta_{\pm}$ are each whole-plane ${\rm SLE}_{\kappa}(2)$
processes in $\mathbf{C}$ from $0$ to $\infty$ [26, Theorem 1.4]. Moreover,
the conditional law of $\eta_{-}$ (resp. $\eta_{+}$) given $\eta_{+}$ (resp.
$\eta_{-}$) is that of a chordal ${\rm SLE}_{\kappa}$ process in
$\mathbf{C}\setminus\eta_{+}$ (resp. $\mathbf{C}\setminus\eta_{-}$) from $0$
to $\infty$. That is, the concatenation of the time-reversal of $\eta_{-}$
together with $\eta_{+}$ is a two-sided whole-plane ${\rm SLE}_{\kappa}$
process from $\infty$ to $\infty$ through $0$ and the same is true with the
roles of $\eta_{\pm}$ swapped. We assume that $\eta_{\pm}$ are parameterized
according to quantum length according to $h$. Let $\mathcal{H}_{+}$ (resp.
$\mathcal{H}_{-}$) be the quantum surface parameterized by the component of
$\mathbf{C}\setminus(\eta_{-}\cup\eta_{+})$ part of whose boundary consists of
the left side of $\eta_{+}$. By Theorems 2.1, 2.2 we have that
$\mathcal{H}_{\pm}$ are independent quantum half-planes, which we assume to be
marked by $0$ and $\infty$. Let $\Gamma_{+}$ be a ${\rm CLE}_{\kappa}$ in
$\mathcal{H}_{+}$ and let $\Upsilon_{+}$ be the carpet of $\Gamma_{+}$. As in
Section 1.3, for each $\epsilon>0$ and $p\in(0,1)$ we let
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$ denote the $p$th quantile of
$\sup_{z,w\in\eta_{+}([0,1])}\mathfrak{d}_{\epsilon}(z,w;\Gamma_{+})$ and let
${\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}={{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(1/2)$

Figure 3.1.  Illustration of the setup and statement of Lemma 3.1 (left) and
Lemma 3.2 (right). Not shown are the loops of $\Gamma$, which are in
particular dense in $\mathcal{H}_{+}\cap\partial\mathcal{N}$,
$\mathcal{H}_{+}\cap\partial\mathcal{N}_{0}$, and
$\mathcal{H}_{+}\cap\partial\mathcal{N}_{1}$.

######  Lemma 3.1.

For each $p_{0},\zeta_{0},p\in(0,1)$ there exists $\xi_{0},\delta_{0}>0$ so
that for all $\delta\in(0,\delta_{0})$ there exists $\epsilon_{0}>0$ so that
for all $\epsilon\in(0,\epsilon_{0})$ the following is true. Suppose that
$\eta$ is an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ in $\mathcal{H}_{+}$ from $0$
to $\infty$ which is coupled with $\Gamma_{+}$ as a CPI and parameterized by
the quantum natural time of its trunk. Let
$\sigma=\inf\\{t\geq\delta^{4/\kappa}:\eta(t)\in\partial\mathcal{H}_{+}\\}$
and let $\mathcal{N}$ be the quantum surface disconnected from $\infty$ by
$\eta|_{[0,\sigma]}$. Let $d_{0}=\mathrm{diam}(\eta([0,\sigma]))$ and let
$z_{R}$ be the rightmost point on the bottom of $\mathcal{N}$. Let $G$ be the
event that for each interval $I$ on the top of $\partial\mathcal{N}$ with
quantum length at least $\zeta_{0}\delta$ and which intersects $\Upsilon_{+}$
there is a path $\omega$ in $\Upsilon_{+}\cap\mathcal{N}$ which

  1. (i)

connects $\partial B(z_{R},\zeta_{0}d_{0})$ to $\widetilde{I}$ where
$\widetilde{I}$ is the interval on the top of $\partial\mathcal{N}$ with the
same center as $I$ but with twice the quantum length,

  2. (ii)

$\mathrm{dist}(\omega,\partial\mathcal{N}\setminus\widetilde{I})\geq\xi_{0}d_{0}$,
and

  3. (iii)

satisfies
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.

Then $\mathbf{P}[G]\geq p_{0}$.

######  Lemma 3.2.

For each $p_{0},\zeta_{0},p\in(0,1)$ there exists $\xi_{0},\delta_{0}>0$ so
that for all $\delta\in(0,\delta_{0})$ there exists $\epsilon_{0}>0$ so that
for all $\epsilon\in(0,\epsilon_{0})$ the following is true. Suppose that
$\eta_{0}$ is an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ in $\mathcal{H}_{+}$ from
$0$ to $\infty$ which is coupled with $\Gamma_{+}$ as a CPI and parameterized
by the quantum natural time of its trunk. Let
$\sigma_{0}=\inf\\{t\geq\delta^{4/\kappa}:\eta_{0}(t)\in\partial\mathcal{H}_{+}\\}$.
Let $\mathcal{N}_{0}$ be the quantum surface disconnected from $\infty$ by
$\eta_{0}|_{[0,\sigma_{0}]}$. Let $\eta_{1}$ be an ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ process in
$\mathcal{H}_{+,1}=\mathcal{H}_{+}\setminus\mathcal{N}_{0}$ starting from a
uniformly random point chosen from the set of points on the top
$\mathcal{H}_{+}\cap\partial\mathcal{N}_{0}$ of $\partial\mathcal{N}_{0}$
whose clockwise boundary length distance to $\partial\mathcal{H}_{+}$ is an
integer multiple of $\zeta_{0}\delta$ and are within boundary length distance
$\zeta_{0}\delta$ of a point in $\Upsilon_{+}\cap\partial\mathcal{N}_{0}$ and
coupled with the loops of $\Gamma_{+}$ contained in $\mathcal{H}_{+,1}$ as a
CPI. We assume that $\eta_{1}$ is parameterized according to the quantum
natural time of its trunk. Let
$\sigma_{1}=\inf\\{t\geq\delta^{4/\kappa}:\eta_{1}(t)\in\partial\mathcal{H}_{+,1}\\}$.
Let $d_{0}=\mathrm{diam}(\eta_{0}([0,\sigma_{0}]))$ and let $z_{R}$ be the
rightmost point on the bottom of $\mathcal{N}_{0}$. Let $\mathcal{N}$ be the
quantum surface parameterized by
$\mathrm{int}(\mathrm{cl}(\mathcal{N}_{0}\cup\mathcal{N}_{1}))$. Let $G$ be
the event that for each interval $I$ on the top of $\partial\mathcal{N}_{1}$
with quantum length at least $\zeta_{0}\delta$ and which intersects
$\Upsilon_{+}$ there is a path $\omega$ in $\Upsilon_{+}\cap\mathcal{N}$ which

  1. (i)

connects $\partial B(z_{R},\zeta_{0}d_{0})$ to $\widetilde{I}$ where
$\widetilde{I}$ is the interval on the top of $\partial\mathcal{N}$ with the
same center as $I$ but with twice the quantum length,

  2. (ii)

$\mathrm{dist}(\omega,\partial\mathcal{N}\setminus\widetilde{I})\geq\xi_{0}d_{0}$,
and

  3. (iii)

satisfies
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.

Then $\mathbf{P}[G]\geq p_{0}$.

Let us now explain the main steps used to prove Lemmas 3.1 and 3.2.

  1. Step 1.

Show that a CPI in $\Gamma_{+}$ has a positive chance given $\Upsilon_{+}$ of
being close to any fixed path in $\Upsilon_{+}$ (Proposition 3.3). This result
will be used several other times in this article. Its proof does not need to
be read in order to understand the proofs of Lemmas 3.1 and 3.2 or the rest of
the article.

  2. Step 2.

We will construct a domain in $\mathcal{H}_{+}$ using a pair of CPIs of
$\Gamma_{+}$ whose boundary has positive distance from
$\partial\mathcal{H}_{+}$. We will argue that if we make this domain
sufficiently small then it is likely to contain a crossing $\omega$ in
$\Upsilon_{+}$ satisfying
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.
As we mentioned in Section 1.3, the key point in the proof where we define the
quantiles using
$\sup_{z,w\in\eta_{+}([0,1])}\mathfrak{d}_{\epsilon}(z,w;\Gamma_{+})$ (rather
than using the length of a crossing in $\Upsilon_{+}$ connecting two disjoint
boundary arcs) will be contained in this step and it is used to show that the
probability that the (Euclidean) diameter of the set of points
$z\in\Upsilon_{+}$ with
$\mathfrak{d}_{\epsilon}(0,z;\Gamma_{+})\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$
is at least $\delta>0$ tends to $1$ as $\delta\to 0$ provided $\epsilon>0$ is
sufficiently small relative to $\delta$.

  3. Step 3.

Complete the proofs of Lemmas 3.1 and 3.2. In the case of Lemma 3.1, we will
use Proposition 3.3 to argue that given $\mathcal{N}$ there is a positive
chance that some additional CPIs construct a domain as in Step 2 which
contains a crossing $\omega$ in $\Upsilon_{+}$ satisfying
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$
where the shape of the auxiliary domain is such that the crossing is forced to
connect $\partial B(z_{R},\zeta_{0}d_{0})$ to $\widetilde{I}$ while staying at
distance at least $\xi_{0}d_{0}$ from
$\partial\mathcal{N}\setminus\widetilde{I}$. The proof of Lemma 3.2 is the
same except we want the crossing to traverse
$\mathrm{int}(\mathrm{cl}(\mathcal{N}_{0}\cup\mathcal{N}_{1}))$ instead of a
single ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunk so we will not explain the
argument in this case in detail.

###  3.2. A CPI has a positive chance of being close to any path in the carpet

In [32] it was proved that a CPI inside of a ${\rm CLE}_{\kappa}$ carpet
$\Upsilon$ is a.s. not determined by $\Upsilon$. That is, the conditional law
of a CPI in $\Upsilon$ given $\Upsilon$ is a.s. non-atomic. In what follows,
we will upgrade this statement to show that if $\omega$ is any continuous path
in $\Upsilon$ connecting distinct boundary points then a CPI connecting the
same boundary points has positive conditional probability given $\Upsilon$ of
staying in the $\epsilon$-neighborhood of $\omega$. We will phrase the
following statement for a deterministic path $\omega$ but by considering the
finite collection of piecewise polygonal paths which connect the vertices in
$(\epsilon\mathbf{Z}^{2})\cap\mathbf{D}$ the same statement a.s. holds for all
$\omega$ simultaneously.

######  Proposition 3.3.

Let $\omega\colon[0,1]\to\mathbf{D}$ be a continuous curve with $\omega(0)=-i$
and $\omega(1)=i$. Fix $\epsilon>0$, let $\omega^{\epsilon}$ be the
$\epsilon$-neighborhood of $\omega$, and let
$K=\mathbf{D}\setminus\omega^{\epsilon}$. Suppose that $\Gamma$ is a ${\rm
CLE}_{\kappa}$ on $\mathbf{D}$, $\Upsilon$ its carpet, and let $K^{*}$ be the
closure of the union of $K$ and the loops of $\Gamma$ which intersect $K$. Let
$\eta$ be an ${\rm SLE}_{\kappa}^{1}(\kappa-6)$ in $\mathbf{D}$ from $-i$ to
$i$ coupled with $\Gamma$ as a CPI and let $\eta^{\prime}$ be its trunk. Let
$E$ be the event that $K^{*}$ does not disconnect $-i$ from $i$ in
$\omega^{\epsilon}$. Then

$\mathbf{P}[\eta^{\prime}\subseteq\omega^{\epsilon}\,|\,\Upsilon]>0\quad\text{a.s.\
on}\quad E.$

In order to complete the proof of Proposition 3.3 we need to recall a few
things from [32]. Suppose that $\Gamma^{\prime}$ is a ${\rm
CLE}_{\kappa^{\prime}}$ on $\mathbf{D}$. Let $\eta^{\prime}$ be the branch of
the exploration tree of $\Gamma^{\prime}$ from $-i$ to $i$ and let
$\overline{\eta}^{\prime}$ be its time-reversal. Suppose that $t>0$. On the
event that $\eta^{\prime}|_{[0,t]}$ has not yet reached $i$, we have that
$\eta^{\prime}$ is a.s. exploring some loop
$\mathcal{L}^{\prime}\in\Gamma^{\prime}$ at the time $t$. Let $t_{0}$ be such
that $[t_{0},t]$ is the interval of time in which $\eta^{\prime}|_{[0,t]}$ is
exploring $\mathcal{L}^{\prime}$. Suppose that $\overline{t}>0$ and we are on
the event that $\overline{\eta}^{\prime}|_{[0,\overline{t}]}$ has not hit
$\eta^{\prime}([0,t])$. Then $\overline{\eta}^{\prime}$ is a.s. exploring some
loop $\overline{\mathcal{L}}^{\prime}\in\Gamma^{\prime}$ at the time
$\overline{t}$. Let $\overline{t}_{0}$ be such that
$[\overline{t}_{0},\overline{t}]$ is the interval of time in which
$\overline{\eta}^{\prime}|_{[0,\overline{t}]}$ is exploring
$\overline{\mathcal{L}}^{\prime}$. Let $D_{t,\overline{t}}$ be the component
of
$\mathbf{D}\setminus(\eta^{\prime}([0,t])\cup\overline{\eta}^{\prime}([0,\overline{t}]))$
with $\eta^{\prime}(t)$, $\overline{\eta}^{\prime}(\overline{t})$ on its
boundary. Then it is shown in [32, Lemma 3.1] that the conditional law of the
part of $\Gamma^{\prime}$ contained in $D_{t,\overline{t}}$ is a conformally
invariant function of $D_{t,\overline{t}}$ and the marked points
$\eta^{\prime}(t)$, $\eta^{\prime}(t_{0})$,
$\overline{\eta}^{\prime}(\overline{t})$,
$\overline{\eta}^{\prime}(\overline{t}_{0})$. Moreover, let
$\eta_{0}^{\prime}$ be equal to $\eta^{\prime}$ starting from
$\eta^{\prime}(t)$, $\eta_{1}^{\prime}$ the time-reversal of
$\eta_{0}^{\prime}$ starting from $\eta^{\prime}(t_{0})$,
$\overline{\eta}_{0}^{\prime}$ be equal to $\overline{\eta}^{\prime}$ starting
from $\overline{\eta}^{\prime}(\overline{t})$, and
$\overline{\eta}_{1}^{\prime}$ the time-reversal of
$\overline{\eta}_{0}^{\prime}$ starting from
$\overline{\eta}^{\prime}(\overline{t}_{0})$. Then $\eta_{0}^{\prime}$ can
terminate at either $\eta^{\prime}(t_{0})$, in which case
$\mathcal{L}^{\prime}\neq\overline{\mathcal{L}}^{\prime}$, or at
$\overline{\eta}^{\prime}(\overline{t})$, in which case
$\mathcal{L}^{\prime}=\overline{\mathcal{L}}^{\prime}$. (The probability of
the two ways of the paths hooking up was subsequently computed explicitly in
[35].) Then it is shown in [32, Section 4.1] that conditionally on the ranges
of $\eta_{0}^{\prime}$, $\eta_{1}^{\prime}$, $\overline{\eta}_{0}^{\prime}$,
$\overline{\eta}_{1}^{\prime}$ and the event
$\mathcal{L}^{\prime}\cap\overline{\mathcal{L}}^{\prime}\neq\emptyset$, either
possibility has positive probability.

We recall that there is a natural exploration path $\Lambda$ associated with
the iterated $\mathrm{BCLE}$ construction of a ${\rm CLE}_{\kappa}$ [31] which
we will now describe. For simplicity, we will assume that the ${\rm
CLE}_{\kappa}$ is in $\mathbf{D}$. We first let $\Lambda_{0}$ be given by the
path which follows $\partial\mathbf{D}$ counterclockwise starting from $-i$.
Let $\Lambda_{1}$ be given by $\Lambda_{0}$ except whenever $\Lambda_{0}$ hits
a $\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}(0)$ loop then it
follows it clockwise in its entirety before continuing along $\Lambda_{0}$. We
then let $\Lambda_{2}$ be given by $\Lambda_{1}$ except whenever it hits a
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(-\kappa/2)$ loop it follows it
counterclockwise in its entirety before continuing along $\Lambda_{1}$. We
then continue iterating this construction in the false loops of the
$\mathrm{BCLE}^{\bm{\circlearrowright}}_{\kappa^{\prime}}(0)$ and of the
$\mathrm{BCLE}^{\bm{\circlearrowleft}}_{\kappa}(-\kappa/2)$’s to obtain a path
which follows the entire iterated $\mathrm{BCLE}$ construction of the ${\rm
CLE}_{\kappa}$ instance. It is explained in [31] that this a.s. defines a
continuous path, which we call $\Lambda$. We note that if we take $\Lambda$
and target it at any point in $\partial\mathbf{D}$ then we obtain a CPI in
$\Gamma$ starting from $-i$ (and which has the law of an ${\rm
SLE}_{\kappa}^{1}(\kappa-6)$ process).

Finally, let us recall that a trunk $\eta^{\prime}$ of an ${\rm
SLE}_{\kappa}^{1}(\kappa-6)$ process in $\mathbf{H}$ from $0$ to $\infty$ is
an ${\rm SLE}_{\kappa^{\prime}}(\kappa^{\prime}-6)$ process where the force
point is at $0^{+}$. Moreover, by [24, Theorem 1.4] (see also [36, Figure
2.5]) the law of the left (resp. right) boundary of the trunk is that of an
${\rm SLE}_{\kappa}(\kappa-4;2-\kappa)$ (resp. ${\rm
SLE}_{\kappa}(\kappa/2-2;-\kappa/2)$) process. In particular, the right
boundary a.s. does not hit $(-\infty,0)$ since $\kappa/2-2$ is the critical
value at or above which an ${\rm SLE}_{\kappa}(\rho)$ process does not hit the
boundary. On the other hand, since $\kappa\in(8/3,4)$ we have that
$2-\kappa<\kappa/2-2$ so that the left boundary a.s. does hit $(0,\infty)$.
Finally, $\eta^{\prime}$ hits the points in its left (resp. right) boundary in
order so can be decomposed into a collection of it excursions from the left
(resp. right) boundary.

###### Proof of Proposition 3.3.

We consider the following Markovian resampling operation which preserves the
law of $\Lambda$. Let $z\in\mathbf{D}$ and $U\in[0,1]$ be independently
sampled from Lebesgue measure and independently of $\Lambda$. For each $t\geq
0$, we let $D_{t}$ be the component of $\mathbf{D}\setminus\Lambda([0,t])$
which contains $z$. We let $\tau_{0}$ be the first time $t$ that the conformal
radius ${\rm CR}(z,D_{t})$ of $D_{t}$ as seen from $z$ is at most $U$ or
$\partial D_{t}$ is a loop of $\Gamma$. In the latter case we set
$\tau=\tau_{0}$. In the former case, if $\Lambda$ is not drawing a ${\rm
CLE}_{\kappa}$ loop at time $\tau_{0}$ then we also set $\tau=\tau_{0}$. If
$\Lambda$ is drawing a ${\rm CLE}_{\kappa}$ loop at time $\tau_{0}$, we let
$\tau$ be the time after $\tau_{0}$ that $\Lambda$ completes the ${\rm
CLE}_{\kappa}$ loop. Suppose that we are working on the event that $\partial
D_{\tau}$ is not a loop of $\Gamma$. Let $\varphi\colon D_{\tau}\to\mathbf{D}$
be the unique conformal map with $\varphi(z)=0$ and
$\varphi(\Lambda(\tau))=-i$. Let $\sigma>\tau$ be such that $[\tau,\sigma]$ is
the interval of time in which $\Lambda$ is in $D_{\tau}$. By the definition of
$\Lambda$ and the iterated $\mathrm{BCLE}$ construction, modulo time
parameterization we have that $\varphi(\Lambda|_{[\tau,\sigma]})$ has the same
law as $\Lambda$. Let $\overline{x}\in\partial\mathbf{D}$ be sampled from
Lebesgue measure and $t,\overline{t}\sim\exp(1)$ independently of each other
and everything else. Let $\eta_{0}^{\prime}$, $\eta_{1}^{\prime}$,
$\overline{\eta}_{0}^{\prime}$, $\overline{\eta}_{1}^{\prime}$ be the
$4$-tuple of paths as described above defined in terms of the branch of the
exploration tree of the ${\rm CLE}_{\kappa^{\prime}}$ in $\mathbf{D}$ from
$-i$ to $\overline{x}$ described by $\varphi(\Lambda|_{[\tau,\sigma]})$ and
the times $t$, $\overline{t}$. We then resample $\eta_{0}^{\prime}$,
$\eta_{1}^{\prime}$, $\overline{\eta}_{0}^{\prime}$,
$\overline{\eta}_{1}^{\prime}$ from their conditional law given their range.
As we have explained above, by [32, Section 4.1] we have that on the event
that the ${\rm CLE}_{\kappa^{\prime}}$ loops described by $\eta_{0}^{\prime}$,
$\eta_{1}^{\prime}$, $\overline{\eta}_{0}^{\prime}$,
$\overline{\eta}_{1}^{\prime}$ intersect it is a positive conditional
probability event that after resampling $\eta_{0}^{\prime}$,
$\overline{\eta}_{0}^{\prime}$ describe part of the same ${\rm
CLE}_{\kappa^{\prime}}$ loop or a different ${\rm CLE}_{\kappa^{\prime}}$
loop.

Suppose that $N$ is sampled from the geometric distribution with parameter
$1/2$ independently of everything else. In the remainder of the proof, we will
show that if we run this Markov resampling step $N$ times then on the event
$E$ from the statement it is a positive probability event given $\Upsilon$
that a CPI in $\Upsilon$ from $-i$ to $i$ reaches $i$ before leaving
$\omega^{\epsilon}$. Upon showing this, the proof of the lemma will be
complete.

We assume that we are working on $E$. Let $V$ be the component of
$\mathbf{D}\setminus K^{*}$ with $-i,i$ on its boundary, $\Gamma_{V}$ the
loops of $\Gamma$ contained in $V$, and let
$I_{-}=[a_{-},b_{-}]_{\partial\mathbf{D}}^{\bm{\circlearrowleft}}$ (resp.
$I_{+}=[a_{+},b_{+}]_{\partial\mathbf{D}}^{\bm{\circlearrowright}}$) be the
arc of $\partial\mathbf{D}\cap\partial V$ which contains $-i$ (resp. $i$). Fix
$x_{-}$ (resp. $x_{+}$) in the interior of $I_{-}$ (resp. $I_{+}$). Let
$\Gamma_{V}^{\prime}$ be a ${\rm CLE}_{\kappa^{\prime}}$ in $V$ coupled with
$\Gamma_{V}$ as in the $\mathrm{BCLE}$ construction of ${\rm CLE}$. Let
$\eta^{\prime}$ be the branch of the exploration tree of $\Gamma_{V}^{\prime}$
from $x_{-}$ to $x_{+}$. Then $\eta^{\prime}$ is the trunk of an ${\rm
SLE}_{\kappa}^{1}(\kappa-6)$ process from $x_{-}$ to $x_{+}$ in $V$ coupled
with $\Gamma_{V}$ as a CPI. As explained above, the right boundary of
$\eta^{\prime}$ a.s. does not hit $(x_{-},x_{+})_{\partial
V}^{\bm{\circlearrowright}}$. Let
$x_{-}^{\prime}\in(a_{-},x_{-})_{\partial\mathbf{D}}^{\bm{\circlearrowleft}}$
and
$x_{+}^{\prime}\in(a_{+},x_{+})_{\partial\mathbf{D}}^{\bm{\circlearrowright}}$.
Let $n$ be the number of excursions that $\eta^{\prime}$ makes from its right
boundary which intersect $[x_{-}^{\prime},x_{+}^{\prime}]_{\partial
V}^{\bm{\circlearrowright}}$ and note that $n$ is a.s. finite by the
continuity of $\eta^{\prime}$.

More generally, let $\eta_{j}$, $\eta_{j}^{\prime}$ be the paths after
performing the resampling step $j$ times and let $n_{j}$ be the number of
excursions that $\eta_{j}^{\prime}$ makes from its right boundary which
intersect $[x_{-}^{\prime},x_{+}^{\prime}]_{\partial
V}^{\bm{\circlearrowright}}$. Let us assume that $n\geq 1$ and let
$\eta_{0,1}^{\prime},\ldots,\eta_{0,n}^{\prime}$ be the $n$ excursions that
$\eta^{\prime}=\eta_{0}^{\prime}$ makes from its right boundary which hit
$[x_{-}^{\prime},x_{+}^{\prime}]_{\partial V}^{\bm{\circlearrowright}}$
ordered chronologically. We assume that $\eta_{0,1}^{\prime}$ starts from
where $\eta^{\prime}$ first starts drawing it and let
$\overline{\eta}_{0,1}^{\prime}$ be its time-reversal. Let $\tau_{0,1}$ (resp.
$\overline{\tau}_{0,1}$) be the first time that $\eta_{0,1}^{\prime}$ (resp.
$\overline{\eta}_{0,1}^{\prime}$) hits
$[x_{-}^{\prime},x_{+}^{\prime}]_{\partial V}^{\bm{\circlearrowright}}$ and
let $W$ be a component of
$V\setminus(\eta_{0,1}^{\prime}([0,\tau_{0,1}])\cup\overline{\eta}_{0,1}^{\prime}([0,\overline{\tau}_{0,1}]))$
whose boundary is contained in and has non-empty intersection with both
$\eta_{0,1}^{\prime}([0,\tau_{0,1}])$ and in
$\overline{\eta}_{0,1}^{\prime}([0,\overline{\tau}_{0,1}])$. We note that such
a component a.s. exists.

Given $\eta$ and $\Upsilon$, it is a positive probability event that $z\in W$
and $U$ is such that $\Lambda|_{[0,\tau]}$ is in $\partial
W\setminus\eta_{0,1}^{\prime}$ and has not drawn all of $\partial
W\cap\eta_{0,1}^{\prime}$. Given this, there is a positive chance that
$\overline{x}$, $t$, $\overline{t}$ are such that
$\varphi^{-1}(\eta_{0}^{\prime}\cup\eta_{1}^{\prime}\cup\overline{\eta}_{0}^{\prime}\cup\overline{\eta}_{1}^{\prime})$
and the loops of $\Gamma$ it intersects contains $\partial W$. On this event,
as we have explained above, the Markovian resampling step has positive chance
of making it so that $n_{1}=n-1$. By repeating this $n-1$ further times, we
see that there is a positive chance that $\eta_{n}^{\prime}$ does not hit
$[x_{-}^{\prime},x_{+}^{\prime}]_{\partial V}^{\bm{\circlearrowright}}$.

The above implies that on $E$ it is a positive conditional probability event
given $\Upsilon$ that $\eta^{\prime}$ does not hit
$[x_{-}^{\prime},x_{+}^{\prime}]_{\partial V}^{\bm{\circlearrowright}}$. We
note that by definition $\eta^{\prime}$ is given by targeting at $x_{+}$ the
path which visits the loops of $\Gamma_{V}^{\prime}$ ordered according to when
they are hit by $[x_{-},x_{+}]_{\partial V}^{\bm{\circlearrowleft}}$. Thus
what we have shown is that on $E$ it is a positive conditional probability
event given $\Upsilon$ that the loops of $\Gamma_{V}^{\prime}$ which hit
$[x_{-},x_{+}]_{\partial V}^{\bm{\circlearrowleft}}$ do not hit
$[x_{-}^{\prime},x_{+}^{\prime}]_{\partial V}^{\bm{\circlearrowright}}$.
Applying this principle a second time implies that the following is true.
Suppose that
$x_{-}^{\prime\prime}\in(x_{-},b_{-})_{\partial\mathbf{D}}^{\bm{\circlearrowleft}}$
and
$x_{+}^{\prime\prime}\in(x_{+},b_{+})_{\partial\mathbf{D}}^{\bm{\circlearrowright}}$.
Then on $E$ it is a positive conditional probability event given $\Upsilon$
that the loops of $\Gamma_{V}^{\prime}$ which hit either
$[x_{-}^{\prime},x_{+}^{\prime}]_{\partial V}^{\bm{\circlearrowright}}$ or
$[x_{-}^{\prime\prime},x_{+}^{\prime\prime}]_{\partial
V}^{\bm{\circlearrowleft}}$ do not disconnect $x_{-}$ from $x_{+}$ in $V$. Let
$W$ be the complementary component in $V$ of these loops which has
$x_{-},x_{+}$ on its boundary. We note that the loops $\Gamma_{W}^{\prime}$ of
$\Gamma_{V}^{\prime}$ which are contained in $\mathrm{cl}(W)$ have the law of
a ${\rm CLE}_{\kappa^{\prime}}$ in $W$.

On the above event, we note that there a.s. exists a loop
$\mathcal{L}_{-}^{\prime}$ (resp. $\mathcal{L}_{+}^{\prime}$) of
$\Gamma_{W}^{\prime}$ which disconnects $x_{-}$ (resp. $x_{+}$) from $x_{+}$
(resp. $x_{-}$) and does not hit either
$[x_{-}^{\prime},x_{+}^{\prime}]_{\partial V}^{\bm{\circlearrowright}}$ or
$[x_{-}^{\prime\prime},x_{+}^{\prime\prime}]_{\partial
V}^{\bm{\circlearrowleft}}$. Since the graph of ${\rm CLE}_{\kappa^{\prime}}$
loops is a.s. connected, there a.s. exists $n$ loops
$\mathcal{L}_{1}^{\prime},\ldots,\mathcal{L}_{n}^{\prime}$ in
$\Gamma_{W}^{\prime}$ so that
$\mathcal{L}_{1}^{\prime}\cap\mathcal{L}_{-}^{\prime}\neq\emptyset$,
$\mathcal{L}_{n}^{\prime}\cap\mathcal{L}_{+}^{\prime}\neq\emptyset$, and
$\mathcal{L}_{j}^{\prime}\cap\mathcal{L}_{j-1}^{\prime}\neq\emptyset$ for each
$2\leq j\leq n$. Thus applying the resampling procedure one more time, it is a
positive probability event that
$\mathcal{L}_{1}^{\prime},\ldots,\mathcal{L}_{n}^{\prime}$ get connected into
a single ${\rm CLE}_{\kappa^{\prime}}$ loop in $\mathrm{cl}(W)$ which
disconnects $x_{-}$ from $x_{+}$ and vice-versa. On this event, the branch of
the exploration tree of the resulting ${\rm CLE}_{\kappa^{\prime}}$ does not
hit either $[x_{-}^{\prime},x_{+}^{\prime}]_{\partial
V}^{\bm{\circlearrowright}}$ or
$[x_{-}^{\prime\prime},x_{+}^{\prime\prime}]_{\partial
V}^{\bm{\circlearrowleft}}$ and is a CPI of $\Gamma_{V}$. Altogether, this
completes the proof. ∎

###  3.3. Crossing a disk bounded by CPIs away from the boundary

Figure 3.2. Illustration of the statement of Lemma 3.4. The light blue curve
is the trunk of $\widetilde{\eta}_{0}$ and the green curve is the trunk of
$\widetilde{\eta}_{1}$. The left boundary ($\eta_{0}$ and its time-reversal
$\overline{\eta}_{0}$) of $\widetilde{\eta}_{0}$ (as viewed from
$\eta_{+}(Y_{0})$) is shown in dark blue and the left boundary ($\eta_{1}$) of
$\widetilde{\eta}_{1}$ (as viewed from $\overline{\eta}_{0}(Y_{1})$) is shown
in dark green. The ${\rm CLE}_{\kappa}$ loops visited by
$\widetilde{\eta}_{0}$, $\widetilde{\eta}_{1}$ are not shown as we have taken
them to be ${\rm SLE}_{\kappa}^{-1}(\kappa-6)$ processes (i.e., ${\rm
CLE}_{\kappa}$ loops are only on the left side of the trunk) so that
$\partial\mathcal{D}$ is only composed of parts of the boundaries of the
trunks of $\widetilde{\eta}_{0}$, $\widetilde{\eta}_{1}$ and not of ${\rm
CLE}_{\kappa}$ loops.

######  Lemma 3.4.

For each $\zeta_{0},p\in(0,1)$ there exists $\delta_{0}>0$ so that for all
$\delta\in(0,\delta_{0})$ there exists $\epsilon_{0}>0$ so that for all
$\epsilon\in(0,\epsilon_{0})$ the following is true. Let $X_{0}$ be an
exponential random variable with mean $\delta$ and let $Y_{0}$ be uniform in
$[0,X_{0}]$, independently of everything else. Let $\widetilde{\eta}_{0}$ be
an ${\rm SLE}_{\kappa}^{-1}(\kappa-6)$ in $\mathcal{H}_{+}$ from
$\eta_{+}(X_{0})$ to $\eta_{+}(Y_{0})$ coupled with $\Gamma_{+}$ as a CPI, let
$\eta_{0}$ be its left boundary, assume that $\eta_{0}$ is parameterized
according to quantum length, and let $L$ be the total quantum length of
$\eta_{0}$. Given $L$, let $X_{1},Y_{1}$ be i.i.d. uniform in $[0,L]$
independently of everything else. Let $\overline{\eta}_{0}$ be the time-
reversal of $\eta_{0}$. On the event that
$\eta_{0}([0,X_{1}])\cap\overline{\eta}_{0}([0,Y_{1}])=\emptyset$ and the part
of $\eta_{0}$ from $\eta_{0}(X_{1})$ to $\overline{\eta}_{0}(Y_{1})$ (i.e.,
$\eta_{0}|_{[X_{1},L-Y_{1}]}$) does not hit $\partial\mathcal{H}_{+}$, let
$\widetilde{\eta}_{1}$ be an ${\rm SLE}_{\kappa}^{-1}(\kappa-6)$ process in
the unbounded component of $\mathcal{H}_{+}\setminus\eta_{0}$ from
$\eta_{0}(X_{1})$ to $\overline{\eta}_{0}(Y_{1})$, coupled as a CPI of the
loops of $\Gamma_{+}$ in this component. Let $\eta_{1}$ be the left boundary
of $\widetilde{\eta}_{1}$. Let $Z_{1}$ be uniform in $[X_{1},L-Y_{1}]$ and let
$\mathcal{D}$ be the component of
$\mathcal{H}_{+}\setminus(\eta_{0}\cup\eta_{1})$ with $\eta_{0}(Z_{1})$ on its
boundary. Let $E$ be the event that there does not exist a path $\omega$ in
$\mathcal{D}\cap\Upsilon_{+}$ from $\eta_{0}(Z_{1})$ to $\eta_{1}$ with
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.
Then $\mathbf{P}[E]\leq\zeta_{0}$.

The strategy to prove Lemma 3.4 is the following. First, we will prove a
technical result (Lemma 3.5) which gives that the conditional law of the
intermediate part of an ${\rm SLE}_{\kappa}(\rho_{1};\rho_{2})$ process is
absolutely continuous with respect to ordinary ${\rm SLE}_{\kappa}$ on the
event it does not intersect the domain boundary. This is purely a statement
about ${\rm SLE}_{\kappa}(\rho_{1};\rho_{2})$ processes whose proof can be
skipped on a first reading of the proof of Lemma 3.4. Next, we will prove in
Lemma 3.6 that if we consider an ${\rm SLE}_{\kappa}^{-1}(\kappa-6)$ process
$\eta$ on $\mathcal{H}_{+}$ from $\eta_{-}(\delta)$ to $\infty$ then provided
$\delta>0$ is sufficiently small it is very likely that there is a path
$\omega$ in $\mathcal{H}_{+}$ from $0$ to the top of the quantum surface
disconnected by $\eta$ from $\infty$ with $0$ on its boundary satisfying
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.
We will then complete the proof of Lemma 3.4 by using that the law of the part
of $\partial\mathcal{D}$ on $\eta_{0}$ near $\eta_{0}(Z_{1})$ is comparable to
the law of $\partial\mathcal{H}_{+}$ near $0$.

######  Lemma 3.5.

Suppose that $D\subseteq\mathbf{C}$ is a simply connected domain and
$x,y\in\partial D$ are distinct. Fix $\rho_{1},\rho_{2}>-2$. Let $\eta$ be an
${\rm SLE}_{\kappa}(\rho_{1};\rho_{2})$ process in $D$ from $x$ to $y$. Let
$\tau$ (resp. $\overline{\tau}$) be a stopping time (resp. reverse stopping
time) for $\eta$. The conditional law of $\eta|_{[\tau,\overline{\tau}]}$
given $\eta|_{[0,\tau]}$ and $\eta|_{[\overline{\tau},\infty)}$ on the event
that $\eta|_{[\tau,\overline{\tau}]}$ does not intersect $\partial D$ is
absolutely continuous with respect to that of an ${\rm SLE}_{\kappa}$ from
$\eta(\tau)$ to $\eta(\overline{\tau})$ in the component of
$D\setminus(\eta([0,\tau])\cup\eta([\overline{\tau},\infty))$ with
$\eta(\tau)$ on its boundary.

###### Proof.

It suffices to prove the result in the case that $D=\mathbf{H}$, $x=0$, and
$y=\infty$. We can view $\eta$ as a flow line of a GFF $h$ on $\mathbf{H}$
with boundary conditions given by $-\lambda(1+\rho_{1})$ on $\mathbf{R}_{-}$
and $\lambda(1+\rho_{2})$ on $\mathbf{R}_{+}$. Let $\eta_{L}^{\prime}$ be the
counterflow line of $h+\pi\chi/2$ from $\infty$ to $0$. Then $\eta$ is equal
to the right boundary of $\eta_{L}^{\prime}$. Let $\eta_{R}^{\prime}$ be the
counterflow line of $h-\pi\chi/2$ from $\infty$ to $0$. Then $\eta$ is also
equal to the left boundary of $\eta_{R}^{\prime}$. In other words, $\eta$ is
equal to the common boundary of $\eta_{L}^{\prime}$ and $\eta_{R}^{\prime}$.

Let $\tau_{L}$ (resp. $\tau_{R}$) be the first time that $\eta_{L}^{\prime}$
(resp. $\eta_{R}^{\prime}$) hits $\eta(\overline{\tau})$. Then the common
boundary of $\eta_{L}^{\prime}([0,\tau_{L}])$ and
$\eta_{R}^{\prime}([0,\tau_{R}])$ is equal to
$\eta([\overline{\tau},\infty))$. It is shown in the proof of [25, Lemma 5.13]
that
$A_{\overline{\tau}}=\eta_{L}^{\prime}([0,\tau_{L}])\cup\eta_{R}^{\prime}([0,\tau_{R}])$
is a local set for $h$. Let $\mathcal{F}_{\tau,\overline{\tau}}$ be the
$\sigma$-algebra generated by $\eta|_{[0,\tau]}$,
$\eta_{L}^{\prime}|_{[0,\tau_{L}]}$, $\eta_{R}^{\prime}|_{[0,\tau_{R}]}$. Then
we moreover have that the conditional law of $\eta|_{[\tau,\overline{\tau}]}$
given $\mathcal{F}_{\tau,\overline{\tau}}$ is that of an ${\rm
SLE}_{\kappa}(\rho_{1},\tfrac{\kappa}{2}-2-\rho_{1};\rho_{2},\tfrac{\kappa}{2}-2-\rho_{2})$
process in the component $D_{\tau,\overline{\tau}}$ of
$\mathbf{H}\setminus(\eta([0,\tau])\cup A_{\overline{\tau}})$ with
$\eta(\tau)$ on its boundary from $\eta(\tau)$ to $\eta(\overline{\tau})$. The
force points are located at the leftmost and rightmost intersections of
$\eta|_{[0,\tau]}$ with $\partial\mathbf{H}$ and the intersections of
$\eta_{L}^{\prime}|_{[0,\tau_{L}]}$ and $\eta_{R}^{\prime}|_{[0,\tau_{R}]}$ on
$\mathbf{R}_{+}$ and $\mathbf{R}_{-}$, respectively, which are closest to $0$.

Let $D_{L}$ (resp. $D_{R}$) be the component of $\mathbf{H}\setminus\eta$
which is to the left (resp. right) of $\eta$ with $\eta(\overline{\tau})$ on
its boundary. Let $\eta_{L}$ (resp. $\eta_{R}$) denote the left (resp. right)
boundary of $\eta_{L}^{\prime}([0,\tau_{L}])$ (resp.
$\eta_{R}^{\prime}([0,\tau_{R}])$). Then $\eta_{L}$ (resp. $\eta_{R}$) is the
flow line of $h$ restricted to $D_{L}$ (resp. $D_{R}$) starting from
$\eta(\overline{\tau})$ with angle $\pi$ (resp. $-\pi$). In particular, the
conditional law of $\eta_{L}$ given $\eta$ is that of an ${\rm
SLE}_{\kappa}(\kappa/2-2,\rho_{1};-\kappa/2)$ in $D_{L}$ starting from
$\eta(\overline{\tau})$ and the conditional law of $\eta_{R}$ given $\eta$ is
that of an ${\rm SLE}_{\kappa}(-\kappa/2;\kappa/2-2,\rho_{2})$ in $D_{R}$
starting from $\eta(\overline{\tau})$. Let
$\mathcal{G}_{\tau,\overline{\tau}}$ be the $\sigma$-algebra generated by
$\eta|_{[0,\tau]}$, $\overline{\eta}|_{[\overline{\tau},\infty)}$, $\eta_{L}$,
and $\eta_{R}$. It is moreover shown in [25] that the conditional law of
$\eta$ given $\mathcal{G}_{\tau,\overline{\tau}}$ is the same as the
conditional law of $\eta$ given $\mathcal{F}_{\tau,\overline{\tau}}$.

Fix $\epsilon>0$. Let $\overline{\tau}_{\epsilon}$ be the first time $t$ that
$\eta$ gets within distance $\epsilon$ of $A_{\overline{\tau}}$. Then it
follows that the law of $\eta|_{[\tau,\overline{\tau}_{\epsilon}]}$ on the
event that it does not hit $\partial D$ is absolutely continuous with respect
to the law of an ${\rm SLE}_{\kappa}$ in $D_{\tau,\overline{\tau}}$ from
$\eta(\tau)$ to $\eta(\overline{\tau})$ stopped upon getting within distance
$\epsilon$ of $A_{\overline{\tau}}$. Since $\eta$ hits $A_{\overline{\tau}}$
exactly at $\eta(\overline{\tau})$, it follows that if we let
$\tau_{\epsilon}$ be the first time $t$ that $\eta$ gets within distance
$\epsilon$ of $\eta(\overline{\tau})$ then the law of
$\eta|_{[\tau,\tau_{\epsilon}]}$ on the event that it does not hit $\partial
D$ is absolutely continuous with respect to the law of an ${\rm SLE}_{\kappa}$
in $D_{\tau,\overline{\tau}}$ from $\eta(\tau)$ to $\eta(\overline{\tau})$
stopped upon getting within distance $\epsilon$ of $\eta(\overline{\tau})$.
Let $U_{\tau,\overline{\tau}}$ be the component of
$\mathbf{H}\setminus(\eta([0,\tau])\cup\eta([\overline{\tau},\infty))$ with
$\eta(\tau)$ on its boundary. Then it follows from [17, Proposition 5.3] that
the law of $\eta|_{[\tau,\tau_{\epsilon}]}$ on the event that it does not hit
$\partial D$ is absolutely continuous with respect to the law of an ${\rm
SLE}_{\kappa}$ in $U_{\tau,\overline{\tau}}$ from $\eta(\tau)$ to
$\eta(\overline{\tau})$ stopped at the first time it gets within distance
$\epsilon$ of $\eta(\overline{\tau})$.

By reversing the roles of $\tau$ and $\overline{\tau}$ and using the
reversibility of ${\rm SLE}_{\kappa}(\rho_{1};\rho_{2})$ established in [25],
this implies that the following is true. Let $\overline{\eta}$ be the time-
reversal of $\eta$ and let $\overline{\sigma},\sigma$, respectively, be the
times for $\overline{\eta}$ which correspond to $\overline{\tau},\tau$,
respectively, for $\eta$. Let $\overline{\sigma}_{\epsilon}$ be the first time
that $\overline{\eta}$ gets within distance $\epsilon$ of $\eta(\tau)$. Then
the law of
$\overline{\eta}|_{[\overline{\sigma},\overline{\sigma}_{\epsilon}]}$ on the
event that it does not hit $\partial D$ is absolutely continuous with respect
to the law of an ${\rm SLE}_{\kappa}$ in $U_{\tau,\overline{\tau}}$ from
$\eta(\overline{\tau})$ to $\eta(\tau)$ stopped at the first time it gets
within distance $\epsilon$ of $\eta(\tau)$.

For $\epsilon,\delta>0$, we let $E_{\epsilon,\delta}$ be the event that
$\overline{\eta}|_{[\overline{\sigma}_{\epsilon},\sigma]}$ does not leave
$B(\eta(\tau),\delta)$ and $\eta_{L},\eta_{R}$ do not hit
$B(\eta(\tau),2\delta)$. Then for every $\zeta>0$ there exists $\delta_{0}>0$
so that for all $\delta\in(0,\delta_{0})$ there exists $\epsilon_{0}>0$ so
that for all $\epsilon\in(0,\epsilon_{0})$ we have that the probability of
$E_{\epsilon,\delta}$ is at least $1-\zeta$. Let
$(\widetilde{\eta},\widetilde{\eta}_{L},\widetilde{\eta}_{R})$ be a triple
where

  1. (i)

$\widetilde{\eta}$ is an ${\rm SLE}_{\kappa}$ in $U_{\tau,\overline{\tau}}$
from $\eta(\overline{\tau})$ to $\eta(\tau)$ and

  2. (ii)

the conditional law of $(\widetilde{\eta}_{L},\widetilde{\eta}_{R})$ given
$\widetilde{\eta}$ is the same as the conditional law of $(\eta_{L},\eta_{R})$
given $\eta$.

On $E_{\epsilon,\delta}$ and the event that
$\overline{\eta}|_{[\overline{\sigma},\overline{\sigma}_{\epsilon}]}$ does not
hit $\partial D$, we have that the joint law of the triple consisting of
$\overline{\eta}|_{[\overline{\sigma},\overline{\sigma}_{\epsilon}]}$,
$\eta_{L}$, and $\eta_{R}$ is absolutely continuous with respect to the triple
consisting of $\widetilde{\eta}$ stopped upon getting within distance
$\epsilon$ of $\eta(\tau)$, $\widetilde{\eta}_{L}$, and
$\widetilde{\eta}_{R}$.

Let $\sigma_{\epsilon}^{L}$ (resp. $\sigma_{\epsilon}^{R}$) be the first time
that $\eta_{L}$ (resp. $\eta_{R}$) gets within distance $\epsilon$ of
$\partial\mathbf{H}$. Then the conditional law of
$\eta_{L}|_{[0,\sigma_{\epsilon}^{L}]}$ given $\eta$ is absolutely continuous
with respect to the law of an ${\rm SLE}_{\kappa}(\kappa/2-2;-\kappa/2)$
stopped at the corresponding time. Likewise, the conditional law of
$\eta_{R}|_{[0,\sigma_{\epsilon}^{R}]}$ given $\eta$ is absolutely continuous
with respect to the law of an ${\rm SLE}_{\kappa}(-\kappa/2;\kappa/2-2)$
stopped at the corresponding time. Let
$(\widehat{\eta},\widehat{\eta}_{L},\widehat{\eta}_{R})$ be a triple where

  1. (i)

$\widehat{\eta}$ is an ${\rm SLE}_{\kappa}$ in $U_{\tau,\overline{\tau}}$ from
$\eta(\overline{\tau})$ to $\eta(\tau)$,

  2. (ii)

the conditional law of $\widehat{\eta}_{L}$ given $\widehat{\eta}$ is that of
an ${\rm SLE}_{\kappa}(-\kappa/2;\kappa/2-2)$ starting at
$\overline{\eta}(\overline{\tau})$, and

  3. (iii)

the conditional law of $\widehat{\eta}_{R}$ given $\widehat{\eta}$ is that of
an ${\rm SLE}_{\kappa}(\kappa/2-2;-\kappa/2)$ starting at
$\overline{\eta}(\overline{\tau})$.

On the event that
$\overline{\eta}|_{[\overline{\sigma},\overline{\sigma}_{\epsilon}]}$ does not
hit $\partial\mathbf{H}$, we have that the law of the triple consisting of
$\overline{\eta}|_{[\overline{\sigma},\overline{\sigma}_{\epsilon}]}$,
$\eta_{L}|_{[0,\sigma_{\epsilon}^{L}]}$, and
$\eta_{R}|_{[0,\sigma_{\epsilon}^{R}]}$ is absolutely continuous with respect
to the law of the triple consisting of $\widehat{\eta}$ stopped at the first
time it gets within distance $\epsilon$ of $\eta(\tau)$ and
$\widehat{\eta}_{L},\widehat{\eta}_{R}$ both stopped at the first time they
get within distance $\epsilon$ of $\partial\mathbf{H}$. In particular, on the
event that
$\overline{\eta}|_{[\overline{\sigma},\overline{\sigma}_{\epsilon}]}$ does not
hit $\partial\mathbf{H}$, the law of the pair consisting of
$\eta_{L}|_{[0,\sigma_{\epsilon}^{L}]}$ and
$\eta_{R}|_{[0,\sigma_{\epsilon}^{R}]}$ is absolutely continuous with respect
to the law of the pair $(\widehat{\eta}_{L},\widehat{\eta}_{R})$ both stopped
at the corresponding time.

Let $F_{\epsilon,\delta}$ be the event that $\eta$ does not get within
distance $\delta$ of $\eta_{L}([\sigma_{\epsilon}^{L},\infty))$ or
$\eta_{R}([\sigma_{\epsilon}^{R},\infty))$. Then for every $\zeta>0$ there
exists $\delta_{0}>0$ so that for all $\delta\in(0,\delta_{0})$ there exists
$\epsilon_{0}>0$ so that for all $\epsilon\in(0,\epsilon_{0})$ we have that
the probability of $F_{\epsilon,\delta}$ is at least $1-\zeta$. On
$F_{\epsilon,\delta}$, it follows that the conditional law of
$\eta|_{[\tau,\overline{\tau}]}$ given $\eta_{L}$, $\eta_{R}$ and on the event
that $\eta|_{[\tau,\overline{\tau}]}$ does not hit $\partial\mathbf{H}$ is
absolutely continuous with respect to the conditional law of $\widehat{\eta}$
given $\widehat{\eta}_{L}$, $\widehat{\eta}_{R}$. Altogether, this implies
that the conditional law of $\eta|_{[\tau,\overline{\tau}]}$ given $\eta_{L}$,
$\eta_{R}$ and on the event that $\eta|_{[\tau,\overline{\tau}]}$ does not hit
$\partial\mathbf{H}$ is absolutely continuous with respect to the conditional
law of $\widehat{\eta}$ given $\widehat{\eta}_{L}$, $\widehat{\eta}_{R}$.
Therefore the law of $\eta|_{[\tau,\overline{\tau}]}$ on the event that it
does not hit $\partial\mathbf{H}$ is absolutely continuous with respect to the
law of $\widehat{\eta}$. ∎

Figure 3.3.  Illustration of the proof of Lemma 3.6; the light blue ball is
$B(w_{j},\xi/32)$.

######  Lemma 3.6.

Suppose that $\eta$ is an ${\rm SLE}_{\kappa}^{-1}(\kappa-6)$ in
$\mathcal{H}_{+}$ from $\eta_{-}(\delta)$ to $\infty$ which is coupled with
$\Gamma_{+}$ as a CPI and parameterized by the quantum natural time of its
trunk. For each $p,\zeta_{0}\in(0,1)$ there exists $\delta_{0},\epsilon_{0}>0$
so that for all $\delta\in(0,\delta_{0})$ and $\epsilon\in(0,\epsilon_{0})$
the following is true. Let $\mathcal{D}$ be the component of
$\mathcal{H}_{+}\setminus\eta$ with $\eta_{+}(0)=0$ on its boundary. Let
$X_{\epsilon}=\inf_{\omega}{\mathfrak{N}}_{\epsilon}(\omega)$ where the
infimum is over all paths $\omega$ in $\mathcal{H}_{+}$ contained in
$\Upsilon_{+}\cap\mathcal{D}$ and which connect $0$ to
$\eta\cap\partial\mathcal{D}$. Then
$\mathbf{P}[X_{\epsilon}\geq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)]\leq\zeta_{0}$.

###### Proof.

Step 1. Setup. Fix $p\in(0,1)$. It suffices to show that for every
$\zeta_{0}>0$ there exists $\xi,\epsilon_{0}>0$ so that for all
$\epsilon\in(0,\epsilon_{0})$ the following is true. Let $Y_{\epsilon}$ be the
Euclidean diameter of the set of points in $\mathcal{H}_{+}$ which are
connected to $0$ by paths $\omega$ in $\Upsilon_{+}$ with
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.
Then $\mathbf{P}[Y_{\epsilon}\leq\xi]\leq\zeta_{0}$. Suppose that this is not
true. Then there exists $\zeta_{0}>0$ and a sequence $(\epsilon_{n})$ of
positive numbers with $\epsilon_{n}\to 0$ as $n\to\infty$ so that
$\mathbf{P}[Y_{\epsilon_{n}}\leq 1/n]\geq\zeta_{0}$ for every
$n\in\mathbf{N}$. We are going to assume that this is true throughout the rest
of the proof in order to show that

$\mathbf{P}\\!\left[\sup_{z,w\in\eta_{+}([0,1])}\mathfrak{d}_{\epsilon_{n}}(z,w;\Gamma_{+})\geq{{\mathfrak{q}}}_{\epsilon_{n}}^{\mathrm{HP}}(p)\right]\to
1\quad\text{as}\quad n\to\infty.$

This, in turn, will contradict the definition of
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.

Step 2. First observation. Fix $\rho_{L},\rho_{R}>-2$ and suppose that
$\widetilde{\eta}$ is an ${\rm SLE}_{\kappa}(\rho_{L};\rho_{R})$ in
$\mathbf{D}$ from $-i$ to $i$. Let $\widetilde{\Gamma}$ be a collection of
loops in the components of $\mathbf{D}\setminus\widetilde{\eta}$ which are to
the left of $\widetilde{\eta}$ which in each such component is a conditionally
independent ${\rm CLE}_{\kappa}$. Let $\widetilde{\Upsilon}$ be the closure of
the union of the loops in $\widetilde{\Gamma}$. Fix $z\in\mathbf{D}$ and $r>0$
so that $B(z,2r)\subseteq\mathbf{D}$. Then it follows from our assumption
above that there exists a sequence $(\epsilon_{n})$ of positive numbers with
$\epsilon_{n}\to 0$ as $n\to\infty$ so that the following is true. Let
$\widetilde{\sigma}=\inf\\{t\geq 0:\widetilde{\eta}(t)\in B(z,r)\\}$. On
$\widetilde{\sigma}<\infty$, let $\widetilde{Y}_{\epsilon_{n}}$ be the
Euclidean diameter of the set of points to the left of $\widetilde{\eta}$
which are accessible by paths $\omega$ in $\widetilde{\Upsilon}$ starting from
$\widetilde{\eta}(\widetilde{\sigma})$ with
${\mathfrak{N}}_{\epsilon_{n}}(\omega)\leq{{\mathfrak{q}}}_{\epsilon_{n}}^{\mathrm{HP}}(p)$.
Then $\mathbf{P}[\widetilde{Y}_{\epsilon_{n}}\leq 1/n,\
\widetilde{\sigma}<\infty]$ is bounded from below by a positive constant which
depends only on $\rho_{L}$, $\rho_{R}$, $z$, $r$, and $\zeta_{0}$ from Step 1.
Indeed, this follows from absolute continuity and our assumption in Step 1.

Step 3. Approximate ${\rm CLE}_{\kappa}$-metric ball has small Euclidean
diameter in many places on the interface. We are now going to show that the
event that the “$\mathfrak{d}_{\epsilon_{n}}(\cdot,\cdot;\Gamma_{+})$-metric
ball” at a point along $\eta_{+}$ has small diameter is likely to happen at
many points. In this step, we are going to focus on proving a version of this
statement in the setting of $\mathbf{H}$; in the next step we will
subsequently transfer the result to the setting of $\eta_{+}$. To this end, we
suppose that $h$ is a GFF on $\mathbf{H}$ with boundary conditions given by
$-\lambda$ on $\mathbf{R}_{-}$ and $\lambda$ on $\mathbf{R}_{+}$. Let $\eta$
be the flow line of $h$ from $0$ to $\infty$. Then $\eta$ is an ${\rm
SLE}_{\kappa}$ curve in $\mathbf{H}$ from $0$ to $\infty$. Fix $\xi>0$ and
$\theta\in(0,\pi)$. For each $j$, we let $\tau_{j}=\inf\\{t\geq
0:\eta(t)\notin B(0,\xi j)\\}$. We then let $\eta_{j,1}$ (resp. $\eta_{j,2}$)
be the flow line of $h$ starting from $\eta_{j}(\tau_{j})$ with angle
$2\lambda^{\prime}/\chi-\pi$ (resp. $-\theta$). Then $\eta_{j,1}$ is equal to
the right boundary of the counterflow line $\eta_{j,1}^{\prime}$ of
$h+2\lambda^{\prime}-\pi\chi/2$ from $\infty$ to $\eta(\tau_{j})$. We note
that the angle difference between $\eta_{j,1}$ and $\eta_{j,2}$ is equal to
$2\lambda^{\prime}/\chi-\pi+\theta$ and that flow lines intersect whenever the
angle difference is between $0$ and $2\lambda^{\prime}/\chi$. That is, with
any choice of $\theta\in(0,\pi)$ we have that $\eta_{j,1}$, $\eta_{j,2}$ a.s.
intersect each other.

Given $\eta$, we let $\Gamma_{L}$ be a ${\rm CLE}_{\kappa}$ in the component
of $\mathbf{H}\setminus\eta$ which is to the left of $\eta$. We assume that
$\Gamma_{L}$ is generated from $h$ using the coupling of ${\rm
SLE}_{\kappa}(\kappa-6)$ with the GFF so that $\eta_{j,1}^{\prime}$ is a CPI
in $\Gamma_{L}$ for each $j$. Then the conditional law of $\Gamma_{L}$ in any
component $U$ of $\mathbf{H}\setminus(\eta\cup\eta_{1,j})$ which is to the
left of $\eta$ and to the right of $\eta_{1,j}$ is that of a ${\rm
CLE}_{\kappa}$ in $U$.

For each $j\in\mathbf{N}$ and $k\in\\{1,2\\}$, we let $\tau_{j,k}$ be the
first time that $\eta_{j,k}$ leaves $B(\eta_{j}(\tau_{j}),\xi/16)$. We let
$\mathcal{F}_{j}$ be the $\sigma$-algebra generated by $\eta|_{[0,\tau_{j}]}$
and $\eta_{i,k}|_{[0,\tau_{i,k}]}$ for $1\leq i\leq j$ and $k\in\\{1,2\\}$. We
also let $E_{j}$ be the event that:

  1. (i)

There exists $w\in\mathbf{H}$ so that $\eta_{j,1}|_{[0,\tau_{j,1}]}$,
$\eta_{j,2}|_{[0,\tau_{j,2}]}$ do not intersect $B(w,\xi/32)$ and disconnect
$B(w,\xi/32)$ from $\infty$ and the harmonic measure of
$\eta_{j,k}([0,\tau_{j,k}])$ in
$\mathbf{H}\setminus\cup_{k=1}^{2}\eta_{j,k}([0,\tau_{j,k}])$ as seen from $w$
is at least $1/4$ for $k\in\\{1,2\\}$.

  2. (ii)

Let $V_{j}$ be the component of
$\mathbf{H}\setminus\cup_{k=1}^{2}\eta_{j,k}([0,\tau_{j,k}])$ disconnected
from $\infty$ as in (i) which is first visited by $\eta_{j,1}$. Let $w_{j}$ be
the point $w$ as in (i) with the smallest real part, breaking ties by taking
the point with smallest imaginary part. Let $\Gamma_{j,L}$ be the loops of
$\Gamma_{L}$ which are contained in $V_{j}$. Let $\sigma_{j}$ be the first
time that $\eta$ visits $B(w_{j},\xi/64)$. Then $\sigma_{j}<\infty$ and
$\mathfrak{d}_{\epsilon_{n}}(\eta(\sigma_{j}),\partial
B(w_{j},\xi/32);\Gamma_{j,L})\geq{{\mathfrak{q}}}_{\epsilon_{n}}^{\mathrm{HP}}(p)$.

We claim that there exists $p_{0}>0$ which does not depend on $j$ so that

(3.1) $\displaystyle\mathbf{P}[E_{j}\,|\,\mathcal{F}_{j-1}]\geq
p_{0}\quad\text{for each}\quad j\in\mathbf{N}.$

To start to prove (3.1), for each $j$ we let $z_{j}$ be the point on
$\mathbf{H}\cap\partial B(0,\xi j)$ with
$\arg(z_{j})=((\mathrm{Im}(\eta(\tau_{j-1}))\wedge(\pi-\xi))\vee\xi$. Let
$A_{j}$ be the union of $\eta([0,\tau_{j}])$ and $\eta_{i,k}([0,\tau_{i,k}])$
for $1\leq i\leq j$ and $k\in\\{1,2\\}$. Let $\varphi_{j}$ be the unique
conformal map from the unbounded component of $\mathbf{H}\setminus A_{j-1}$ to
$\mathbf{H}$ which fixes $\infty$ and takes $z_{j}$ to $i$. Then there exists
a constant $M>0$ so that $\varphi(\eta(\tau_{j-1}))$ is contained in $[-M,M]$
(as the harmonic measure in $\mathbf{H}\setminus\eta([0,\tau_{j-1}])$ as seen
from $z_{j}$ of the left side of $\eta$ and $\mathbf{R}_{-}$ is bounded from
below and the same is also true for the and right side of $\eta$ and
$\mathbf{R}_{+}$). [36, Lemmas 2.3–2.5] thus imply that there exists $p_{0}>0$
which does not depend on $j$ such that the conditional probability given
$\mathcal{F}_{j-1}$ of part (i) of $E_{j}$ is at least $p_{0}$.

We are now going to show that, by possibly decreasing the value of $p_{0}>0$,
we have that (3.1) holds. Let $w_{j}$, $V_{j}$ be as in (ii). Let $\psi_{j}$
be the unique conformal transformation $V_{j}\to\mathbf{D}$ which takes the
first (resp. last) point on $\partial V$ visited by $\eta$ to $-i$ (resp. $i$)
and is such that $\mathrm{Im}(\psi_{j}(w_{j}))=0$. Then there exists
$\rho_{L},\rho_{R}>-2$ so that $\widetilde{\eta}_{j}=\psi_{j}(\eta)$ is an
${\rm SLE}_{\kappa}(\rho_{L};\rho_{R})$ in $\mathbf{D}$ from $-i$ to $i$.
Since the harmonic measure of the clockwise (resp. counterclockwise) arc of
$\partial\mathbf{D}$ from $-i$ to $i$ as seen from
$\widetilde{w}_{j}=\psi_{j}(w_{j})$ is at least $1/4$, it follows that there
exists $r_{0}>0$ so that $B(\widetilde{w}_{j},2r_{0})\subseteq\mathbf{D}$. Let
$\widetilde{\sigma}_{j}$ be the first time that $\widetilde{\eta}_{j}$ visits
$B(\widetilde{w}_{j},r_{0})$. Let $\widetilde{Y}_{\epsilon_{n}}$ be the
Euclidean diameter of the set of points to the left of $\widetilde{\eta}_{j}$
which are accessible by paths in $\omega$ in $\psi_{j}(\Gamma_{j,L})$ starting
from $\widetilde{\eta}_{j}(\widetilde{\sigma}_{j})$ with
${\mathfrak{N}}_{\epsilon_{n}}(\omega)\leq{{\mathfrak{q}}}_{\epsilon_{n}}^{\mathrm{HP}}(p)$.
Step 2 implies that there exists $q_{0}>0$ so that
$\mathbf{P}[\widetilde{Y}_{\epsilon_{n}}\leq 1/n,\
\widetilde{\sigma}_{j}<\infty]\geq q_{0}$. We note that the restriction of
$(\psi_{j}^{-1})^{\prime}$ to $\psi_{j}(B(w_{j},\xi/32))$ is bounded from
below by a constant times $\xi$. Therefore (3.1) follows by conformally
mapping back.

Step 4. Contradiction to the definition of
${{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$. Suppose that we have the setup
described in Step 3. Fix $N\in\mathbf{N}$ and let $F_{N}=\cup_{j=1}^{N}E_{j}$
be the event that $E_{j}$ occurs for some $1\leq j\leq N$. It follows from
(3.1) that $\mathbf{P}[F_{N}]\to 1$ as $N\to\infty$. Let
$K_{\xi}=[-\xi^{1/2},\xi^{1/2}]\times[\xi/128,\xi^{1/2}]$ and let $G_{N}$ be
the event that $E_{j}$ occurs and $\eta(\tau_{j})\in K_{\xi}$. Then it further
follows that for each $p_{0}\in(0,1)$ we can choose $\xi>0$ sufficiently small
so that with $N=\xi^{-1/2}$ we have that $\mathbf{P}[G_{N}]\geq p_{0}$. Let
$\varphi$ be the unique conformal transformation from $\mathbf{H}$ to
$\mathbf{C}\setminus\eta_{-}$ which fixes $0$ and $\infty$ and takes $-1$ to
the prime end corresponding to the point $\eta_{-}(1)$ which is on
$\partial\mathcal{H}_{+}$. Then there exists a constant $c\geq 1$ depending
only on $\eta_{-}$ so that the restriction of $|\varphi^{\prime}|$ to
$K_{\xi}$ is between $c^{-1}\xi^{-1}$ and $c\xi^{-1/2}$. We assume that
$\eta_{+}=\varphi(\eta)$, modulo parameterization. It therefore follows that
on $G_{N}$ we have that the
“$\mathfrak{d}_{\epsilon_{n}}(\cdot,\cdot;\Gamma_{+})$-metric ball” centered
at $0$ of radius ${{\mathfrak{q}}}_{\epsilon_{n}}^{\mathrm{HP}}(p)$ does not
contain all of the image under $\varphi$ of $\eta$ up until time $\tau_{N}$.
As the probability that $\varphi(\eta([0,\tau_{N}]))$ is contained in
$\eta_{+}([0,1])$ (recall that $\eta_{+}$ is parameterized by quantum length)
tends to $1$ as $\xi\to 0$, the desired contradiction to the definition of
${{\mathfrak{q}}}_{\epsilon_{n}}^{\mathrm{HP}}(p)$ follows. ∎

###### Proof of Lemma 3.4.

In order to start to prove the lemma, we will first explain why a simpler
version of the statement holds. Let $\eta$ be an ${\rm
SLE}_{\kappa}^{-1}(\kappa-6)$ process from $0$ to $\infty$ in
$\mathcal{H}_{+}$ coupled as a CPI of $\Gamma_{+}$. Let $X$ be an exponential
random variable with mean $1$ which is independent of everything else and let
$\mathcal{D}_{0}$ be the component of $\mathcal{H}_{+}\setminus\eta$ with
$\eta_{+}(X\delta)$ on its boundary. It follows from the argument used to
prove Lemma 3.6 that for each $\zeta_{0}\in(0,1)$ there exists
$\delta_{0}\in(0,1)$ so that for each $\delta\in(0,\delta_{0})$ there exists
$\epsilon_{0}\in(0,1)$ so that for every $\epsilon\in(0,\epsilon_{0})$ the
probability that there is a path $\omega$ in $\Upsilon_{+}$ from
$\eta_{+}(X\delta)$ to the part of $\partial\mathcal{D}_{0}$ which is
contained in $\eta$ with
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$
is at least $1-\zeta_{0}$.

Let now describe the law of the ensemble of paths which make up
$\partial\mathcal{D}_{0}$. We know that the conditional law of $\eta_{+}$
given $\eta_{-}$ is that of an ${\rm SLE}_{\kappa}$ curve in
$\mathbf{C}\setminus\eta_{-}$ from $0$ to $\infty$. Let $\eta_{R}$ be the
right boundary of $\eta$. Note that $\eta_{R}$ is also the right boundary of
the trunk of $\eta$. Recall that the trunk is an ${\rm
SLE}_{\kappa^{\prime}}(\kappa^{\prime}-6)$ process in $\mathcal{H}_{+}$ from
$0$ to $\infty$ with its force point located at $0^{-}$. We thus know from
[24, Theorem 1.5] that the law of $\eta_{R}$ is equal to that of an ${\rm
SLE}_{\kappa}(\kappa-4;2-\kappa)$ process from $\infty$ to $0$ in
$\mathcal{H}_{+}$. Also, note that $\mathcal{D}_{0}$ is the component of
$\mathcal{H}_{+}\setminus\eta_{R}$ with $\eta_{+}(X\delta)$ on its boundary.

We are now going to make a comparison between the law of $\mathcal{D}$ as
defined in the lemma statement to the law of $\mathcal{D}_{0}$ as described
just above. Let $L$ be the quantum length of $\eta_{0}$. Let $G$ be the event
that $\eta_{0}([0,X_{1}])\cap\overline{\eta}_{0}([0,Y_{1}])=\emptyset$ and the
part of $\eta_{0}$ from $\eta_{0}(X_{1})$ to $\overline{\eta}_{0}(Y_{1})$,
i.e., $\eta_{0}|_{[X_{1},L-Y_{1}]}$ does not hit $\partial\mathcal{H}_{+}$. On
$G$, Lemma 3.5 implies that the conditional law of
$\eta_{0}|_{[X_{1},L-Y_{1}]}$ is absolutely continuous with respect to the law
of an ${\rm SLE}_{\kappa}$ in the unbounded component $U$ of
$\mathcal{H}_{+}\setminus(\eta_{0}([0,X_{1}])\cup\overline{\eta}_{0}([0,Y_{1}]))$
from $\eta_{0}(X_{1})$ to $\overline{\eta}_{0}(Y_{1})$. We also have that the
law of $\eta_{1}$ given $\eta_{0}$ is that of an ${\rm
SLE}_{\kappa}(\kappa-4;2-\kappa)$ from $\overline{\eta}_{0}(Y_{1})$ to
$\eta_{0}(X_{1})$.

The previous two paragraphs imply that the following is true. Let $\varphi$ be
the unique conformal transformation from $\mathbf{C}\setminus\eta_{-}$ to $U$
which takes $0$ to $\eta_{0}(X_{1})$ and $\infty$ to
$\overline{\eta}_{0}(Y_{1})$. Then it follows from the above that the law of
the pair $(\eta_{0},\eta_{1})$ on $G$ is absolutely continuous with respect to
the law of the pair $(\varphi(\eta_{+}),\varphi(\eta_{R}))$. Moreover, since
$\eta_{+}(X\delta)$ has a positive chance of being in the boundary of any
fixed component of $\mathcal{H}_{+}\setminus\eta$ which intersects $\eta_{+}$,
it follows that the law of $\mathcal{D}$ is absolutely continuous with respect
to the law of $\varphi(\mathcal{D}_{0})$. The result then follows since we
know that $|\varphi^{\prime}|$ is a.s. bounded on a neighborhood of the
closure of $\mathcal{D}_{0}$ (recall (1.1)). ∎

###  3.4. Proofs of main crossing statements

Figure 3.4.  Illustration of the proof of Lemma 3.1. Shown are just the left
boundaries of $\widetilde{\eta}_{0}$ ($\eta_{0}$, dark blue) as viewed from
$\eta_{+}(Y_{0})$ and $\widetilde{\eta}_{1}$ ($\eta_{1}$, dark green) as
viewed from $\eta_{0}(X_{1})$, using the notation from Lemma 3.4. Proposition
3.3 implies that on the event that $I\cap\Upsilon\neq\emptyset$, it is a
positive conditional probability event given $\mathcal{N}$ that a path in
$\mathcal{D}$ from $\eta_{0}(Z_{1})$ to $\eta_{1}$ in $\Upsilon$ passes
through $I$ and $\eta_{1}(Z_{1})\in B(z_{R},\zeta_{0}d_{0})$, as shown.

###### Proof of Lemma 3.1.

See Figure 3.4 for an illustration of the argument. Let $\eta$ be an ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ process in $\mathcal{H}_{+}$ from $0$ to $\infty$
coupled with $\Gamma_{+}$ as a CPI. We assume that $\eta$ is parameterized
according to the quantum natural time of its trunk. Let
$\tau_{\delta}=\inf\\{t\geq\delta^{4/\kappa}:\eta(t)\in\partial\mathcal{H}_{+}\\}$
and let $\mathcal{N}$ be the quantum surface parameterized by the region
disconnected from $\infty$ by $\eta([0,\tau_{\delta}])$. Let also
$d_{0}=\mathrm{diam}(\mathcal{N})$.

Let $\widetilde{\eta}_{0}$, $\widetilde{\eta}_{1}$, $\eta_{0}$, $\eta_{1}$,
$X_{0}$, $Y_{0}$, $X_{1}$, $Y_{1}$, $Z_{1}$ be as in the statement of Lemma
3.4, taken to be conditionally independent of everything else given
$\Gamma_{+}$. On
$\eta_{0}([0,X_{1}])\cap\overline{\eta}_{0}([0,Y_{1}])=\emptyset$ let also
$\mathcal{D}$ be the component of
$\mathcal{H}_{+}\setminus(\eta_{0}\cup\eta_{1})$ between $\eta_{0}$ and
$\eta_{1}$ with $\eta_{0}(Z_{1})$ on its boundary as in Lemma 3.4. Let $Q$ be
the event that
$\eta_{0}([0,X_{1}])\cap\overline{\eta}_{0}([0,Y_{1}])\neq\emptyset$, the part
of $\eta_{0}$ from $\eta_{0}(X_{1})$ to $\overline{\eta}_{0}(Y_{1})$ does not
hit $\partial\mathcal{H}_{+}$, and there does not exist a path $\omega$ in
$\Upsilon_{+}\cap\mathcal{D}$ which connects $\eta_{0}(Z_{1})$ to $\eta_{1}$
with
${\mathfrak{N}}_{\epsilon}(\omega)\leq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.
Fix $\zeta_{1}>0$ and assume that we have chosen $\delta>0$ sufficiently small
so that Lemma 3.4 implies that $\mathbf{P}[Q]\leq\zeta_{1}$.

Let $Z$ be chosen uniformly from the boundary measure on the top of
$\mathcal{N}$ and let $I$ (resp. $\widetilde{I}$) be the interval of quantum
length $\zeta_{0}\delta$ (resp. $3\zeta_{0}\delta/2$) on the top of
$\mathcal{N}$ centered at $Z$. Let $E=\\{I\cap\Upsilon_{+}\neq\emptyset\\}$.
Let $F$ be the event that
$\eta_{0}([0,X_{1}])\cap\overline{\eta}_{0}([0,Y_{1}])=\emptyset$, the part of
$\eta_{0}$ from $\eta_{0}(X_{1})$ to $\overline{\eta}_{0}(Y_{1})$ does not hit
$\partial\mathcal{H}_{+}$, and every path in $\Upsilon_{+}\cap\mathcal{D}$
from $\eta_{0}(Z_{1})$ to $\eta_{1}$ passes through $I$, $\eta_{0}(Z_{1})\in
B(z_{R},\zeta_{0}d_{0})$, and $\partial\mathcal{D}$ has distance at least
$\xi_{0}d_{0}$ from $\partial\mathcal{N}\setminus\widetilde{I}$. Let
$\mathcal{F}$ be the $\sigma$-algebra generated by $\Gamma_{+}$,
$\mathcal{N}$, and $Z$. Note that $E$ is $\mathcal{F}$-measurable. It follows
from Proposition 3.3 that

$\mathbf{P}\big{[}\mathbf{P}[F\,|\,\mathcal{F}]>0\,|\,E\big{]}\to
1\quad\text{as}\quad\xi_{0}\to 0.$

Fix $p_{0}>0$ and let $G=\\{\mathbf{P}[F\,|\,\mathcal{F}]{\bf 1}_{E}\geq
p_{0}{\bf 1}_{E}\\}$. Fix $\zeta_{2}>0$. By making $p_{0},\xi_{0}>0$
sufficiently small we have that $\mathbf{P}[G^{c}\cap E]\leq\zeta_{2}$. We
note that by scale invariance the choice of $p_{0}$ does not depend on
$\delta$. Therefore we can take $\zeta_{1}=p_{0}\zeta_{2}$ and assume that
$\delta>0$ is sufficiently small so that $\mathbf{P}[Q]\leq\zeta_{1}$.

Let $H$ be the event that $E$ occurs and every path $\omega$ in
$\Upsilon_{+}\cap\mathcal{N}$ connecting $B(z_{R},\zeta_{0}d_{0})$ to
$\widetilde{I}$ with distance at least $\xi_{0}d_{0}$ from
$\partial\mathcal{N}\setminus\widetilde{I}$ satisfies
${\mathfrak{N}}_{\epsilon}(\omega)\geq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.
Then we have that

$\displaystyle\mathbf{P}[H]$ $\displaystyle=\mathbf{E}[{\bf 1}_{E}{\bf
1}_{H}]\leq\frac{1}{p_{0}}\mathbf{E}[{\bf 1}_{E}{\bf
1}_{H}\mathbf{P}[F\,|\,\mathcal{F}]]+\mathbf{P}[G^{c}\cap E]$
$\displaystyle=\frac{1}{p_{0}}\mathbf{P}[E,\ H,\ F]+\mathbf{P}[G^{c}\cap
E]\quad\text{($E,H\in\mathcal{F}$)}$
$\displaystyle\leq\frac{1}{p_{0}}\mathbf{P}[Q]+\mathbf{P}[G^{c}]\leq\frac{\zeta_{1}}{p_{0}}+\zeta_{2}\leq
2\zeta_{2}.$

Consider the event that there exists $J$ on the top of $\partial\mathcal{N}$
with quantum length $\zeta_{0}$, intersects $\Upsilon_{+}$, and such that if
$\widetilde{J}$ is the interval on the top of $\partial\mathcal{N}$ with the
same center as $J$ but twice the quantum length every path $\omega$ from
$B(z_{R},\zeta_{0}d_{0})$ to $\widetilde{J}$ in $\Upsilon_{+}\cap\mathcal{N}$
and with distance at least $\xi_{0}d_{0}$ to
$\partial\mathcal{N}\setminus\widetilde{J}$ satisfies
${\mathfrak{N}}_{\epsilon}(\omega)\geq{{\mathfrak{q}}}_{\epsilon}^{\mathrm{HP}}(p)$.
On this event, there is a positive chance that
$\widetilde{I}\subseteq\widetilde{J}$. Therefore the probability of this event
is at most a constant times $\zeta_{2}$, from which the result follows. ∎

###### Proof of Lemma 3.2.

This follows from the same argument used to prove Lemma 3.1. ∎

##  4\. Percolation exploration

The purpose of this section is to describe a type of supercritical percolation
exploration inside of a ${\rm CLE}_{\kappa}$ carpet drawn on top of a quantum
half-plane. In particular, we will decompose the ${\rm CLE}_{\kappa}$ carpet
using “chunks” of ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ processes coupled as
CPIs. Our ultimate aim is to show that two points on the boundary of the
quantum half-plane are likely to be connected by a collection of such chunks
which are in a certain sense “good”. The first step in this (Section 4.1) is
to define a version of the exploration where we start with chunks which each
consist of order $2^{-K}$ units of quantum natural time where $K\in\mathbf{N}$
large, with the chunk size growing larger until we see chunks with of order
$2^{-J}$ units of quantum natural time where $J\in\mathbf{N}$ with $J\leq K$
is fixed, and then getting smaller with chunks with quantum natural time
tending to $0$. One can think of the chunks in this path as being analogous to
a path between two boundary points on $\partial\mathbf{H}$ which passes
through at most a fixed number of squares of each scale in a Whitney cube
decomposition of $\mathbf{H}$. We will then explain in Section 4.2 that it is
likely that this exploration “succeeds” in the sense that it is likely that we
can construct a path of such chunks all of which satisfy an event which allows
us to say that the chunk is “good”. Finally, in Section 4.3 we will explain
how one can take a (subsequential) limit as $K\to\infty$ to construct an
exploration which starts with arbitrarily small chunks and connects two
boundary points.

###  4.1. Definition of the exploration

Suppose that $\mathcal{H}=(\mathbf{H},h,0,\infty)$ is a quantum half-plane.
Suppose that we have fixed $\delta_{0},a_{0}\in(0,1)$ and
$\alpha_{\mathrm{PERC}}\in(1-\kappa/4,1)$. Let $x_{-1}<0<x_{1}$ be such that
$\nu_{h}([x_{-1},0])=\nu_{h}([0,x_{1}])=1$. We assume that $\Gamma$ is a ${\rm
CLE}_{\kappa}$ on $\mathbf{H}$. We will describe below an adaptive exploration
of $\Gamma$ and $\mathcal{H}$. We will assume that the various ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ curves in what follows are coupled with $\Gamma$
as CPIs. Suppose that $\eta$ is an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ in
$\mathbf{H}$ from $0$ to $\infty$ which is parameterized by the quantum
natural time of its trunk, $\mathcal{N}_{t}$ is the quantum surface
parameterized by the domain disconnected by $\eta|_{[0,t]}$ from $\infty$, and
$\sigma$ is a stopping time for the filtration generated by $\mathcal{N}_{t}$.
Then the _top_ (resp.  _bottom_) of $\mathcal{N}_{t}$ is
$\partial\mathcal{N}_{t}\cap\mathbf{H}$ (resp.
$\partial\mathcal{N}_{t}\cap\partial\mathbf{H}$).

Figure 4.1.  Illustration of the first few chunks in the percolation
exploration of a ${\rm CLE}_{\kappa}$ decorated quantum half-plane. The chunks
which are “good” are indicated in green and the chunks which are “bad” are
indicated in red. Part of our definition of a chunk being good is that the
quantum surface it disconnects from $\infty$ is simply connected, which is why
the green chunks are homeomorphic to $\mathbf{D}$ while the bad chunks are not
(there are also other ways in which a chunk can be bad). The arrows indicate
the starting points of the ${\rm SLE}_{\kappa}^{0}(\kappa-6)$’s which cut out
the chunks and the squares indicate the points on the tops of the good chunks
whose clockwise boundary length distance to the wedge boundary in which they
are drawn is an integer multiple of $a_{0}2^{-(\kappa/4)K}$; we always take
chunks to start at such points.

####  4.1.1. Increasing chunk sizes

Fix $J,K\in\mathbf{N}$ with $J\leq K$. We consider the following exploration
of $\mathcal{H}$ by ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ curves. Let
$\eta_{1,K}^{\uparrow}$ be an independent ${\rm SLE}_{\kappa}^{0}(\kappa-6)$
curve in $\mathcal{H}$ from $0$ to $\infty$ and let
$\sigma_{1,K}^{\uparrow}=\inf\\{t\geq
2^{-K}\delta_{0}:\eta_{1,K}^{\uparrow}(t)\in\partial\mathcal{H}\\}\wedge
2^{-K}$. Let $\mathcal{N}_{1,K}^{\uparrow}$ be the quantum surface
disconnected from $\infty$ by
$\eta_{1,K}^{\uparrow}([0,\sigma_{1,K}^{\uparrow}])$ and let
$\mathcal{T}_{1,K}^{\uparrow}$ be the top of $\mathcal{N}_{1,K}^{\uparrow}$.
We also let $\mathcal{F}_{1,K}^{\uparrow}$ be the $\sigma$-algebra generated
by the quantum surface $\mathcal{N}_{1,K}^{\uparrow}$ decorated by the path
$\eta_{1,K}^{\uparrow}|_{[0,\sigma_{1,K}^{\uparrow}]}$ and let
$E_{1,K}^{\uparrow}$ be an $\mathcal{F}_{1,K}^{\uparrow}$-measurable event. We
then inductively define $\sigma$-algebras $\mathcal{F}_{j,K}^{\uparrow}$,
quantum half-planes $\mathcal{H}_{j,K}^{\uparrow}$, ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ curves $\eta_{j,K}^{\uparrow}$, events
$E_{j,K}^{\uparrow}$, stopping times $\sigma_{j,K}^{\uparrow}$, and quantum
surfaces $\mathcal{N}_{j,K}^{\uparrow}$ with tops
$\mathcal{T}_{j,K}^{\uparrow}$ as follows. We let
$\mathcal{H}_{j+1,K}^{\uparrow}$ be the quantum surface parameterized by the
unbounded component of
$\mathcal{H}_{j,K}^{\uparrow}\setminus\eta_{j,K}^{\uparrow}([0,\sigma_{j,K}^{\uparrow}])$
and marked by the points $\eta_{j,K}^{\uparrow}(\sigma_{j,K}^{\uparrow})$ and
$\infty$. If $E_{j,K}^{\uparrow}$ occurs, we let $\eta_{j+1,K}^{\uparrow}$ be
an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process in
$\mathcal{H}_{j+1,K}^{\uparrow}$ starting from the rightmost intersection of
$\eta_{j,K}^{\uparrow}([0,\sigma_{j,K}^{\uparrow}])$ with
$\partial\mathcal{H}_{j,K}^{\uparrow}$. For $1\leq i\leq j$, let
$\mathcal{X}_{i,K}^{\uparrow}$ be the set of points in
$\mathcal{T}_{i,K}^{\uparrow}$ whose clockwise boundary length distance to
$\partial\mathcal{H}_{i,K}^{\uparrow}$ is an integer multiple of
$a_{0}2^{-(\kappa/4)K}$. If $E_{j,K}^{\uparrow}$ does not occur, we let
$\eta_{j+1,K}^{\uparrow}$ be an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process in
$\mathcal{H}_{j+1,K}^{\uparrow}$ starting from the rightmost point
$z_{j,K}^{\uparrow}$ on $\partial\mathcal{H}_{j+1,K}^{\uparrow}$ which is:

  * •

to the left of the leftmost point of $\mathcal{T}_{j,K}^{\uparrow}$,

  * •

in $\mathcal{X}_{i,K}^{\uparrow}$ with $i<j$ such that the part of
$\mathcal{T}_{i,K}^{\uparrow}\cap\partial\mathcal{H}_{j+1,K}^{\uparrow}$ to
the left of $z_{j,K}^{\uparrow}$ has quantum length at least
$a_{0}2^{-(\kappa/4)K}$ and with boundary length distance at most
$a_{0}2^{-(\kappa/4)K}$ from $\Upsilon_{+}$.

In the case that there is no such point, we take the starting point of
$\eta_{j+1,K}^{\uparrow}$ to be
$\eta_{j,K}^{\uparrow}(\sigma_{j,K}^{\uparrow})$. We let
$\sigma_{j+1,K}^{\uparrow}=\inf\\{t\geq
2^{-K}\delta_{0}:\eta_{j+1,K}^{\uparrow}(t)\in\partial\mathcal{H}_{j+1,K}^{\uparrow}\\}\wedge
2^{-K}$, let $\mathcal{N}_{j+1,K}^{\uparrow}$ be the quantum surface in
$\mathcal{H}_{j+1,K}^{\uparrow}$ disconnected by
$\eta_{j+1,K}^{\uparrow}([0,\sigma_{j+1,K}^{\uparrow}])$ from $\infty$,
$\mathcal{F}_{j+1,K}^{\uparrow}$ be the $\sigma$-algebra generated by the
quantum surface parameterized by
$\mathrm{int}(\mathrm{cl}(\cup_{i=1}^{j+1}\mathcal{N}_{i,K}^{\uparrow}))$ and