decorated by the paths $\eta_{i,K}^{\uparrow}|_{[0,\sigma_{i,K}^{\uparrow}]}$
for $1\leq i\leq j+1$. Finally, we let $E_{j+1,K}^{\uparrow}$ be an
$\mathcal{F}_{j+1,K}^{\uparrow}$-measurable event. Let $n_{K}^{\uparrow}$ be
the first $j$ so that either

  * •

$x_{1}$ is disconnected from $\infty$, or

  * •

$E_{j,K}^{\uparrow}$ occurs and the boundary length distance along
$\partial\mathcal{H}_{j+1,K}^{\uparrow}$ from $\eta_{j+1,K}^{\uparrow}(0)$ to
$x_{1}$ is at most $1-2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}$.

Suppose that we have defined the exploration procedure for some $J+1\leq M\leq
K$ and we are on the event that $x_{1}$ has not been disconnected from
$\infty$. We then define it for $M-1$ as follows. Let
$\eta_{1,M-1}^{\uparrow}$ be an independent ${\rm SLE}_{\kappa}^{0}(\kappa-6)$
curve in
$\mathcal{H}_{1,M-1}^{\uparrow}=\mathcal{H}_{n_{M}^{\uparrow}+1,M}^{\uparrow}$
from the rightmost intersection of
$\eta_{n_{M},M}^{\uparrow}([0,\sigma_{n_{M},M}^{\uparrow}])$ with
$\partial\mathcal{H}_{n_{M},M}^{\uparrow}$ to $\infty$. Let
$\sigma_{1,M-1}^{\uparrow}=\inf\\{t\geq
2^{-(M-1)}\delta_{0}:\eta_{1,M-1}^{\uparrow}(t)\in\partial\mathcal{H}_{1,M-1}^{\uparrow}\\}\wedge
2^{-(M-1)}$. We then inductively define quantum half-planes
$\mathcal{H}_{j,M-1}^{\uparrow}$, ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ curves
$\eta_{j,M-1}^{\uparrow}$, events $E_{j,M-1}^{\uparrow}$, stopping times
$\sigma_{j,M-1}^{\uparrow}$, and quantum surfaces
$\mathcal{N}_{j,M-1}^{\uparrow}$ with tops $\mathcal{T}_{j,M-1}^{\uparrow}$ as
follows. We let $\mathcal{H}_{j+1,M-1}^{\uparrow}$ be the quantum surface
parameterized by the unbounded component of
$\mathcal{H}_{j,M-1}^{\uparrow}\setminus\eta_{j,M-1}^{\uparrow}([0,\sigma_{j,M-1}^{\uparrow}])$
and marked by the points $\eta_{j,M-1}^{\uparrow}(\sigma_{j,M-1}^{\uparrow})$
and $\infty$. If $E_{j,M-1}^{\uparrow}$ occurs, we let
$\eta_{j+1,M-1}^{\uparrow}$ be an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process
in $\mathcal{H}_{j+1,M-1}^{\uparrow}$ starting from the rightmost intersection
of $\eta_{j,M-1}^{\uparrow}([0,\sigma_{j,M-1}^{\uparrow}])$ with
$\partial\mathcal{H}_{j,M-1}^{\uparrow}$. For $1\leq i\leq j$, let
$\mathcal{X}_{i,M-1}^{\uparrow}$ be the set of points in
$\mathcal{T}_{i,M-1}^{\uparrow}$ whose clockwise boundary length distance to
$\partial\mathcal{H}_{i,M-1}^{\uparrow}$ is an integer multiple of
$a_{0}2^{-(\kappa/4)(M-1)}$. If $E_{j,M-1}^{\uparrow}$ does not occur, we let
$\eta_{j+1,M-1}^{\uparrow}$ be an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process
in $\mathcal{H}_{j+1,M-1}^{\uparrow}$ starting from the rightmost point
$z_{j,M-1}^{\uparrow}$ on $\mathcal{H}_{j+1,M-1}^{\uparrow}$ which is:

  * •

to the left of the leftmost point of $\mathcal{T}_{j,M-1}^{\uparrow}$ and

  * •

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

In the case that there is no such point, we take the starting point of
$\eta_{j+1,M-1}^{\uparrow}$ to be
$\eta_{j,M-1}^{\uparrow}(\sigma_{j,M-1}^{\uparrow})$. We let
$\sigma_{j+1,M-1}^{\uparrow}=\inf\\{t\geq
2^{-(M-1)}\delta_{0}:\eta_{j+1,M-1}^{\uparrow}(t)\in\partial\mathcal{H}_{j+1,M-1}^{\uparrow}\\}\wedge
2^{-(M-1)}$, let $\mathcal{N}_{j+1,M-1}^{\uparrow}$ be the quantum surface in
$\mathcal{H}_{j+1,M-1}^{\uparrow}$ disconnected by
$\eta_{j+1,M-1}^{\uparrow}([0,\sigma_{j+1,M-1}^{\uparrow}])$ from $\infty$,
$\mathcal{F}_{j+1,M-1}^{\uparrow}$ the $\sigma$-algebra generated by
$\mathcal{F}_{j,M}^{\uparrow}$ and the quantum surface parameterized by
$\mathrm{int}(\mathrm{cl}(\cup_{i=1}^{j+1}\mathcal{N}_{i,M-1}^{\uparrow}))$
and decorated by the paths
$\eta_{i,M-1}^{\uparrow}|_{[0,\sigma_{i,M-1}^{\uparrow}]}$ for $1\leq i\leq
j+1$. Finally, we let $E_{j+1,M-1}^{\uparrow}$ be an
$\mathcal{F}_{j+1,M-1}^{\uparrow}$-measurable event. Let $n_{M-1}^{\uparrow}$
be the first $j$ so that either

  * •

$x_{1}$ is disconnected from $\infty$, or

  * •

$E_{j,M-1}^{\uparrow}$ occurs and the boundary length distance in
$\mathcal{H}_{j+1,M-1}^{\uparrow}$ from $\eta_{j+1,M-1}^{\uparrow}(0)$ to
$x_{1}$ is at most $1-2^{-\alpha_{\mathrm{PERC}}(\kappa/4)(M-1)}$.

####  4.1.2. Decreasing chunk sizes

We will now define the continuation of the exploration after scale $J$ as
above. The definition will be similar except the chunk sizes will be
decreasing (rather than increasing) as the exploration gets closer to $x_{1}$.

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

  * •

to the left of the leftmost point of $\mathcal{T}_{j,J}^{\downarrow}$ and

  * •

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

In the case that there is no such point, we take the starting point of
$\eta_{j+1,J}^{\downarrow}$ to be
$\eta_{j,J}^{\downarrow}(\sigma_{j,J}^{\downarrow})$. We let
$\sigma_{j+1,J}^{\downarrow}=\inf\\{t\geq
2^{-J}\delta_{0}:\eta_{j+1,J}^{\downarrow}(t)\in\partial\mathcal{H}_{j+1,J}^{\downarrow}\\}\wedge
2^{-J}$, let $\mathcal{N}_{j+1,J}^{\downarrow}$ be the quantum surface in
$\mathcal{H}_{j+1,J}^{\downarrow}$ disconnected by
$\eta_{j+1,J}^{\downarrow}([0,\sigma_{j+1,J}^{\downarrow}])$ from $\infty$,
$\mathcal{F}_{j+1,J}^{\downarrow}$ the $\sigma$-algebra generated by
$\mathcal{F}_{1,J}^{\downarrow}$ and the quantum surface parameterized by
$\mathrm{int}(\mathrm{cl}(\cup_{i=1}^{j+1}\mathcal{N}_{i,J}^{\downarrow}))$
and decorated by the paths
$\eta_{i,J}^{\downarrow}|_{[0,\sigma_{i,J}^{\downarrow}]}$ for $1\leq i\leq
j+1$. Finally, we let $E_{j+1,J}^{\downarrow}$ be an
$\mathcal{F}_{j+1,J}^{\downarrow}$-measurable event. Let $n_{J}^{\downarrow}$
be the first $j$ so that either

  * •

$x_{1}$ is disconnected from $\infty$, or

  * •

$E_{j,J}^{\downarrow}$ occurs and the boundary length distance in
$\mathcal{H}_{j+1,J}^{\downarrow}$ from $\eta_{j+1,J}^{\downarrow}(0)$ to
$x_{1}$ is at most $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)J}$.

Suppose that we have defined the downward exploration procedure for some
$M\geq J$. We then define it for $M+1$ as follows. Let
$\eta_{1,M+1}^{\downarrow}$ be an independent ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ curve in
$\mathcal{H}_{1,M+1}^{\downarrow}=\mathcal{H}_{n_{M}^{\downarrow}+1,M}^{\downarrow}$
from the rightmost intersection of
$\eta_{j,M}^{\downarrow}([0,\sigma_{j,M}^{\downarrow}])$ with
$\partial\mathcal{H}_{j,M}^{\uparrow}$ to $\infty$. Let
$\sigma_{1,M+1}^{\downarrow}=\inf\\{t\geq
2^{-(M+1)}\delta_{0}:\eta_{1,M+1}^{\downarrow}(t)\in\partial\mathcal{H}_{1,M+1}^{\downarrow}\\}\wedge
2^{-(M+1)}$. We then inductively define quantum half-planes
$\mathcal{H}_{j,M+1}^{\downarrow}$, ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ curves
$\eta_{j,M+1}^{\downarrow}$, events $E_{j,M+1}^{\downarrow}$, stopping times
$\sigma_{j,M+1}^{\downarrow}$, and quantum surfaces
$\mathcal{N}_{j,M+1}^{\downarrow}$ with tops
$\mathcal{T}_{j,M+1}^{\downarrow}$ as follows. We let
$\mathcal{H}_{j+1,M+1}^{\downarrow}$ be the quantum surface parameterized by
the unbounded component of
$\mathcal{H}_{j,M+1}^{\downarrow}\setminus\eta_{j,M+1}^{\downarrow}([0,\sigma_{j,M+1}^{\downarrow}])$
and marked by the points
$\eta_{j,M+1}^{\uparrow}(\sigma_{j,M+1}^{\downarrow})$ and $\infty$. If
$E_{j,M+1}^{\downarrow}$ occurs, we let $\eta_{j+1,M+1}^{\downarrow}$ be an
${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process in
$\mathcal{H}_{j+1,M+1}^{\downarrow}$ starting from the rightmost intersection
of $\eta_{j,M+1}^{\downarrow}([0,\sigma_{j,M+1}^{\downarrow}])$ with
$\partial\mathcal{H}_{j,M+1}^{\downarrow}$. For $1\leq i\leq j$, let
$\mathcal{X}_{i,M+1}^{\downarrow}$ be the set of points in
$\mathcal{T}_{i,M+1}^{\downarrow}$ whose clockwise boundary length distance to
$\partial\mathcal{H}_{i,M+1}^{\downarrow}$ is an integer multiple of
$a_{0}2^{-(\kappa/4)(M+1)}$. If $E_{j,M+1}^{\downarrow}$ does not occur, we
let $\eta_{j+1,M+1}^{\downarrow}$ be an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$
process in $\mathcal{H}_{j+1,M+1}^{\downarrow}$ starting from the rightmost
point $z_{j,M+1}^{\downarrow}$ on $\partial\mathcal{H}_{j+1,M+1}^{\downarrow}$
which is:

  * •

to the left of the leftmost point of $\mathcal{T}_{j,M+1}^{\downarrow}$ and

  * •

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

In the case that there is no such point, we take the starting point of
$\eta_{j+1,M+1}^{\downarrow}$ to be
$\eta_{j,M}^{\downarrow}(\sigma_{j,M}^{\downarrow})$. We let
$\sigma_{j+1,M+1}^{\downarrow}=\inf\\{t\geq
2^{-(M+1)}\delta_{0}:\eta_{j+1,M+1}^{\downarrow}(t)\in\partial\mathcal{H}_{j+1,M+1}^{\downarrow}\\}\wedge
2^{-(M+1)}$, let $\mathcal{N}_{j+1,M+1}^{\downarrow}$ be the quantum surface
in $\mathcal{H}_{j+1,M+1}^{\downarrow}$ disconnected by
$\eta_{j+1,M+1}^{\downarrow}([0,\sigma_{j+1,M+1}^{\downarrow}])$ from
$\infty$, $\mathcal{F}_{j+1,M+1}^{\downarrow}$ the $\sigma$-algebra generated
by $\mathcal{F}_{j+1,M}^{\downarrow}$ and the quantum surface parameterized by
$\mathrm{int}(\mathrm{cl}(\cup_{i=1}^{j+1}\mathcal{N}_{i,M+1}^{\downarrow}))$
and decorated by the paths
$\eta_{i,M+1}^{\downarrow}|_{[0,\sigma_{i,M+1}^{\downarrow}]}$ for $1\leq
i\leq j+1$. Finally, we let $E_{j+1,M+1}^{\downarrow}$ be an
$\mathcal{F}_{j+1,M+1}^{\downarrow}$-measurable event. Let
$n_{M+1}^{\downarrow}$ be the first $j$ so that either

  * •

$x_{1}$ is disconnected from $\infty$, or

  * •

$E_{j,M-1}^{\downarrow}$ occurs and the boundary length distance in
$\mathcal{H}_{j+1,M+1}^{\downarrow}$ from $\eta_{j+1,M+1}^{\downarrow}(0)$ to
$x_{1}$ is at most $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)(M+1)}$.

####  4.1.3. Definition of events

Figure 4.2.  Illustration of two of the ways in which the exploration can
fail. The blue region illustrates the quantum surface parameterized by the
exploration before discovering the chunk which led to the failure. On the
left, the exploration failed due to a large downward jump (which also
disconnected $x_{1}$ from $\infty$. On the right, the exploration failed due
to a large upward jump (corresponding to the discovery of a large ${\rm
CLE}_{\kappa}$ loop).

Fix $c_{F}>1$; we will choose its value later. We let $F_{K}^{\uparrow}$ be
the event that

  * •

$x_{-1}$ or $x_{1}$ is disconnected from $\infty$ by one of
$\eta_{j,K}^{\uparrow}|_{[0,\sigma_{j,K}^{\uparrow}]}$ for $1\leq j\leq
n_{K}^{\uparrow}$,

  * •

there exists $1\leq j\leq n_{K}^{\uparrow}$ so that
$\eta_{j,K}^{\uparrow}|_{[0,\sigma_{j,K}^{\uparrow}]}$ makes a jump (downward
or upward) of size at least $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}$, or

  * •

$n_{K}^{\uparrow}\geq c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}$.

For each $J\leq M\leq K-1$, we let $F_{M}^{\uparrow}$ be the event that

  * •

$x_{-1}$ or $x_{1}$ is disconnected from $\infty$ by one of
$\eta_{j,M}^{\uparrow}|_{[0,\sigma_{j,M}^{\uparrow}]}$ for $1\leq j\leq
n_{M}^{\uparrow}$,

  * •

there exists $1\leq j\leq n_{M}^{\uparrow}$ so that
$\eta_{j,M}^{\uparrow}|_{[0,\sigma_{j,M}^{\uparrow}]}$ makes a jump (downward
or upward) of size at least $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$,

  * •

$n_{M}^{\uparrow}\geq c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$, or

  * •

the curves $\eta_{j,M}^{\uparrow}|_{[0,\sigma_{j,M}^{\uparrow}]}$ for $1\leq
j\leq n_{M}^{\uparrow}$ disconnect all of the tops
$\mathcal{T}_{j,M+1}^{\uparrow}$ for $1\leq j\leq n_{M+1}^{\uparrow}$ from
$\infty$.

For each $J\leq M$, we let $F_{M}^{\downarrow}$ be the event that

  * •

$x_{-1}$ or $x_{1}$ is disconnected from $\infty$ by one of
$\eta_{j,M}^{\downarrow}|_{[0,\sigma_{j,M}^{\downarrow}]}$ for $1\leq j\leq
n_{M}^{\downarrow}$,

  * •

there exists $1\leq j\leq n_{M}^{\downarrow}$ so that
$\eta_{j,M}^{\downarrow}|_{[0,\sigma_{j,M}^{\downarrow}]}$ makes a jump
(downward or upward) of size at least
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$,

  * •

$n_{M}^{\downarrow}\geq c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$
($M\geq J+1$) or $n_{M}^{\downarrow}\geq c_{F}2^{(\kappa/4)M}$ ($M=J$), or

  * •

the curves $\eta_{j,M}^{\downarrow}|_{[0,\sigma_{j,M}^{\downarrow}]}$ for
$1\leq j\leq n_{M}^{\downarrow}$ disconnect all of the tops

    * –

(if $M\geq J+1$) $\mathcal{T}_{j,M-1}^{\downarrow}$ for $1\leq j\leq
n_{M-1}^{\downarrow}$

    * –

(if $M=J$) $\mathcal{T}_{j,J}^{\uparrow}$ for $1\leq j\leq n_{J}^{\uparrow}$

from $\infty$.

We say that the exploration _fails_ if one of $F_{M}^{\uparrow}$ for $J\leq
M\leq K$ or $F_{M}^{\downarrow}$ for $J\leq M$ occurs.

###  4.2. Estimates

######  Lemma 4.1.

Fix $C_{0}>0$. There exist constants $c,\delta_{0},a_{0}\in(0,1)$ and
$c_{F}>0$ so that with $p_{0}=1-a_{0}$ the following is true for every
$J\in\mathbf{N}$. Assume that

  1. (i)

$\mathbf{P}[E_{j,M}^{\uparrow}\,|\,\mathcal{F}_{j-1,M}^{\uparrow}]\geq p_{0}$
for all $j\geq 1$ and $J\leq M\leq K$,

  2. (ii)

$\mathbf{P}[E_{j,M}^{\downarrow}\,|\,\mathcal{F}_{j-1,M}^{\downarrow}]\geq
p_{0}$ for all $j\geq 1$ and $J\leq M$ and

  3. (iii)

$E_{j,M}^{\uparrow}$ (resp. $E_{j,M}^{\downarrow}$) implies that the top
boundary length of $\mathcal{N}_{j,M}^{\uparrow}$ (resp.
$\mathcal{N}_{j,M}^{\downarrow}$) is at most $C_{0}2^{-(\kappa/4)M}$ for all
$j\geq 1$ and $J\leq M\leq K$ (resp. $J\leq M$).

Then the probability that the exploration as defined in Section 4.1 fails is
$O(2^{-cJ})$.

###### Proof.

We are going to show that the following is true. There exists a constant $c>0$
so that for each $J\leq M\leq K$ we have
$\mathbf{P}[F_{M}^{\uparrow}]=O(2^{-cM})$ and for $M\geq J$ we have
$\mathbf{P}[F_{M}^{\downarrow}]=O(2^{-cM})$. Then the result follows by
applying a union bound. We will explain carefully how to obtain the upper
bound for $\mathbf{P}[F_{K}^{\uparrow}]$ and then subsequently explain the
necessary modifications to bound $\mathbf{P}[F_{M}^{\uparrow}]$ for $J\leq
M\leq K$ and $\mathbf{P}[F_{M}^{\downarrow}]$ for $J\leq M$.

Step 1. Definition of boundary length processes. For each $j$ and $M$, we let
$T_{j,M}^{\uparrow}$ denote the quantum length of the top
$\mathcal{T}_{j,M}^{\uparrow}$ of $\mathcal{N}_{j,M}^{\uparrow}$. We also let
$B_{L,j,M}^{\uparrow}$ (resp. $B_{R,j,M}^{\uparrow}$) denote the quantum
length of the part of the bottom of $\mathcal{N}_{j,M}^{\uparrow}$ which is to
the left (resp. right) of $\eta_{j,M}^{\uparrow}(0)$. Finally, we define
$L_{j,M}^{\uparrow}$ (resp. $R_{j,M}^{\uparrow}$) to be equal to the quantum
length of the part of
$\partial\mathcal{H}_{j,M}^{\uparrow}\setminus\partial\mathcal{H}$ to the left
(resp. right) of $\eta_{j,M}^{\uparrow}(0)$ minus the quantum length of the
part of $\partial\mathcal{H}\setminus\partial\mathcal{H}_{j,M}^{\uparrow}$ to
the left (resp. right) of $0$. We define
$L_{0,M}^{\uparrow}=L_{n_{M-1},M-1}^{\uparrow}$ and
$R_{0,M}^{\uparrow}=R_{n_{M-1},M-1}^{\uparrow}$. On
$(E_{j,M}^{\uparrow})^{c}$, we let $S_{j,M}^{\uparrow}$ be the quantum length
of the interval on $\partial\mathcal{H}_{j+1,M}^{\uparrow}$ from
$\eta_{j+1,M}^{\uparrow}(0)$ to the leftmost point on the top of
$\mathcal{N}_{j,M}^{\uparrow}$. For each $j\geq 0$, we then have that

(4.1)
$L_{j+1,M}^{\uparrow}-L_{j,M}^{\uparrow}=(T_{j,M}^{\uparrow}-B_{L,j,M}^{\uparrow}){\bf
1}_{E_{j,M}^{\uparrow}}-(B_{L,j,M}^{\uparrow}+S_{j,M}^{\uparrow}){\bf
1}_{(E_{j,M}^{\uparrow})^{c}}.$

We similarly have that

(4.2) $R_{j+1,M}^{\uparrow}-R_{j,M}^{\uparrow}=-B_{R,j,M}^{\uparrow}{\bf
1}_{E_{j,M}^{\uparrow}}+(T_{j,M}^{\uparrow}-B_{R,j,M}^{\uparrow}+S_{j,M}^{\uparrow}){\bf
1}_{(E_{j,M}^{\uparrow})^{c}}.$

Step 2. Dominating the boundary length processes. We are now going to define a
sequence $\widetilde{L}_{j,M}^{\uparrow}$ which we can use to dominate
$L_{j,M}^{\uparrow}$ from above and a sequence
$\widetilde{R}_{j,M}^{\uparrow}$ which we can use to dominate
$R_{j,M}^{\uparrow}$ from below.

We set $\widetilde{L}_{0,K}^{\uparrow}=0$. We define
$\widetilde{L}_{j,K}^{\uparrow}$ using the recurrence relation for $j\geq 0$

(4.3)
$\widetilde{L}_{j+1,K}^{\uparrow}-\widetilde{L}_{j,K}^{\uparrow}=(T_{j,K}^{\uparrow}-B_{L,j,K}^{\uparrow}-a_{0}2^{-(\kappa/4)K}){\bf
1}_{E_{j,K}^{\uparrow}}-B_{L,j,K}^{\uparrow}{\bf
1}_{(E_{j,K}^{\uparrow})^{c}}.$

We then inductively define $\widetilde{L}_{j,M}^{\uparrow}$ for $M\leq K-1$ by
setting
$\widetilde{L}_{0,M}^{\uparrow}=\widetilde{L}_{n_{M-1}^{\uparrow},M-1}^{\uparrow}$
and we define $\widetilde{L}_{j,M}^{\uparrow}$ using the recurrence relation
for $j\geq 0$

(4.4)
$\widetilde{L}_{j+1,M}^{\uparrow}-\widetilde{L}_{j,M}^{\uparrow}=(T_{j,M}^{\uparrow}-B_{L,j,M}^{\uparrow}-a_{0}2^{-(\kappa/4)M}){\bf
1}_{E_{j,M}^{\uparrow}}-B_{L,j,M}^{\uparrow}{\bf
1}_{(E_{j,M}^{\uparrow})^{c}}.$

Let $N_{j,M}^{\uparrow}=\sum_{i=1}^{j}{\bf 1}_{(E_{i,M}^{\uparrow})^{c}}$ and
let $\widetilde{\Delta}_{j,M}^{\uparrow}$ be the sum of
$N_{j,M}^{\uparrow}(a_{0}2^{-(\kappa/4)M})$ and the $N_{j,M}^{\uparrow}$
largest upward jumps made by either the left or the right boundary length
processes up to the time that $\mathcal{N}_{j,M}^{\uparrow}$ has been
explored. One can see that
$\widetilde{L}_{j,M}^{\uparrow}-\widetilde{\Delta}_{j,M}^{\uparrow}\leq
L_{j,M}^{\uparrow}$ for all $j,M$ because in the definition of the
exploration, we have a slide to the left of size at most
$a_{0}2^{-(\kappa/4)M}$ on the top of each good chunk while in the definition
of $\widetilde{L}_{j,M}^{\uparrow}$ we always have such a shift. Moreover, in
the definition of $L_{j,M}^{\uparrow}$ we slide to the left over
$N_{j,M}^{\uparrow}$ loops of $\Gamma$ previous discovered by the exploration
as well as at most $N_{j,M}^{\uparrow}(a_{0}2^{-(\kappa/4)M})$ units of extra
length while $\widetilde{\Delta}_{j,M}^{\uparrow}$ gives the sum of the
lengths of the longest $N_{j,M}^{\uparrow}$ loops discovered by the
exploration together with this extra length.

We also set $\widetilde{R}_{0,K}^{\uparrow}=0$. We define
$\widetilde{R}_{j,K}^{\uparrow}$ using the recurrence relation for $j\geq 0$

(4.5)
$\widetilde{R}_{j+1,K}^{\uparrow}-\widetilde{R}_{j,K}^{\uparrow}=(a_{0}2^{-(\kappa/4)K}-B_{R,j,K}^{\uparrow}){\bf
1}_{E_{j,K}^{\uparrow}}+(T_{j,K}^{\uparrow}-B_{R,j,K}^{\uparrow}){\bf
1}_{(E_{j,K}^{\uparrow})^{c}}.$

We then inductively define $\widetilde{R}_{j,M}^{\uparrow}$ for $M\leq K-1$ by
setting
$\widetilde{R}_{0,M}^{\uparrow}=\widetilde{R}_{n_{M-1}^{\uparrow},M-1}^{\uparrow}$
and we define $\widetilde{R}_{j,M}^{\uparrow}$ using the recurrence relation
for $j\geq 0$

(4.6)
$\widetilde{R}_{j+1,M}^{\uparrow}-\widetilde{R}_{j,M}^{\uparrow}=(a_{0}2^{-(\kappa/4)M}-B_{R,j,M}^{\uparrow}){\bf
1}_{E_{j,M}^{\uparrow}}+(T_{j,M}^{\uparrow}-B_{R,j,M}^{\uparrow}){\bf
1}_{(E_{j,M}^{\uparrow})^{c}}.$

One can similarly see that
$\widetilde{R}_{j,M}^{\uparrow}+\widetilde{\Delta}_{j,M}^{\uparrow}\geq
R_{j,M}^{\uparrow}$ for all $j,M$.

Step 3. Analysis of dominating boundary length processes.

Lemma C.2 implies that
$\mathbf{E}[B_{R,j,K}^{\uparrow}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]\gtrsim(\log\delta_{0}^{-1})\delta_{0}^{\kappa/4}2^{-(\kappa/4)K}$.
We also have that

$\displaystyle\mathbf{E}[T_{j,K}^{\uparrow}{\bf
1}_{(E_{j,K}^{\uparrow})^{c}}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]$
$\displaystyle\leq\mathbf{E}[T_{j,K}^{\uparrow}{\bf
1}_{\\{\sigma_{j,K}^{\uparrow}=2^{-K}\\}}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]+\mathbf{E}[T_{j,K}^{\uparrow}{\bf
1}_{(E_{j,K}^{\uparrow})^{c}}{\bf
1}_{\\{\sigma_{j,K}^{\uparrow}<2^{-K}\\}}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}].$

Lemma C.4 implies that the first term on the right hand side is at most a
constant times $\delta_{0}^{\kappa/4}2^{-(\kappa/4)K}$. For the second term on
the right hand side, fix $p\in(1,4/\kappa)$ and let $q>1$ be such that
$p^{-1}+q^{-1}=1$. Then

$\displaystyle\mathbf{E}[T_{j,K}^{\uparrow}{\bf
1}_{(E_{j,K}^{\uparrow})^{c}}{\bf
1}_{\\{\sigma_{j,K}^{\uparrow}<2^{-K}\\}}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]$
$\displaystyle\leq\mathbf{E}[(T_{jK}^{\uparrow})^{p}{\bf
1}_{\\{\sigma_{j,K}^{\uparrow}<2^{-K}\\}}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]^{1/p}\mathbf{P}[(E_{j,K}^{\uparrow})^{c}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]^{1/q}.$

The first factor on the right hand side is at most a constant times
$2^{-(\kappa/4)K}$ by [4, Chapter VIII, Proposition 4]. The second term can be
made arbitrarily small by making $a_{0}>0$ sufficiently small. In particular,
we choose $a_{0}>0$ sufficiently small so that the second term is at most
$\delta_{0}^{\kappa/4}$.

Therefore by choosing $\delta_{0}>0$ sufficiently small and then $a_{0}>0$
sufficiently small given our choice of $\delta_{0}$ we see for a constant
$d_{0}>0$ that

(4.7)
$\displaystyle\mathbf{E}[\widetilde{R}_{j,K}^{\uparrow}-\widetilde{R}_{j-1,K}^{\uparrow}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]\leq-
d_{0}2^{-(\kappa/4)K}.$

As the increments
$\widetilde{R}_{j,K}^{\uparrow}-\widetilde{R}_{j-1,K}^{\uparrow}-\mathbf{E}[\widetilde{R}_{j,K}^{\uparrow}-\widetilde{R}_{j-1,K}^{\uparrow}\,|\,\mathcal{F}_{j-1,K}^{\uparrow}]$
form a martingale difference sequence with a bounded $p$th moment for every
$p\in(1,4/\kappa)$, it follows from (4.7) and [6] that for every
$p\in(1,4/\kappa)$ there exist constants $c_{0},c_{1},c_{2}>0$ so that

(4.8)
$\mathbf{P}[\widetilde{R}_{N,K}^{\uparrow}\notin[-c_{0}2^{-(\kappa/4)K}N,-c_{1}2^{-(\kappa/4)K}N]]\leq
c_{2}N^{1-p}$

and by Doob’s maximal inequality the same holds $\min_{1\leq n\leq
N}\widetilde{R}_{n,K}^{\uparrow}$ in place of
$\widetilde{R}_{N,K}^{\uparrow}$. A similar analysis implies that there exist
constants $c_{3},c_{4},c_{5}>0$ so that

(4.9)
$\mathbf{P}[\widetilde{L}_{N,K}^{\uparrow}\notin[c_{3}2^{-(\kappa/4)K}N,c_{4}2^{-(\kappa/4)K}N]]\leq
c_{5}N^{1-p}$

and by Doob’s maximal inequality the same holds $\max_{1\leq n\leq
N}\widetilde{L}_{n,K}^{\uparrow}$ in place of
$\widetilde{L}_{n,K}^{\uparrow}$.

Let us now collect a few observations from the above. Let
$\widetilde{n}_{K}^{\uparrow}$ be the first $j$ so that
$\widetilde{R}_{j,K}^{\uparrow}$ is at most
$-2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}$.

  1. (i)

By (4.8), we have that

$\mathbf{P}[c_{0}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}\leq\widetilde{n}_{K}^{\uparrow}\leq
c_{1}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}]=1-O(2^{-cK}).$

  2. (ii)

By (4.8) and (4.9), we have that

$\mathbf{P}[\widetilde{L}_{\widetilde{n}_{K}^{\uparrow},K}^{\uparrow}\geq
c_{3}2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}]=1-O(2^{-cK}).$

  3. (iii)

On $\widetilde{n}_{K}^{\uparrow}\leq
c_{1}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}$, the amount of quantum natural
time elapsed by the first $\widetilde{n}_{K}^{\uparrow}$ chunks is at most
$c_{1}2^{((1-\alpha_{\mathrm{PERC}})(\kappa/4)-1)K}$. Therefore (from the
explicit form of the Lévy measure) on $\widetilde{n}_{K}^{\uparrow}\leq
c_{1}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}$, the number of jumps that
either the left or right boundary length processes make of size at least
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}$ is stochastically dominated by a
Poisson random variable with mean a constant times

$(2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K})^{-4/\kappa}\times
2^{((1-\alpha_{\mathrm{PERC}})(\kappa/4)-1)K}=2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4-1)K}.$

As $\kappa\in(8/3,4)$ we note that $(1-\alpha_{\mathrm{PERC}})(\kappa/4-1)<0$.
Altogether, we conclude that the probability that either the left or right
boundary length processes make a jump of size at least
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}$ is $O(2^{-cK})$.

Step 4. Analysis of boundary length processes. Since we have that
$\widetilde{L}_{j,M}^{\uparrow}-\widetilde{\Delta}_{j,M}^{\uparrow}\leq
L_{j,M}^{\uparrow}$ and
$\widetilde{R}_{j,M}^{\uparrow}+\widetilde{\Delta}_{j,M}^{\uparrow}\geq
R_{j,M}^{\uparrow}$, it follows from the above and Lemma C.5 that (possibly
changing the values of $c,c_{0},c_{1}$) we have

  1. (i)

$\mathbf{P}[c_{0}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}\leq
n_{K}^{\uparrow}\leq
c_{1}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}]=1-O(2^{-cK}).$

  2. (ii)

We have that

$\mathbf{P}[L_{n_{K}^{\uparrow},K}^{\uparrow}\geq
c_{3}2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}]=1-O(2^{-cK}).$

  3. (iii)

The probability that either the left or right boundary length processes make a
jump of size at least $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}$ is
$O(2^{-cK})$.

Step 5. Completion of the proof. On the event that the above hold, we consider
the exploration starting at the next scale. As in the case $M=K$, we consider
the dominating boundary length processes. We note that when we start the
exploration for $M=K-1$, on the event described in (ii), the amount of
boundary length to the left of the starting point which is contained in chunks
for which $E_{j,K}^{\uparrow}$ occurs is at least a constant times
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}$. On the event described in (i), the
number of such chunks is at most
$c_{1}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}$. Since the top boundary
length of each such chunk is at most $C_{0}2^{-(\kappa/4)K}$, if $A$ is the
number of chunks with top boundary length at most $a_{0}2^{-(\kappa/4)K}$ and
$B$ is the number of the remaining chunks, we have that

$c_{3}2^{-\alpha_{\mathrm{PERC}}(\kappa/4)K}\leq a_{0}2^{-(\kappa/4)K}\times
A+C_{0}2^{-(\kappa/4)K}\times B.$

By decreasing the value of $a_{0}>0$ if necessary, the only way that this
inequality can hold is if $B$ is at least a constant times
$2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)K}$.

The same analysis as for $M=K$ thus implies for $M=K-1$ that

  1. (i)

By (4.8), we have that

$\mathbf{P}[c_{0}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}\leq
n_{M}^{\uparrow}\leq
c_{1}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}]=1-O(2^{-cK}).$

  2. (ii)

By (4.8) and (4.9), we have that

$\mathbf{P}[L_{n_{M}^{\uparrow},M}^{\uparrow}\geq
c_{3}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}]=1-O(2^{-cM}).$

  3. (iii)

On $n_{M}^{\uparrow}\leq c_{1}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$, the
amount of quantum natural time elapsed by the chunks is at most
$c_{1}2^{((1-\alpha_{\mathrm{PERC}})(\kappa/4)-1)M}$. Therefore (from the
explicit form of the Lévy measure) the probability that either the left or
right boundary length processes make a jump of size at least a constant times
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$ is $O(2^{-cM})$.

We can therefore iterate this for the rest of the steps in the upward
exploration and likewise for the downward exploration to obtain the result. ∎

###  4.3. Limiting exploration

The exploration described in Section 4.1 and the result of Lemma 4.1 from
Section 4.2 shows that with probability $1-O(e^{-cJ})$ there exists a path of
“good chunks” in a percolation exploration which starts with a chunk
corresponding to an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ run for time of order
$2^{-K}$ which forms a neighborhood of $0$ and terminates precisely at
$x_{1}$, the point on $\mathbf{R}_{+}$ so that $\nu_{h}([0,x_{1}])=1$. The
purpose of this section is to show that one can construct such an exploration
which starts precisely at $0$ and ends precisely at $x_{1}$ by taking a limit
as $K\to\infty$ (Lemma 4.2). We will then extend the construction to the
setting of a quantum disk in Lemma 4.4.

######  Lemma 4.2.

Suppose that $\mathcal{H}=(\mathbf{H},h,0,\infty)$ is a quantum half-plane.
Let $x_{1}\in\mathbf{R}_{+}$ be such that $\nu_{h}([0,x_{1}])=1$. Fix
$J\in\mathbf{N}$ and suppose that we have families of events
$E_{i,M}^{\uparrow}$, $E_{i,M}^{\downarrow}$ for path-decorated quantum
surfaces for each $i\in\mathbf{N}$ and $M\geq J$ so that if we fix $K\geq J$
and perform the exploration as in Section 4.1 with $J\leq M\leq K$ then the
conditions of Lemma 4.1 are satisfied.

There exist families of quantum surfaces
$(\mathcal{E}_{M,J}^{\uparrow})_{M\geq J}$ and
$(\mathcal{E}_{M,J}^{\downarrow})_{M\geq J}$ coupled with $\mathcal{H}$ so
that the following are true.

  1. (i)

$\mathcal{E}_{M,J}^{\uparrow}$ is decreasing in $M$ and
$\mathcal{E}_{M,J}^{\downarrow}$ is increasing in $M$.

  2. (ii)

For each $M\geq J$, the conditional law of
$\mathcal{H}\setminus\mathcal{E}_{M,J}^{\uparrow}$ given
$\mathcal{E}_{M,J}^{\uparrow}$ is a quantum half-plane. The same holds with
$\mathcal{E}_{M,J}^{\downarrow}$ in place of $\mathcal{E}_{M,J}^{\uparrow}$.

  3. (iii)

Fix $K\geq J$. The conditional law of the surfaces
$\mathcal{E}_{M,J}^{\uparrow}\setminus\mathcal{E}_{K,J}^{\uparrow}$ for $J\leq
M\leq K$ and
$\mathcal{E}_{M,J}^{\downarrow}\setminus\mathcal{E}_{K,J}^{\uparrow}$ for
$J\leq M$ given $\mathcal{E}_{K,J}^{\uparrow}$ is described by the exploration
procedure from Section 4.1 where we abuse notation and write
$E_{i,M}^{\uparrow}$ for the event that $E_{i,M}^{\uparrow}$ holds for the
path-decorated quantum surface up until when the $i$th ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ chunk is explored in
$\mathcal{E}_{M,J}^{\uparrow}$. (In particular, each
$\mathcal{E}_{M,J}^{\uparrow}\setminus\mathcal{E}_{M+1,J}^{\uparrow}$ and
$\mathcal{E}_{M,J}^{\downarrow}\setminus\mathcal{E}_{M-1,J}^{\downarrow}$ can
be further decomposed into a collection of ${\rm SLE}_{\kappa}^{0}(\kappa-6)$
chunks.)

Moreover, there exists a constant $c>0$ (which does not depend on $J$) so that
the probability that any of the following occur is $O(2^{-cJ})$:

  * •

There exists $M\geq J$ so that the number of ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ chunks which make up
$\mathcal{E}_{M,J}^{\uparrow}\setminus\mathcal{E}_{M+1,J}^{\uparrow}$ for
$J\leq M$ or
$\mathcal{E}_{M,J}^{\downarrow}\setminus\mathcal{E}_{M-1,J}^{\downarrow}$ for
$J\leq M+2$ is at least $c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$. The
number of ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunks which make up
$\mathcal{E}_{J,J}^{\downarrow}\setminus\mathcal{E}_{J,J}^{\uparrow}$ is at
least $c_{F}2^{(\kappa/4)J}$.

  * •

There exists $M\geq J$ so that the boundary length process for an ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ chunk which is part of either
$\mathcal{E}_{M,J}^{\uparrow}\setminus\mathcal{E}_{M+1,J}^{\uparrow}$ or
$\mathcal{E}_{M,J}^{\downarrow}\setminus\mathcal{E}_{M-1,J}^{\downarrow}$
makes a jump of size at least $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$.

  * •

Either $0$ or $x_{1}$ is disconnected from $\infty$ by
$\mathcal{E}_{M,J}^{\uparrow}\setminus\mathcal{E}_{M+1,J}^{\uparrow}$ or
$\mathcal{E}_{M,J}^{\downarrow}\setminus\mathcal{E}_{M-1,J}^{\downarrow}$.

  * •

The difference in the boundary length of
$\partial\mathcal{E}_{M,J}^{\uparrow}\setminus\partial\mathbf{H}$ and
$\partial\mathcal{E}_{M,J}^{\uparrow}\cap\partial\mathbf{H}$ is at least
$2^{-cM}$. The difference in the boundary length of
$\partial\mathcal{E}_{M,J}^{\downarrow}\setminus\partial\mathbf{H}$ and
$\partial\mathcal{E}_{M,J}^{\downarrow}\cap\partial\mathbf{H}$ is at least
$2^{-cJ}$.

Moreover, the probability that any of the above three items occur for some
fixed value of $M\geq J$ is $O(2^{-cM})$ (for the same constant $c$).

Before we proceed to the proof of Lemma 4.2, we first need to collect the
following lemma which will be used in order to show that the law of the
diameter of the approximations to the $(\mathcal{E}_{M,J}^{\uparrow})_{M\geq
J}$ and $(\mathcal{E}_{M,J}^{\downarrow})_{M\geq J}$ is tight. The proof of
this lemma can be skipped on a first reading.

Figure 4.3.  Illustration of the setup and proof of Lemma 4.3.

######  Lemma 4.3.

Suppose that $-\infty<a_{1}<b_{1}<a_{2}<0<b_{2}<\infty$. For every $p\in(0,1)$
there exists $d_{0}\in(0,\infty)$ so that the following is true. Let $A$ be a
compact $\mathbf{H}$-hull (i.e., $A\subseteq\mathbf{H}$,
$A=\mathrm{cl}(A)\cap\mathbf{H}$, $\mathrm{cl}(A)$ is compact, and
$\mathbf{H}\setminus A$ is simply connected) so that
$\mathrm{cl}(A)\cap\partial\mathbf{H}\subseteq[a_{2},b_{2}]$. Let $\eta$ be an
${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process in $\mathbf{H}\setminus A$ from
$x\in[a_{1},b_{1}]$ to $\infty$. If the probability that $\eta$ hits $A$ is at
most $p$ then $\mathrm{diam}(A)\leq d_{0}$.

###### Proof.

See Figure 4.3 for an illustration of the setup. Let $\psi$ be the unique
conformal transformation from $\mathbf{H}\setminus A$ to $\mathbf{H}$ which
fixes $\infty$, takes $a_{1}$ to $0$, and $b_{1}$ to $1$. Then $\psi(\eta)$ is
an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process in $\mathbf{H}$ from
$\psi(x)\in[0,1]$ to $\infty$. Let $I=\psi(\mathrm{cl}(\mathbf{H}\cap\partial
A))=[u,v]$. By assumption, the probability that $\psi(\eta)$ hits $I$ is at
most $p$.

We claim that there exists $M\in(0,\infty)$ depending only on $a_{1}$,
$b_{1}$, $a_{2}$, $b_{2}$ so that $u\leq M$. Indeed, suppose that
$\eta^{\prime}$ is an ${\rm SLE}_{6}$ in $\mathbf{H}$ from $b_{1}$ to
$\infty$. Then there exists $q\in(0,1)$ so that the probability that
$\eta^{\prime}$ hits $[b_{2},\infty)$ before hitting $(-\infty,a_{1}]$ is at
least $q$. Therefore the probability that $\eta^{\prime}$ hits $A$ or
$[b_{2},\infty)$ before hitting $(-\infty,a_{1}]$ is also at least $q$. Now
assume that $\eta^{\prime}$ is an ${\rm SLE}_{6}$ in $\mathbf{H}\setminus A$
instead of $\mathbf{H}$. By the locality property for ${\rm SLE}_{6}$, the
same statement holds. Therefore the probability that $\psi(\eta^{\prime})$, an
${\rm SLE}_{6}$ in $\mathbf{H}$ from $1$ to $\infty$, hits
$\psi(\mathbf{H}\cap\partial A)\cup\psi([b_{2},\infty))\subseteq[u,\infty)$
before hitting $(-\infty,0]$ is at least $q$. This proves the claim.

Let $N\geq M$ be such that the probability that $\psi(\eta)$ hits $[M,N]$ is
at least $p$. We note that such an $N$ exists as the trunk of $\eta$ is an
${\rm SLE}_{\kappa^{\prime}}(\kappa^{\prime}/2-3;\kappa^{\prime}/2-3)$ process
in $\mathbf{H}$ from $x$ to $\infty$ and the probability that such a process
hits any infinite boundary interval is $1$. Note that $N$ depends only on
$a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$, and $p$. Since the probability that
$\psi(\eta)$ hits $I$ is at most $p$ and $u\leq M$ it must be that $v\leq N$.

To complete the proof, it suffices to show that for each $R\in(0,\infty)$
there exists $d\in(0,\infty)$ so that if $\mathrm{diam}(A)\geq d$ then $v\geq
R$. To see that this is the case, let $\eta^{\prime}$ be as above. Suppose
that we have $k\in\mathbf{N}$ so that $A$ crosses the annulus
$A_{k}=B(0,2^{k})\setminus B(0,2^{k-1})$. Let $\tau_{k}=\inf\\{t\geq
0:\eta^{\prime}(t)\in\partial B(0,3\cdot 2^{k-2})\\}$ (i.e., $\tau_{k}$ is the
first time that $\eta^{\prime}$ hits the circle in the middle of $A_{k}$) and
let $\sigma_{k}=\inf\\{t\geq\tau_{k}:\eta(t)\notin A_{k}\\}$. If
$\eta^{\prime}|_{[\tau_{k},\sigma_{k}]}$ hits $[-2^{k},-2^{k-1}]$ and then
hits $[2^{k-1},2^{k}]$, then $\eta^{\prime}|_{[\tau_{k},\sigma_{k}]}$ hits
$A$. As there exists $p_{0}>0$ so that the conditional probability of this
event given $\eta^{\prime}|_{[0,\sigma_{k-1}]}$ is at least $p_{0}$, it
follows that if $\mathrm{diam}(A)\geq d$ then the probability that
$\eta^{\prime}$ hits $A$ is at least $1-O(d^{-a})$ for a constant $a>0$.
Arguing as above, this implies that $v\to\infty$ as
$\mathrm{diam}(A)\to\infty$. ∎

###### Proof of Lemma 4.2.

We are going to construct the family of quantum surfaces described in the
lemma statement using the exploration that we have defined in Section 4.1 and
by taking an appropriate subsequential limit. The first step is to control the
change in the boundary length process (Step 1), then we will obtain a diameter
bound for the approximations (Step 2), then deduce the existence of the
subsequential limit (Step 3), and then finally describe the form of the
conditional law given the exploration (Step 4).

Fix $K\geq J$. Suppose that we perform the exploration as in Section 4.1 with
$J\leq M\leq K$. Let $\mathcal{E}_{M,J,K}^{\uparrow}$ (resp.
$\mathcal{E}_{M,J,K}^{\downarrow}$) be the quantum surface which consists of
all of the ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunks for the exploration up to
when the increasing (resp. decreasing) part with chunks with quantum natural
time in $[\delta_{0}2^{-M},2^{-M}]$ has been completed.

Step 1. Change in boundary length. Let $F$ be the event that the exploration
fails so that $F^{c}$ is the event that it does not fail. For each $J\leq
M\leq K$, let $T_{M}^{\uparrow}$ denote the quantum length of
$\partial\mathcal{E}_{M,J,K}^{\uparrow}\setminus\partial\mathcal{H}$. For each
$J\leq M$ we also let $T_{M}^{\downarrow}$ denote the quantum length of
$\partial\mathcal{E}_{M,J,K}^{\downarrow}\setminus\partial\mathcal{H}$. We are
now going to show that for each $\epsilon>0$ there exists a constant
$C\in(0,\infty)$ so that

$\mathbf{P}\\!\left[\left(\left\\{\max_{J\leq M\leq K}T_{M}^{\uparrow}\geq
C\right\\}\cup\left\\{\sup_{J\leq M}T_{M}^{\downarrow}\geq
C\right\\}\right)\cap F^{c}\right]\leq\epsilon.$

Consider the exploration when we have parameterized it by the quantum natural
time of the individual ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunks. Let $X_{t}$
be the overall boundary length process. That is, $X_{t}$ is equal to the
quantum length of the top boundary (part in $\mathbf{H}$) of the exploration
minus the quantum length of the bottom boundary (part in $\partial\mathbf{H}$)
of the exploration when $t$ units of quantum natural time has elapsed. Note
that the quantum length of the top boundary at time $t$ is equal to $X_{t}$
plus the quantum length of the bottom at time $t$. On the event that the
exploration does not fail, i.e., $F^{c}$ occurs, we have that the bottom is
contained in $[x_{-1},x_{1}]$. As $\nu_{h}([x_{-1},x_{1}])=2$, the quantum
length of the bottom boundary is at most $2$ on $F^{c}$. Therefore to give an
upper bound to the quantum length of the top boundary on this event it
suffices to give an upper bound to $X_{t}$.

Recall that $X_{t}$ evolves as a $4/\kappa$-stable Lévy process. On $F^{c}$,
we have that $n_{M}^{\uparrow}\leq
c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$ for each $J\leq M\leq K$,
$n_{M}^{\downarrow}\leq c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$ for
each $M\geq J+1$, and $n_{J}^{\downarrow}\leq c_{F}2^{(\kappa/4)J}$. At the
$M$th step of the upward or downward exploration (i.e., when exploring chunks
with quantum natural time in $[\delta_{0}2^{-M},2^{-M}]$), on $F^{c}$ the
amount of quantum natural time involved is thus at most
$c_{F}2^{((1-\alpha_{\mathrm{PERC}})(\kappa/4)-1)M}$ for $J\leq M\leq K$
(upward exploration) or $M\geq J+1$ (downward exploration) or at most
$c_{F}2^{(\kappa/4-1)J}$ (first part of the downward exploration). Note that
$(1-\alpha_{\mathrm{PERC}})(\kappa/4)-1<0$ as $\alpha_{\mathrm{PERC}}\in(0,1)$
and $\kappa\in(8/3,4)$ and also that $\kappa/4-1<0$. By summing over $M\geq
J$, we thus see that the total amount of quantum natural time involved in the
exploration on $F^{c}$ is at most

(4.10)
$T=c_{F}2^{(\kappa/4-1)J}+2c_{F}\sum_{M=J}^{\infty}2^{((1-\alpha_{\mathrm{PERC}})(\kappa/4)-1)M}.$

We emphasize here that $T$ is be taken to be uniform in $K$. By [4, Chapter
VIII, Proposition 4], we have that $\sup_{0\leq t\leq T}X_{t}$ is a.s. finite.
Altogether, what we have shown implies that the law of the top boundary length
of the exploration is tight as $K\to\infty$ on $F^{c}$.

Step 2. Diameter bound. Fix $M\geq J$. We are going to show that the diameter
of $\mathcal{E}_{M,J,K}^{\uparrow}$ is tight on $F^{c}$ as $K\to\infty$. Let
$x_{-2}<0$ be so that $\nu_{h}([x_{-2},0])=2$. Let $\eta$ be an ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ process in $\mathbf{H}$ from $x_{-2}$ to $\infty$
coupled as a CPI in $\Gamma$ conditionally independently of the rest of the
exploration. Then we know that the conditional law of $\eta$ given
$\mathcal{E}_{M,J,K}^{\uparrow}$ is that of an ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ process in
$\mathbf{H}\setminus\mathcal{E}_{M,J,K}^{\uparrow}$ from $x_{-2}$ to $\infty$.
Note that we can write
$\partial\mathbf{H}\cap\partial\mathcal{E}_{I,J,K}^{\uparrow}$ as $[a,b]$. Let
$u$ (resp. $v$) be the quantum length of $[x_{-2},a]$ (resp. $[x_{-2},a]$ and
$\mathbf{H}\cap\partial\mathcal{E}_{I,J,K}^{\uparrow}$, i.e., the top of
$\mathcal{E}_{I,J,K}^{\uparrow}$). As the quantum surface parameterized by
$\mathbf{H}\setminus\mathcal{E}_{I,J,K}^{\uparrow}$ is a quantum half-plane,
it follows that the conditional probability given
$\mathcal{E}_{M,J,K}^{\uparrow}$ that $\eta$ hits
$\mathcal{E}_{M,J,K}^{\uparrow}$ is equal to the probability that the running
infimum of a $4/\kappa$-stable Lévy process with both upward and downward
jumps starting from $0$ hits $[2-v,2-u]$. As explained above, the quantum
length $v-u$ of the top of $\mathcal{E}_{I,J,K}^{\uparrow}$ is tight on
$F^{c}$ as $K\to\infty$. Therefore the probability that $\eta$ hits
$\mathcal{E}_{M,J,K}^{\uparrow}$ is bounded away from $1$ on $F^{c}$ as
$K\to\infty$. By Lemma 4.3, this implies that the law of the diameter of
$\mathcal{E}_{M,J,K}^{\uparrow}$ is tight on $F^{c}$ as $K\to\infty$.

Step 3. Subsequential limit and law of unexplored region. Let $g_{M,J,K}$ be
the unique conformal map from
$\mathbf{H}\setminus\mathcal{E}_{M,J,K}^{\uparrow}$ to $\mathbf{H}$ with
$g_{I,J,K}(z)-z\to 0$ as $z\to\infty$. Then it follows that the law of
$g_{I,J,K}$ (with the Caratheodory topology) on $F^{c}$ is tight as
$K\to\infty$ since the diameter of $\mathcal{E}_{M,J,K}^{\uparrow}$ on $F^{c}$
is tight as $K\to\infty$. Indeed, this follows from [18]. Suppose that we pass
to a sequence $(K_{n})$ so that the joint law of any finite collection of the
$g_{I,J,K_{n}}$ together with $h$ converges weakly as $n\to\infty$. Let
$g_{I,J}$ denote the limit as $n\to\infty$. We abuse notation and write $h$
for the instance of $h$ which is coupled with $g_{I,J}$ as well as with
$g_{I,J,K_{n}}$ for each $n$. As the quantum surface described by $h\circ
g_{I,J,K_{n}}^{-1}+Q\log|(g_{I,J,K_{n}}^{-1})^{\prime}|$ is a quantum half-
plane for each $n$, it follows that $h\circ
g_{I,J}^{-1}+Q\log|(g_{I,J}^{-1})^{\prime}|$ is a quantum half-plane as well.
This proves that (ii) holds.

Step 4. Form of the conditional law. By the construction of the subsequential
limit, it follows that (iii) holds. That the final points hold is clear from
the construction. ∎

We now deduce the analog of Lemma 4.2 in the case of the quantum disk. We note
that the exploration that we have defined in Section 4.1 makes sense on a
doubly marked quantum disk, although the estimates we have previously
established for its basic properties are for the exploration in the case of
the quantum half-plane. The purpose of the following lemma is to deduce these
properties when we work instead on the disk.

######  Lemma 4.4.

Fix $\ell\geq 2$ and suppose that
$\mathcal{D}=(\mathbf{D},h)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$.
Suppose that $x\in\partial\mathcal{D}$ is sampled from $\nu_{h}$ and
$y\in\partial\mathcal{D}$ is such that
$\nu_{h}([x,y]_{\partial\mathcal{D}}^{\bm{\circlearrowleft}})=1$. Let
$z\in[x,y]_{\partial\mathcal{D}}^{\bm{\circlearrowleft}}$ be such that
$\nu_{h}([x,z]_{\partial\mathcal{D}}^{\bm{\circlearrowright}})=\nu_{h}([z,y]_{\partial\mathcal{D}}^{\bm{\circlearrowright}})=\nu_{h}([x,y]_{\partial\mathcal{D}}^{\bm{\circlearrowright}})/2=(\ell-1)/2$.
For each $J\in\mathbf{N}$, there is a family of quantum surfaces
$\mathcal{E}_{I,J}^{\uparrow}$ and $\mathcal{E}_{I,J}^{\downarrow}$ so that
all of the properties of Lemma 4.2 hold but with the quantum half-plane
replaced by $\mathcal{D}$ and the marked points at $0$, $x_{1}$, $\infty$
replaced by $x$, $y$, and $z$, respectively.

In order to deduce Lemma 4.4 from Lemma 4.2, we first record the following
Radon-Nikodym derivative ([33, Proposition 5.1]) which serves to compare ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ explorations on quantum disks and on quantum
half-planes.

######  Lemma 4.5.

Fix $\ell>0$ and suppose that
$\mathcal{D}=(\mathbf{D},h)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$.
Suppose that $x\in\partial\mathcal{D}$ is sampled from $\nu_{h}$, let
$L_{0},R_{0}>0$ be such that $\ell=L_{0}+R_{0}$, and let
$y\in\partial\mathcal{D}$ be such that
$\nu_{h}([x,y]_{\partial\mathcal{D}}^{\bm{\circlearrowright}})=L_{0}$ and
$\nu_{h}([x,y]_{\partial\mathcal{D}}^{\bm{\circlearrowleft}})=R_{0}$. Suppose
that $\eta$ is an independent ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ in
$\mathcal{D}$ from $x$ to $y$ which is parameterized by the quantum natural
time of its trunk. For each $t\geq 0$, we let $\mathcal{D}_{t}$ denote the
component of $\mathcal{D}\setminus\eta([0,t])$ with $y$ on its boundary and
let $L_{t}$ (resp. $R_{t}$) denote the quantum length of
$[\eta(t),y]_{\partial\mathcal{D}_{t}}^{\bm{\circlearrowright}}$ (resp.
$[\eta(t),y]_{\partial\mathcal{D}_{t}}^{\bm{\circlearrowleft}}$). Let $E_{t}$
be the event that $\inf_{0\leq s\leq t}L_{s}>0$ and $\inf_{0\leq s\leq
t}R_{s}>0$. On $E_{t}$, the Radon-Nikodym derivative between the law of
$(L,R)|_{[0,t]}$ and a pair of independent $4/\kappa$-stable Lévy processes
starting from $(L_{0},R_{0})$ on $[0,t]$ is given by
$(\Delta_{0}/\Delta_{t})^{4/\kappa+1}$ where $\Delta_{t}=L_{t}+R_{t}$.

###### Proof of Lemma 4.4.

Suppose that $\mathcal{D}$, $x$, $y$, $z$ are as in the statement of the
lemma. Then we can consider the same approximation scheme using the
exploration from Section 4.1 as in the proof of Lemma 4.2. Indeed, even though
we defined the exploration in Section 4.1 in the case of the half-plane, it
still makes sense to consider it in the case of the quantum disk. Let
$\Delta_{0}=\ell$ be the boundary length of $\partial\mathcal{D}$. For each
$J\leq M$, we let $\Delta_{M}^{\downarrow}$ be the boundary length of the
complementary component with $z$ on its boundary after we have completed the
part of the exploration where the ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunk
sizes are in $[\delta_{0}2^{-M},2^{-M}]$. Note that the event that the
exploration has not failed up until this point, the analog of the event
$E_{t}$ from Lemma 4.5 up to this time holds. Therefore on this event, Lemma
4.5 implies that the Radon-Nikodym derivative between the half-planar and disk
explorations is of the form
$(\Delta_{0}/\Delta_{M}^{\downarrow})^{4/\kappa+1}$. Due to the definition of
the exploration failing, on the event that has not failed we have that
$\Delta_{M}^{\downarrow}$ is bounded from below. This completes the proof of
the result. ∎

##  5\. Tightness on the boundary

The purpose of this section is to prove the tightness of
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(\cdot,\cdot;\Gamma)$
restricted to the domain boundary. We are going to prove the result in a more
general setting than in the statement of Theorem 1.1 in the sense that we will
not restrict to the case that the domain boundary is given by a ${\rm
CLE}_{\kappa}$ loop. Instead, we will make an assumption on the behavior of
the quantum area measure when we consider a sample from the law
$\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$ parameterized by the domain.
(We note that this is effectively an assumption about the roughness of
$\partial D$.) In what follows, when we write “quantum length metric on
$\partial D$” for $D\subseteq\mathbf{C}$ simply connected we mean the metric
on (the prime ends of) $\partial D$ where the distance between $x,y$ is given
by the minimum of the quantum lengths of $[x,y]_{\partial
D}^{\bm{\circlearrowright}}$ and $[x,y]_{\partial D}^{\bm{\circlearrowleft}}$.

######  Proposition 5.1.

Let $\alpha_{\mathrm{LBD}}>2$ and fix $\beta\in(0,2/\alpha_{\mathrm{LBD}})$.
Suppose that
$\mathcal{D}=(D,h,x,y)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$. Fix
$c_{0},\epsilon_{0}>0$ and let $E$ be the event that for every $z\in D$ and
$\epsilon\in(0,\epsilon_{0})$ so that $B(z,\epsilon)\subseteq D$ we have that
$\mu_{h}(B(z,\epsilon))\geq c_{0}\epsilon^{\alpha_{\mathrm{LBD}}}$. Let
$\Gamma$ be an independent ${\rm CLE}_{\kappa}$ in $D$ and let $\Upsilon$ be
its carpet. Let $X_{\epsilon,\beta}$ be the $\beta$-Hölder norm of
$(z,w)\mapsto({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$
with respect to the quantum length metric on $\partial D$. For every $p>0$ we
have that

(5.1) $\mathbf{P}[E,\ X_{\epsilon,\beta}\geq A]=O(A^{-p})\quad\text{as}\quad
A\to\infty$

where the implicit constants depend on
$p,\epsilon_{0},c_{0},\alpha_{\mathrm{LBD}},\beta$.

Moreover, let $Y_{\beta}$ be the $\beta$-Hölder norm of the Euclidean metric
with respect to the quantum length metric on $\partial D$. Then we have that

(5.2) $\mathbf{P}[E,\ Y_{\beta}\geq A]=O(A^{-p})\quad\text{as}\quad
A\to\infty$

where the implicit constants depend on
$p,\epsilon_{0},c_{0},\alpha_{\mathrm{LBD}},\beta$.

In the setting of Proposition 5.1, truncating on the event that the quantum
area measure is bounded from below serves to give a lower bound on the size of
the ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunks which make up the exploration
described in Lemma 4.4. The reason that this is important is that our
definition of the good event for a chunk which is given just below will
involve the minimal “${\mathfrak{N}}_{\epsilon}$-length” of various crossings
when one has parameterized the chunk by $\mathbf{D}$. Embedding the chunk into
$D$ involves applying a conformal transformation which can lead to an
effective change in the size of the neighborhood used to approximate the
“length” of a path. As we will see in the proof of Proposition 5.1, the lower
bound on the quantum area measure will serve to give an upper bound on the
size of an embedded chunk hence an upper bound on the “length” of an embedded
path.

Let us now explain the main steps used to prove Proposition 5.1.

  1. Step 1.

We will first give the events for the good chunks in Definition 5.2 and then
show in Lemma 5.3 that we can adjust the parameters in the definition to make
the probability that they occur sufficiently close to $1$ so that the results
of Section 4 apply.

  2. Step 2.

The definition of a good chunk in Definition 5.2 will involve quantities of
the form ${\mathfrak{N}}_{\epsilon}(\omega)$ for paths $\omega$ which arise
after we have parameterized the chunk by $\mathbf{D}$. We then need to control
the Lebesgue measure of the $\epsilon$-neighborhood of these paths after
embedding them and this will be the usage of Lemma 5.4.

  3. Step 3.

We complete the proof of Proposition 5.1 using Lemma 4.4. The idea is that we
will start with two boundary points $x$, $y$ and perform the exploration as in
Lemma 4.4. If the exploration succeeds, then our previous estimates and the
definition of a good chunk will allow us to construct a path $\omega$ in
$\Upsilon$ with control on ${\mathfrak{N}}_{\epsilon}(\omega)$. If the
exploration fails, then we will repeat the exploration as in Lemma 4.4 in the
remaining domains. The exploration will be repeated either once or twice
depending on the type of failure. We will refer to these new explorations as
the “children” of the failed exploration so that we get a tree structure on
such explorations. We will show that by properly adjusting the parameters we
can dominate the number of times the exploration has to be repeated by the
size of a subcritical Galton-Watson tree, which will complete the proof.

Let $\mathcal{H}=(\mathbf{H},h,0,\infty)$ be a quantum half-plane. Suppose
that $\eta$ is an ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ curve on $\mathbf{H}$
from $0$ to $\infty$ which is taken to be independent of $h$ and then
parameterized by the quantum natural time of its trunk. For the quantum
surface $\mathcal{N}_{t}$ disconnected from $\infty$ by $\eta|_{[0,t]}$, we
let $z_{L}(\mathcal{N}_{t})$ (resp. $z_{R}(\mathcal{N}_{t})$) denote the
leftmost (resp. rightmost) point on the bottom of $\mathcal{N}_{t}$. We also
let $z_{C}(\mathcal{N}_{t})=\eta(0)$.

Figure 5.1.  Top: Illustration of part (ii) of Definition 5.2. Shown in blue
is the exploration up until $\mathcal{N}_{i,M}^{\uparrow}$ is discovered and
the images of $z_{L}(\mathcal{N}_{i,M}^{\uparrow})$,
$z_{C}(\mathcal{N}_{i,M}^{\uparrow})$, and
$z_{R}(\mathcal{N}_{i,M}^{\uparrow})$ under $\varphi$ are respectively given
by $-1$, $-i$, and $1$, respectively. Bottom: Illustration of part (iv) of
Definition 5.2. The chunk $\mathcal{N}_{\ell,L}^{\uparrow}$ is shown in light
yellow even though it is “good” in order to differentiate it from
$\mathcal{N}_{i,M}^{\uparrow}$. The images of
$z_{L}(\mathcal{N}_{\ell,L}^{\uparrow})$,
$z_{C}(\mathcal{N}_{\ell,L}^{\uparrow})$, and
$z_{R}(\mathcal{N}_{\ell,L}^{\uparrow})$ under $\psi$ are $-1$, $-i$, and $1$,
respectively.

######  Definition 5.2.

We suppose that we have the setup described in Section 4.1. Fix
$R_{1},R_{2},S_{1},S_{2}\geq 1$. For each $i,M$, we define
$E_{i,M}^{\uparrow}$ to be the event for $\mathcal{N}_{i,M}^{\uparrow}$ and
$\eta_{i,M}^{\uparrow}$ that
$\eta_{i,M}^{\uparrow}(\sigma_{i,M}^{\uparrow})\in\partial\mathbf{H}$ and the
following hold.

  1. (i)

The quantum lengths of the top, left side of the bottom, and the right side of
the bottom of $\mathcal{N}_{i,M}^{\uparrow}$ are in
$[R_{1}^{-1}2^{-(\kappa/4)M},R_{1}2^{-(\kappa/4)M}]$.

  2. (ii)

Let $\varphi\colon\mathcal{N}_{i,M}^{\uparrow}\to\mathbf{D}$ be the unique
conformal transformation which takes $z_{L}(\mathcal{N}_{i,M}^{\uparrow})$ to
$-1$, $z_{C}(\mathcal{N}_{i,M}^{\uparrow})$ to $-i$, and
$z_{R}(\mathcal{N}_{i,M}^{\uparrow})$ to $1$. There is a path $\omega$
connecting $B(-1,R_{2}^{-1})$ to $B(1,R_{2}^{-1})$ in
$\varphi(\Upsilon\cap\mathcal{N}_{i,M}^{\uparrow})$ which has distance at
least $R_{1}^{-1}$ from $\partial\mathbf{D}$ and satisfies
${\mathfrak{N}}_{\epsilon}(\omega)\leq
R_{1}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}$.

  3. (iii)

$\mu_{h}(\mathcal{N}_{i,M}^{\uparrow})\leq R_{1}2^{-M}$.

  4. (iv)

Let $\mathcal{N}_{\ell,L}^{\uparrow}$ be the chunk whose top contains
$\eta_{i,M}^{\uparrow}(0)$ (so that the left side of the bottom of
$\mathcal{N}_{i,M}^{\uparrow}$ is glued to the top of
$\mathcal{N}_{\ell,L}^{\uparrow}$) if it exists, otherwise we set
$\mathcal{N}_{\ell,L}^{\uparrow}=\emptyset$. If
$\mathcal{N}_{\ell,L}^{\uparrow}\neq\emptyset$, let $\psi$ be the unique
conformal map which takes
$\mathcal{N}=\mathrm{int}(\mathrm{cl}(\mathcal{N}_{\ell,L}^{\uparrow}\cup\mathcal{N}_{i,M}^{\uparrow}))$
to $\mathbf{D}$ so that $z_{L}(\mathcal{N}_{\ell,L}^{\uparrow})$ is taken to
$-1$, $z_{C}(\mathcal{N}_{\ell,L}^{\uparrow})$ is taken to $-i$, and
$z_{R}(\mathcal{N}_{\ell,L}^{\uparrow})$ is taken to $1$. The harmonic measure
of each of
$[-1,\psi(z_{L}(\mathcal{N}_{i,M}^{\uparrow}))]_{\partial\mathbf{D}}^{\bm{\circlearrowright}}$,
$[\psi(z_{L}(\mathcal{N}_{i,M}^{\uparrow})),\psi(z_{R}(\mathcal{N}_{i,M}^{\uparrow}))]_{\partial\mathbf{D}}^{\bm{\circlearrowright}}$,
and
$[\psi(z_{R}(\mathcal{N}_{i,M}^{\uparrow})),1]_{\partial\mathbf{D}}^{\bm{\circlearrowright}}$
as seen from $0$ is at least $R_{2}^{-1}$. Let $w$ be the midpoint of
$[\psi(z_{L}(\mathcal{N}_{i,M}^{\uparrow})),\psi(z_{R}(\mathcal{N}_{i,M}^{\uparrow}))]_{\partial\mathbf{D}}^{\bm{\circlearrowright}}$.
Then there moreover is a path $\omega$ in $\psi(\Upsilon\cap\mathcal{N})$
which connects $B(-i,S_{2}^{-1})$ to $B(w,S_{2}^{-1})$ which has distance at
least $S_{1}^{-1}$ from $\partial\mathbf{D}$ and satisfies
${\mathfrak{N}}_{\epsilon}(\omega)\leq
S_{1}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}$.

  5. (v)

If $\widehat{\mathcal{N}}$ is another ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunk
which starts from a point on the top of $\mathcal{N}_{i,M}^{\uparrow}$ with
clockwise boundary length distance from $\partial\mathcal{H}_{i,M}^{\uparrow}$
given by an integer multiple of $a_{0}2^{-(\kappa/4)M}$ and with boundary
length distance from a point in $\Upsilon$ at most $a_{0}2^{-(\kappa/4)M}$ and
is run until the first time it hits the domain boundary after
$\delta_{0}2^{-J}$, $J\in\\{M-1,M,M+1\\}$, then the conditional probability
given $\mathcal{F}_{i,M}^{\uparrow}$ that the previous item holds for the pair
$(\mathcal{N}_{i,M}^{\uparrow},\widetilde{\mathcal{N}})$ in place of
$(\mathcal{N}_{\ell,L}^{\uparrow},\mathcal{N}_{i,M}^{\uparrow})$ is at least
$1-R_{2}^{-1}$.

We define $E_{i,M}^{\downarrow}$ analogously.

######  Lemma 5.3.

Suppose that we have the setup described in Section 4.1 and the events
$E_{i,M}^{\uparrow}$, $E_{i,M}^{\downarrow}$ are as in Definition 5.2. For
every $p_{0}\in(0,1)$ there exists $R_{1}\geq R_{2}\geq 1$ and $S_{2}\geq
S_{1}\geq 1$ so that

  1. (i)

On the event that the exploration has not failed before exploring
$\mathcal{N}_{i,M}^{\uparrow}$ (resp. $\mathcal{N}_{i,M}^{\downarrow}$) we
have that
$\mathbf{P}[E_{i,M}^{\uparrow}\,|\,\mathcal{F}_{i-1,M}^{\uparrow}]\geq p_{0}$
(resp.
$\mathbf{P}[E_{i,M}^{\downarrow}\,|\,\mathcal{F}_{i-1,M}^{\downarrow}]\geq
p_{0}$) for all $i,M\in\mathbf{N}$.

  2. (ii)

Suppose that $\mathcal{N}_{\ell,L}^{\uparrow}$ is as in part (iv) of
Definition 5.2, $\omega_{i,M}^{\uparrow}$ (resp. $\omega_{\ell,L}^{\uparrow}$)
is any path for $\mathcal{N}_{i,M}^{\uparrow}$ (resp.
$\mathcal{N}_{\ell,L}^{\uparrow}$) satisfying part (ii) of Definition 5.2.
Then any path as in part (iv) of Definition 5.2 crosses both
$\omega_{i,M}^{\uparrow}$ and $\omega_{\ell,L}^{\uparrow}$.

###### Proof.

It is clear from the scaling properties of $4/\kappa$-stable Lévy process that
(i) holds with probability tending to $1$ as $R_{1}\to\infty$. Lemma 3.1
implies that with any choice of $R_{2}\geq 1$ we have that (ii) holds with
probability tending to $1$ as $R_{1}\to\infty$. We have that
$\mathbf{E}[\mu_{h}(\mathcal{N}_{i,M}^{\uparrow})\,|\,\mathcal{F}_{i-1,M}^{\uparrow}]=\mathbf{E}[\mu_{h}(\mathcal{N}_{i,M}^{\uparrow})]=O(2^{-M})$
as $\mathbf{E}[\mu_{h}(\mathcal{N}_{i,M}^{\uparrow})]$ is equal to a constant
times the sum of the squares of the jumps made by the associated Lévy process
(which are run for time at most $2^{-M}$). Indeed, the range of the ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ itself has zero quantum area so all of the
quantum area is in the quantum disks that it disconnects from $\infty$ and the
expected quantum area associated with a sample from the law
$\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$ is equal to a constant
times $\ell^{2}$. Therefore Markov’s inequality implies that (iii) holds with
probability tending to $1$ as $R_{1}\to\infty$. That (iv) and (v) hold with
probability tending to $1$ as $R_{1},S_{1}\to\infty$ for any fixed choice of
$R_{2},S_{2}$ follows from Lemma 3.2. Finally, for any values $R_{1}\geq
R_{2}\geq 1$ it is clear that the second assertion of the lemma holds by
making $S_{2}\geq 1$ sufficiently large. ∎

Before we proceed to the proof of Proposition 5.1, we need to collect the
following elementary lemma. As in the case of paths, for
$K\subseteq\mathbf{C}$ and $\epsilon>0$ we let ${\mathfrak{N}}_{\epsilon}(K)$
denote the Lebesgue measure of the $\epsilon$-neighborhood of $K$.

######  Lemma 5.4.

There exists a constant $c_{0}>0$ so that the following is true for all
$\epsilon>0$ and $\delta\in(0,1/2)$. Suppose that $K\subseteq\mathbf{C}$ is
connected and compact. Then we have that

${\mathfrak{N}}_{\delta\epsilon}(K)\geq
c_{0}\delta{\mathfrak{N}}_{\epsilon}(K).$

###### Proof.

Fix $\epsilon>0$ and $\delta\in(0,1/2)$. Let $(z_{k})$ be a countable, dense
subset of $K$. By the compactness of $K$, there exists
$z_{j_{1}},\ldots,z_{j_{n}}$ so that
$K\subseteq\cup_{i=1}^{n}B(z_{j_{i}},\epsilon)$. By the Vitali covering lemma,
there exists $i_{1},\ldots,i_{\ell}\subseteq\\{j_{1},\ldots,j_{n}\\}$ so that
the balls $B(z_{i_{j}},\epsilon)$ for $1\leq j\leq\ell$ are pairwise disjoint
and $K\subseteq\cup_{j=1}^{\ell}B(z_{i_{j}},3\epsilon)$. Note that

$\pi\epsilon^{2}\ell\leq{\mathfrak{N}}_{\epsilon}(K)\leq
9\pi\epsilon^{2}\ell.$

Since $K$ is connected, for each $1\leq j\leq\ell$, we can find at least
$\lfloor\delta^{-1}/2\rfloor$ disjoint balls with centers in $K\cap
B(z_{i_{j}},\epsilon/2)$ and radius $\epsilon\delta$. It therefore follows
that

${\mathfrak{N}}_{\epsilon\delta}(K)\geq\pi(\epsilon\delta)^{2}\times\lfloor\delta^{-1}/2\rfloor\times\ell\geq\frac{\delta^{2}\lfloor\delta^{-1}/2\rfloor}{9}{\mathfrak{N}}_{\epsilon}(K).$

This proves the result as $\delta^{2}\lfloor\delta^{-1}/2\rfloor\geq\delta/2$
for all $\delta\in(0,1/2)$. ∎

We now proceed to give the proof of Proposition 5.1.

###### Proof of Proposition 5.1.

We will first explain the proof of (5.1) and then explain at the end how the
same argument gives (5.2). We assume that we have chosen
$\delta_{0},p_{0}\in(0,1)$ so that the assertion of Lemma 4.1 holds. With
these parameters, we then assume that we have chosen $R_{2}\geq R_{1}\geq 1$
and $S_{2}\geq S_{1}\geq 1$ so that the assertion of Lemma 5.3 holds. In what
follows, we may therefore view $R_{1}$, $R_{2}$, $S_{1}$, $S_{2}$ as
constants. We perform the exploration from Lemma 4.4 using the events defined
in Definition 5.2 in a recursive manner as follows.

Fix $\delta,\zeta>0$. Suppose that $x_{0},x_{1}\in\partial\mathcal{D}$ are so
that $\nu_{h}([x,x_{0}]_{\partial\mathcal{D}}^{\bm{\circlearrowleft}})=\zeta$
and
$\nu_{h}([x_{0},x_{1}]_{\partial\mathcal{D}}^{\bm{\circlearrowleft}})=\delta$.
Let $\mathcal{D}_{0}$ be given by taking $\mathcal{D}$ and replacing $h$ with
$h_{0}=h+\tfrac{2}{\gamma}\log\delta^{-1}$. Let $x_{0}^{0},x_{1}^{0}$ be the
corresponding points for $\mathcal{D}_{0}$. Then
$\nu_{h_{0}}([x_{0}^{0},x_{1}^{0}]_{\partial\mathcal{D}_{0}}^{\bm{\circlearrowleft}})=1$.
We then perform the exploration as in Lemma 4.4 in $\mathcal{D}_{0}$ from
$x_{0}^{0}$ to $x_{1}^{0}$ until it fails. We recall that the exploration
fails if it makes either a large upward jump (i.e., of size at least
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$ when exploring chunks with quantum
natural time in $[\delta_{0}2^{-M},2^{-M}]$), a large downward jump (i.e., of
size at least $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$ when exploring chunks
with quantum natural time in $[\delta_{0}2^{-M},2^{-M}]$), requires too many
chunks at a particular scale (i.e., at least
$c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$ when exploring chunks of size
in $[\delta_{0}2^{-M},2^{-M}]$ for $M>J$ or at least $c_{F}2^{(\kappa/4)J}$
when exploring chunks of size in $[\delta_{0}2^{-J},2^{-J}]$), or the
deviations in the boundary length process are too large (i.e., the absolute
value of the quantum length of the top minus the bottom exceeds
$c_{F}2^{(1-4/\kappa)M}$ when exploring chunks of size in
$[\delta_{0}2^{-M},2^{-M}]$). We stop the exploration immediately if either of
these possibilities happens, and then proceed as described below.

Figure 5.2.  Illustration of the exploration in $D$ from $x_{0}=x_{0}^{0}$ to
$x_{1}=x_{1}^{0}$ (blue) up until it fails due to making a large upward jump
(i.e., a large ${\rm CLE}_{\kappa}$ loop is discovered) shown in red. The
prime ends $z_{0}^{0}$ and $z_{1}^{0}$ correspond to where the loop is rooted
on the exploration. When this happens, two new explorations are started in the
remaining domain going from $x_{0}^{0}$ to $z_{0}^{0}$ and from $z_{1}^{0}$ to
$x_{1}^{0}$. Figure 5.3.  Illustration of the exploration in $D$ from
$x_{0}=x_{0}^{0}$ to $x_{1}=x_{1}^{0}$ (blue) up until it fails due to making
a large downward jump with the corresponding chunk shown in red. Left: In the
case that $x_{0}^{0}$ and $x_{1}^{0}$ are not disconnected, a new exploration
is started in the remaining domain with $x_{0}^{0}$, $x_{1}^{0}$ on its
boundary from $x_{0}^{0}$ to $x_{1}^{0}$. Right: In the case that $x_{0}^{0}$
and $x_{1}^{0}$ are disconnected, we let $z_{0}^{0}$ and $z_{1}^{0}$ be the
two prime ends in the two complementary domains corresponding to where the
downward jump occurred. Two new explorations are started, one from $x_{0}^{0}$
to $z_{0}^{0}$ and the other from $z_{1}^{0}$ to $x_{1}^{0}$.

Case 1: large upward jump. See Figure 5.2 for an illustration. If the
exploration fails by making a large upward jump, we let
$\widetilde{\mathcal{D}}_{0}$ be the quantum disk in the complement of the
exploration with $x_{0}^{0}$, $x_{1}^{0}$ on its boundary and let
$\widetilde{h}_{0}$ be the field which describes
$\widetilde{\mathcal{D}}_{0}$. We let $z_{0}^{0}$, $z_{1}^{0}$ be the two
points on $\partial\widetilde{\mathcal{D}}_{00}$ which correspond to where the
loop corresponding to the upward jump is rooted. We then let
$\mathcal{D}_{00}$ be the surface described by
$h_{00}=\widetilde{h}_{0}+\tfrac{2}{\gamma}\log L_{00}^{-1}$ where
$L_{00}=\nu_{\widetilde{h}_{0}}([x_{0}^{0},z_{0}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{00}$, $x_{1}^{00}$ for the corresponding points. We also
let $\mathcal{D}_{01}$ be the surface described by
$h_{01}=\widetilde{h}_{0}+\tfrac{2}{\gamma}\log L_{01}^{-1}$ where
$L_{01}=\nu_{\widetilde{h}_{0}}([z_{1}^{0},x_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{01}$, $x_{1}^{01}$ for the corresponding points. We refer
to $(\mathcal{D}_{0i},x_{0}^{0i},x_{1}^{0i})$ for $i=0,1$ as the _children_ of
$(\mathcal{D}_{0},x_{0}^{0},x_{1}^{0})$. We then perform the exploration
inside of $\mathcal{D}_{0i}$ from $x_{0}^{0i}$ to $x_{1}^{0i}$ for $i=0,1$
starting with $i=0$ and then with $i=1$ with the following exception. If the
exploration with $i=0$ fails with a large downward jump which disconnects
$x_{0}^{01}=z_{1}^{0}$ from $x_{1}^{01}=x_{1}^{0}$ then we do not perform the
exploration from $x_{0}^{01}=z_{1}^{0}$ to $x_{1}^{01}=x_{1}^{0}$ and instead
at the next stage perform the exploration from $x_{0}^{00}$ to $x_{1}^{01}$
(analogously to how the exploration proceeds in the case of a large downward
jump which does not disconnect the initial and terminal points as described
just below).

Case 2: large downward jump. See Figure 5.3 for an illustration. If the
exploration fails by making a large downward jump, there are two
possibilities. Either $x_{0}$ and $x_{1}$ are disconnected or they are not. If
$x_{0}$, $x_{1}$ are not disconnected, then we let
$\widetilde{\mathcal{D}}_{0}$ be the quantum disk in the complement of the
exploration with $x_{0}^{0}$, $x_{1}^{0}$ on its boundary and let
$\widetilde{h}_{0}$ be the field which describes
$\widetilde{\mathcal{D}}_{0}$. We let $\mathcal{D}_{00}$ be the quantum
surface described by $h_{00}=\widetilde{h}_{0}+\tfrac{2}{\gamma}\log
L_{00}^{-1}$ where
$L_{00}=\nu_{\widetilde{h}_{0}}([x_{0}^{0},x_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{00}$, $x_{1}^{00}$ for the corresponding points. We refer
to $(\mathcal{D}_{00},x_{0}^{00},x_{1}^{00})$ as the _child_ of
$(\mathcal{D}_{0},x_{0}^{0},x_{1}^{0})$ and then continue the exploration in
$\mathcal{D}_{00}$ from $x_{0}^{00}$ to $x_{1}^{00}$.

Now suppose that $x_{0}$, $x_{1}$ are disconnected. Let
$\widetilde{\mathcal{D}}_{0}$ (resp. $\widetilde{\mathcal{D}}_{1}$) be the
quantum disk with $x_{0}$ (resp. $x_{1}$) on its boundary. Let
$\widetilde{h}_{0}$ (resp. $\widetilde{h}_{1}$) be the field which describes
$\widetilde{\mathcal{D}}_{0}$ (resp. $\widetilde{\mathcal{D}}_{1}$). Let
$z_{0}$ (resp. $z_{1}$) be the point on $\partial\widetilde{\mathcal{D}}_{0}$
(resp. $\partial\widetilde{\mathcal{D}}_{1}$) which corresponds to where the
downward jump happened. We then let $\mathcal{D}_{00}$ be the quantum surface
described by $h_{00}=\widetilde{h}_{0}+\tfrac{2}{\gamma}\log L_{00}^{-1}$
where
$L_{00}=\nu_{\widetilde{h}_{0}}([x_{0},z_{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and call $x_{0}^{00}$, $x_{1}^{00}$ the corresponding points. We also let
$\mathcal{D}_{01}$ be the quantum surface described by
$h_{01}=\widetilde{h}_{1}+\tfrac{2}{\gamma}\log L_{01}^{-1}$ where
$L_{01}=\nu_{\widetilde{h}_{1}}([z_{1},x_{1}]_{\partial\widetilde{\mathcal{D}}_{1}}^{\bm{\circlearrowleft}})$
and write $x_{0}^{01}$, $x_{1}^{01}$ for the corresponding points. We refer to
$(\mathcal{D}_{0i},x_{0}^{0i},x_{1}^{0i})$ for $i=0,1$ as the children of
$(\mathcal{D}_{0},x_{0}^{0},x_{1}^{0})$. We then perform the exploration
inside of $\mathcal{D}_{0i}$ from $x_{0}^{0i}$ to $x_{1}^{0i}$ for $i=0,1$.

Case 3: too many chunks or top boundary length too long. Finally, we consider
the situation in which the exploration fails by either having
$n_{M}^{\uparrow}$, $n_{M}^{\downarrow}$ too large for $M\geq J$ or the
quantum length of the top of the exploration exceeds $C2^{(1-4/\kappa)J}$ for
a constant $C>0$ (we will adjust the value of $C$ later in the proof; note
that $1-4/\kappa<0$ since $\kappa\in(8/3,4)$). We note that the quantum length
of the top can only exceed $C2^{(1-4/\kappa)J}$ by at most
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)J}$ for otherwise the exploration would
have failed due to a large upward jump. We let $\widetilde{\mathcal{D}}_{0}$
be the quantum disk in the complement of the exploration with $x_{0}^{0}$,
$x_{1}^{0}$ on its boundary. Let $\widetilde{h}_{0}$ be the field which
describes $\widetilde{\mathcal{D}}_{0}$. We let $\mathcal{D}_{00}$ be the
quantum surface described by $h_{00}=\widetilde{h}_{0}+\tfrac{2}{\gamma}\log
L_{00}^{-1}$ where
$L_{00}=\nu_{\widetilde{h}_{0}}([x_{0}^{0},x_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{00}$, $x_{1}^{00}$ for the corresponding points. We refer
to $(\mathcal{D}_{00},x_{0}^{00},x_{1}^{00})$ as the child of
$(\mathcal{D}_{0},x_{0}^{0},x_{1}^{0})$ and then continue the exploration in
$\mathcal{D}_{00}$ from $x_{0}^{00}$ to $x_{1}^{00}$.

Completion of the proof. The above defines a tree structure of explorations
$(\mathcal{D}_{o},h_{o},x_{0}^{o},x_{1}^{o})$ together with scaling factors
$L_{o}$ for $o\in\\{0,1\\}^{n}$ and $n\in\mathbf{N}$ where each node can
either have zero children (exploration does not fail), one child (large
downward jump which does not separate the endpoints, too many chunks at a
fixed stage, or the top boundary length is too large), or two children (large
downward jump which separates the endpoints or a large upward jump). Let
$\mathcal{T}$ be the tree associated with the exploration and for
$o\in\mathcal{T}$ we let $o^{\prime}$ be the parent of $o$. Note that
$x_{0}^{o},x_{1}^{o}$ correspond to points $\widetilde{x}_{0}^{o}$,
$\widetilde{x}_{1}^{o}$ on the boundary of the exploration from
$x_{0}^{o^{\prime}}$ to $x_{1}^{o^{\prime}}$ in $\mathcal{D}_{o^{\prime}}$.
Moreover, $L_{o}$ is the quantum length of the counterclockwise arc from
$\widetilde{x}_{0}^{o^{\prime}}$ to $\widetilde{x}_{1}^{o^{\prime}}$ along the
boundary of the exploration from $x_{0}^{o}$ to $x_{0}^{o^{\prime}}$. In
particular, $L_{0}=\delta$. On the event $E$ that $\mu_{h}(B(z,\epsilon))\geq
c_{0}\epsilon^{\alpha_{\mathrm{LBD}}}$ for all $\epsilon\in(0,\epsilon_{0})$
and $z\in D$ so that $B(z,\epsilon)\subseteq D$ we have that $h_{0}$ satisfies
$\mu_{h_{0}}(B(z,\epsilon))\geq
c_{0}\delta^{-2}\epsilon^{\alpha_{\mathrm{LBD}}}$ for all
$\epsilon\in(0,\epsilon_{0})$ and $z\in D$ so that $B(z,\epsilon)\subseteq D$.
The reason for this is that adding $\tfrac{2}{\gamma}\log\delta^{-1}$ to the
field has the effect of multiplying quantum areas by the factor $\delta^{-2}$.

Lemma 4.4 implies that by choosing $J\in\mathbf{N}$ and $C>0$ large enough we
can ensure that the mean number of children of each node in $\mathcal{T}$ is
strictly smaller than $1$ so that the corresponding Galton-Watson tree is
subcritical. Since the offspring distribution is supported on $\\{0,1,2\\}$,
the size of the Galton-Watson tree also has an exponential tail.

We are now going to use the definition of the $E_{i,M}^{\uparrow}$,
$E_{i,M}^{\downarrow}$ from Definition 5.2 to construct a path $\omega$ from
$x_{0}$ to $x_{1}$ contained in $\Upsilon$ and bound the tail of
${\mathfrak{N}}_{\epsilon}(\omega)$.

Let us first consider the exploration from $x_{0}^{0}$ to $x_{0}^{1}$ in
$\mathcal{D}_{0}$ in the case that it does not fail. We say that two chunks
$\mathcal{N}_{i,M}^{\uparrow}$, $\mathcal{N}_{\ell,L}^{\uparrow}$ for which
$E_{i,M}^{\uparrow}$ and $E_{\ell,L}^{\uparrow}$ occur and $L\geq M$ (also
$\ell\geq i$ if $L=M$) are adjacent if $\eta_{\ell,L}^{\uparrow}(0)$ is on the
top of $\mathcal{N}_{i,M}^{\uparrow}$ (and the same with $\downarrow$ in place
of $\uparrow$). Let $\mathcal{N}_{n}$ for $n\in\mathbf{Z}$ be the bi-infinite
sequence of adjacent chunks. Then parts (ii) and (iv) of Definition 5.2 give
us a way to construct a path connecting $x_{0}^{0}$ to $x_{1}^{0}$ as follows.
For each $n\in\mathbf{Z}$, we let $\omega_{n}$ be the path in
$\mathcal{N}_{n}$ corresponding to part (ii) of Definition 5.2 and we let
$\omega_{n-1,n}$ be the path in
$\mathrm{int}(\mathrm{cl}(\mathcal{N}_{n-1}\cup\mathcal{N}_{n}))$
corresponding to part (iv) of Definition 5.2. Then successively concatenating
the paths $\omega_{n-1,n}$, $\omega_{n}$ leads to a path $\omega$ in
$\Upsilon$ connecting $x_{0}^{0}$ to $x_{1}^{0}$ and we have that

${\mathfrak{N}}_{\epsilon}(\omega)\leq\sum_{n\in\mathbf{Z}}({\mathfrak{N}}_{\epsilon}(\omega_{n})+{\mathfrak{N}}_{\epsilon}(\omega_{n-1,n})).$

Let $\varphi_{n}\colon\mathcal{N}_{n}\to\mathbf{D}$ be the map as defined in
part (ii) of Definition 5.2. Then we know that
${\mathfrak{N}}_{\epsilon}(\varphi_{n}(\omega_{n}))\leq
R_{1}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}$ and $\varphi_{n}(\omega_{n})$
has distance at least $R_{1}^{-1}$ from $\partial\mathbf{D}$. Let

$\overline{d}_{n}=\sup_{z\in
B(0,1-R_{1}^{-1})}|(\varphi_{n}^{-1})^{\prime}(z)|\quad\text{and}\quad\underline{d}_{n}=\inf_{z\in
B(0,1-R_{1}^{-1})}|(\varphi_{n}^{-1})^{\prime}(z)|.$

For each $\zeta>0$, let $A_{n,\zeta}$ (resp. $B_{n,\zeta}$) be the
$\zeta$-neighborhood of $\varphi_{n}(\omega_{n})$ (resp. $\omega_{n}$). Then
as $B_{n,\underline{d}_{n}\epsilon}\subseteq\varphi_{n}(A_{n,\epsilon})$ we
have that

(5.3)
$\displaystyle{\mathfrak{N}}_{\underline{d}_{n}\epsilon}(\omega_{n})\leq\int_{A_{n,\epsilon}}|(\varphi_{n}^{-1})^{\prime}(z)|^{2}dz\leq
R_{1}\overline{d}_{n}^{2}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}.$

By Lemma 5.4 we have for a constant $c_{0}>0$ that

(5.4) ${\mathfrak{N}}_{\underline{d}_{n}\epsilon}(\omega_{n})\geq
c_{0}\underline{d}_{n}{\mathfrak{N}}_{\epsilon}(\omega_{n}).$

Distortion estimates for conformal maps imply that
$\overline{d}_{n}/\underline{d}_{n}$ is bounded from above by a constant which
depends only on $R_{1}$. Consequently, by combining (5.3) with (5.4) we see
for a constant $c_{1}>0$ that

(5.5) ${\mathfrak{N}}_{\epsilon}(\omega_{n})\leq
c_{1}\overline{d}_{n}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}.$

If $\mathcal{N}_{n}=\mathcal{N}_{i,M}^{\uparrow}$ or
$\mathcal{N}_{n}=\mathcal{N}_{i,M}^{\downarrow}$, then we have that
$\mu_{h_{0}}(\mathcal{N}_{n})\leq R_{1}2^{-M}$. Consequently,
$\mu_{h}(\mathcal{N}_{n})\leq R_{1}\delta^{2}2^{-M}$.

It therefore follows that on $E$ we have for a constant $c_{2}>0$ that

(5.6) $\underline{d}_{n}\leq
c_{2}\delta^{2/\alpha_{\mathrm{LBD}}}2^{-M/\alpha_{\mathrm{LBD}}}.$

Combining (5.5) with (5.6) implies for a constant $c_{3}>0$ that

(5.7) ${\mathfrak{N}}_{\epsilon}(\omega_{n})\leq
c_{3}\delta^{2/\alpha_{\mathrm{LBD}}}2^{-M/\alpha_{\mathrm{LBD}}}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}.$

By a similar argument, we have (possibly increasing $c_{3}>0$) that

(5.8) ${\mathfrak{N}}_{\epsilon}(\omega_{n-1,n})\leq
c_{3}\delta^{2/\alpha_{\mathrm{LBD}}}2^{-M/\alpha_{\mathrm{LBD}}}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}.$

Recall from the definition of the exploration succeeding that if it does
succeed then the number of chunks with quantum natural time in
$[\delta_{0}2^{-M},2^{-M}]$ with $M\geq J+1$ is at most a constant times
$2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$ and with $M=J$ is at most a
constant times $2^{(\kappa/4)J}$. We assume that we have chosen
$\alpha_{\mathrm{PERC}}\in(0,1)$ sufficiently close to $1$ so that
$\alpha=(1-\alpha_{\mathrm{PERC}})(\kappa/4)-1/\alpha_{\mathrm{LBD}}<0$. If we
sum (5.7), (5.8) over $n$, we thus see that those terms coming from chunks
with natural time in $[\delta_{0}2^{-M},2^{-M}]$ for $M\geq J+1$ contribute a
constant times $2^{-\alpha
J}\delta^{2/\alpha_{\mathrm{LBD}}}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}$.
With $\alpha=(\kappa/4)-\alpha_{\mathrm{LBD}}^{-1}$, those terms with $M=J$
contribute a constant times $2^{\alpha
J}\delta^{2/\alpha_{\mathrm{LBD}}}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}$.
Altogether, we have for a constant $c_{4}>0$ that

${\mathfrak{N}}_{\epsilon}(\omega)\leq c_{4}2^{\alpha
J}\delta^{2/\alpha_{\mathrm{LBD}}}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}.$

Let us now describe how we recursively construct the path $\omega$ in the case
that the exploration fails at some stage. Consider the tree $\mathcal{T}$
associated with iterating the exploration as defined above. We note that
$\mathcal{T}$ has a natural planar structure which comes from the ordering of
the exploration. We let $\widetilde{\mathcal{T}}$ consist of the nodes in
$\mathcal{T}$ which either have zero or two children and we view
$\widetilde{\mathcal{T}}$ as a planar tree with the tree structure and
ordering coming from that of $\mathcal{T}$. Let $o_{1},\ldots,o_{n}$ be the
leaves of $\widetilde{\mathcal{T}}$ (given according to their contour order).
For each $1\leq j\leq n$, we let $\omega_{j}$ be the path which is defined in
the same manner as $\omega$ just as above except using the exploration
associated with the node $o_{j}$ (which we emphasize does not fail as $o_{j}$
is a leaf). We let $\omega$ be the concatenation of the $\omega_{j}$. Then

${\mathfrak{N}}_{\epsilon}(\omega_{j})\leq c_{4}2^{\alpha
J}\left(\prod_{o}L_{o}^{2/\alpha_{\mathrm{LBD}}}\right){\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}$

where the product is over those $o\in\mathcal{T}$ which are ancestors of
$o_{j}$. The reason for this is that if we add $\tfrac{2}{\gamma}\log
L_{o}^{-1}$ to the field to multiply boundary lengths by $L_{o}^{-1}$ it has
the effect of multiplying areas by $L_{o}^{-2}$. Since $L_{o}\leq C$,
$L_{0}=\delta$, and the number of ancestors of $o_{j}$ is trivially at most
$|\mathcal{T}|$, the above is in turn bounded from above by

$c_{4}2^{\alpha
J}C^{2|\mathcal{T}|/\alpha_{\mathrm{LBD}}}\delta^{2/\alpha_{\mathrm{LBD}}}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}.$

Therefore

${\mathfrak{N}}_{\epsilon}(\omega)\leq c_{4}2^{\alpha
J}|\mathcal{T}|C^{2|\mathcal{T}|/\alpha_{\mathrm{LBD}}}\delta^{2/\alpha_{\mathrm{LBD}}}{\mathfrak{m}}_{\epsilon}^{\mathrm{HP}}.$

We note that making $J$ large has the effect of improving the tail of
$|\mathcal{T}|$. Therefore for every value of $p>0$ we can choose $J$
sufficiently large so that for a constant $c_{5}>0$ we have

$\mathbf{E}[(({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}{\mathfrak{N}}_{\epsilon}(\omega))^{p}{\bf
1}_{E}]\leq c_{5}\delta^{2p/\alpha_{\mathrm{LBD}}}.$

Fix $\beta\in(0,2/\alpha_{\mathrm{LBD}})$. As $p>0$ was arbitrary, the
Kolmogorov-Centsov theorem thus gives that the law of the $\beta$-Hölder norm
of
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(\cdot,\cdot;\Gamma)$
(with respect to the quantum length metric on $\partial D$) satisfies (5.1).

We note that (5.2) follows from the proof of (5.1) except instead of using the
paths $\omega_{n}$, $\omega_{n-1,n}$ we instead use the images of straight
lines under $\varphi_{n}^{-1}$. The same argument used to prove (5.7), (5.8)
implies that the diameter of such a path is at most a constant times
$\delta^{2/\alpha_{\mathrm{LBD}}}2^{-M/\alpha_{\mathrm{LBD}}}$. Therefore we
can string together these paths in place of the $\omega_{n}$, $\omega_{n-1,n}$
to complete the proof of (5.2). ∎

##  6\. Tightness in the interior

The aim of this section is to extend the tightness result from Section 5 to
points in the ${\rm CLE}_{\kappa}$ carpet which are contained in the interior
of the domain. As in Section 5, we will state and prove the result in a more
general setting than described in the statement of Theorem 1.1.

######  Proposition 6.1.

For each $\alpha_{\mathrm{LBD}}>2>\alpha_{\mathrm{UBD}}>0$ there exists
$\alpha_{\mathrm{KC}}>0$ so that the following is true. Suppose that
$\mathcal{D}=(D,h,x,y)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$. Fix
$\epsilon_{0}>0$ and let $E$ be the event that for every $z\in D$ and
$\epsilon\in(0,\epsilon_{0})$ so that $B(z,\epsilon)\subseteq D$ we have that
$\epsilon^{\alpha_{\mathrm{LBD}}}\leq\mu_{h}(B(z,\epsilon))\leq\epsilon^{\alpha_{\mathrm{UBD}}}$.
Let $\Gamma$ be an independent ${\rm CLE}_{\kappa}$ in $D$ and let $\Upsilon$
be its carpet. Let $X_{\epsilon}$ be the $\alpha_{\mathrm{KC}}$-Hölder norm of
$(z,w)\mapsto({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(z,w;\Gamma)$
with respect to the Euclidean metric on $D$. For every $p>0$ we have that

$\mathbf{P}[E,\ X_{\epsilon}\geq A]=O(A^{-p})\quad\text{as}\quad A\to\infty$

where the implicit constants depend only on
$p,\epsilon_{0},\alpha_{\mathrm{LBD}},\alpha_{\mathrm{UBD}}$.

The proof of Proposition 6.1 will involve several steps which we will now
describe.

  1. Step 1.

We fix $\alpha_{\mathrm{NET}}>0$ and for each $j\in\mathbf{N}$ we let
$N_{j}=2^{\alpha_{\mathrm{NET}}j}$ then pick $(z_{n})$ i.i.d. from
$\mu_{h,\Upsilon}$. By Lemma B.1, the points $z_{1},\ldots,z_{N_{j}}$ are very
likely to form a $2^{-j}$-net of $\Upsilon$. We then aim to show in the
remaining steps that if we have $1\leq k,\ell\leq N_{j}$ such that
$|z_{\ell}-z_{k}|\leq 2^{-j}$ then
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(z_{\ell},z_{k};\Gamma)$
is very likely to be at most $2^{-\alpha_{\mathrm{KC}}j}$ for some constant
$\alpha_{\mathrm{KC}}>0$.

  2. Step 2.

We will argue in Lemma 6.2 that there is a constant $\alpha_{\mathrm{PP}}>0$
so that there does not exist $z\in\Upsilon$ and $\mathcal{L}\in\Gamma$ so that
$\mathcal{L}\setminus B(z,\delta)$ has more than one component of diameter at
least $\delta^{1/\alpha_{\mathrm{PP}}}$. (We will also need a version where we
rule out there being more than one component of $\mathcal{L}\setminus
B(z,\delta)$ with quantum length at least $\delta^{1/\alpha_{\mathrm{PP}}}$.)

  3. Step 3.

We then use the percolation exploration from Section 4 to show that for every
$z\in\Upsilon$ with $\mathrm{dist}(z,\partial D)<\delta$ there exists a path
$\eta$ in $\Upsilon$ so that the component $U$ of $D\setminus\eta$ containing
$z$ has diameter at most a power of $\delta$ and every pair of points
$u,v\in\eta\cap\partial U$ satisfy that
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(u,v;\Gamma)$
is also at most a power of $\delta$ (Lemma 6.8). We now aim to combine this
with Steps 1 and 2 to bound
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(z_{\ell},z_{k};\Gamma)$
as described at the end of Step 1.

  4. Step 4.

For each $z\in\Upsilon$ and $r>0$ we let ${\mathfrak{B}}_{\epsilon}(z,r)$ be
the set of $w\in\Upsilon$ such that
$({\mathfrak{m}}_{\epsilon}^{\mathrm{HP}})^{-1}\mathfrak{d}_{\epsilon}(z,w;\Gamma)\leq
r$. We next establish a _lower bound_ on the Euclidean diameter of
${\mathfrak{B}}_{\epsilon}(z_{\ell},r)$ for each $1\leq\ell\leq N_{j}$ (Lemma
6.9). The bound will be so that it implies for a constant
$\alpha_{\mathrm{B}}>0$ that if $|z_{\ell}-z_{k}|\leq 2^{-j}$ then it very
likely that ${\mathfrak{B}}_{\epsilon}(z_{\ell},2^{-\alpha_{\mathrm{B}}j})$,
${\mathfrak{B}}_{\epsilon}(z_{k},2^{-\alpha_{\mathrm{B}}j})$ have diameter
much larger than $2^{-j}$ but it will not be clear at this point that
${\mathfrak{B}}_{\epsilon}(z_{\ell},2^{-\alpha_{\mathrm{B}}j})$,
${\mathfrak{B}}_{\epsilon}(z_{k},2^{-\alpha_{\mathrm{B}}j})$ intersect.

  5. Step 5.

We now fix a value of $1\leq\ell\leq N_{j}$ and consider an ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ exploration from a point on $\partial D$ targeted
at $z_{\ell}$ and coupled with $\Gamma$ as a CPI. We assume we are working on
the event that points within distance $2^{-j}$ of each other are not separated
from each other by a loop of $\Gamma$ of size at least
$2^{-j/\alpha_{\mathrm{PP}}}$. It is a consequence of the results in Appendix
A that if we parameterize $\eta$ according to the amount of quantum area that
its trunk $\eta^{\prime}$ has disconnected from $z_{\ell}$ then
$\eta^{\prime}$ is Hölder continuous with some exponent
$\alpha_{\mathrm{HO}}$. For each $m\in\mathbf{N}$ we let
$t_{m}=m2^{-j/(\alpha_{\mathrm{HO}}\alpha_{\mathrm{PP}})}$. By Step 3, we have
for each $m\in\mathbf{N}$ that the condition of Step 3 holds for the component
of $D\setminus\eta([0,t_{m}])$ which contains $z_{\ell}$ with very high
probability. Let $\tau$ be the first time $t$ that $\eta^{\prime}$ gets within
distance $2^{-j/\alpha_{\mathrm{PP}}}$ of $z_{\ell}$ and let $m_{0}$ be such
that $\tau\in(t_{m_{0}},t_{m_{0}+1}]$. We fix another value of $1\leq k\leq
N_{j}$. On the event that $|z_{\ell}-z_{k}|\leq 2^{-j}$ we have that
$\eta|_{[0,\tau]}$ hence also $\eta|_{[0,t_{m_{0}}]}$ does not disconnect
$z_{\ell}$, $z_{k}$ (as this would only be possible if there were a loop which
violated Step 2). Step 3 implies that we can find paths
$\omega_{\ell},\omega_{k}$ in the component of $D\setminus\eta([0,t_{m_{0}}])$
which contains $z_{\ell}$ which satisfy the properties mentioned above for
$z_{\ell}$, $z_{k}$. The sets
${\mathfrak{B}}_{\epsilon}(z_{k},2^{-\alpha_{\mathrm{B}}j})$,
${\mathfrak{B}}_{\epsilon}(z_{\ell},2^{-\alpha_{\mathrm{B}}j})$ from Step 4
then have to intersect $\omega_{\ell}$, $\omega_{k}$ which gives an overall
bound on $\mathfrak{d}_{\epsilon}(z_{\ell},\eta([0,t_{m_{0}}]);\Upsilon)$ and
$\mathfrak{d}_{\epsilon}(z_{k},\eta([0,t_{m_{0}}]);\Upsilon)$. Proposition 5.1
then gives an upper bound on
$\mathfrak{d}_{\epsilon}(u_{\ell},u_{k};\Upsilon)$ where $u_{\ell}$, $u_{k}$
are where $\omega_{\ell}$, $\omega_{k}$, respectively, hit
$\eta([0,t_{m_{0}}])$. Altogether, this gives an upper bound to
$\mathfrak{d}_{\epsilon}(z_{\ell},z_{k};\Upsilon)$ which is a power of
$2^{-j}$. All of these estimates will be combined carefully in Lemma 6.10.

  6. Step 6.

We complete the proof by applying an argument analogous to that used to prove
the Kolmogorov-Centsov continuity criterion.

###  6.1. ${\rm CLE}$ loop pinch point bound

Suppose that $D\subseteq\mathbf{C}$ is a bounded simply connected domain,
$(D,h,x,y)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$, and $\Gamma$ is
an independent ${\rm CLE}_{\kappa}$ in $D$. Fix
$\epsilon,\alpha_{\mathrm{PP}}>0$. We say that $\mathcal{L}\in\Gamma$ has an
$(\epsilon,\alpha_{\mathrm{PP}})$-pinch point if one of the following hold.

  1. (i)

The quantum length of $\mathcal{L}$ exceeds $2\epsilon$ and there exists
$z\in\mathcal{L}$ so that if $z_{1}$ (resp. $z_{2}$) is the point in
$\mathcal{L}$ so that
$\nu_{h}([z,z_{1}]_{\mathcal{L}}^{\bm{\circlearrowleft}})=\epsilon$ (resp.
$\nu_{h}([z,z_{2}]_{\mathcal{L}}^{\bm{\circlearrowright}})=\epsilon$) then
$[z_{1},z_{2}]_{\mathcal{L}}^{\bm{\circlearrowleft}}\cap
B(z,\epsilon^{\alpha_{\mathrm{PP}}})\neq\emptyset$ or

  2. (ii)

The diameter of $\mathcal{L}$ exceeds $2\epsilon$ and there exists
$z\in\mathcal{L}$ such that if $z_{1}$ (resp. $z_{2}$) is the first
counterclockwise (resp. clockwise) point in $\mathcal{L}$ starting from $z$ so
that $\mathrm{diam}([z,z_{1}]_{\mathcal{L}}^{\bm{\circlearrowleft}})=\epsilon$
(resp.
$\mathrm{diam}([z,z_{2}]_{\mathcal{L}}^{\bm{\circlearrowright}})=\epsilon)$
then $[z_{1},z_{2}]_{\mathcal{L}}^{\bm{\circlearrowleft}}\cap
B(z,\epsilon^{\alpha_{\mathrm{PP}}})\neq\emptyset$.

######  Lemma 6.2.

Suppose that $D\subseteq\mathbf{C}$ is a bounded simply connected domain and
$(D,h,x,y)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$. Suppose that
$\alpha_{\mathrm{UBD}},\epsilon_{0}>0$. There exists
$\alpha_{\mathrm{PP}},\beta>0$ depending only on $\alpha_{\mathrm{UBD}}$ so
that the following is true. Let $E_{1}$ be the event that for every
$\epsilon\in(0,\epsilon_{0})$ and $z\in D$ so that $B(z,\epsilon)\subseteq D$
we have that $\mu_{h}(B(z,\epsilon))\leq\epsilon^{\alpha_{\mathrm{UBD}}}$. Let
$E_{2}$ be the event that there is $\mathcal{L}\in\Gamma$ with an
$(\epsilon,\alpha_{\mathrm{PP}})$-pinch point. Then $\mathbf{P}[E_{1}\cap
E_{2}]=O(\epsilon^{\beta})$ for all $\epsilon\in(0,\epsilon_{0})$.

In what follows, for $\ell>0$ we let
$\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$ be the law on quantum
surfaces $(D,h)$ whose Radon-Nikodym derivative with respect to
$\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$ is given by a normalizing
constant times $\mu_{h}(D)$. As $\mathbf{E}[\mu_{h}(D)]<\infty$ under
$\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$, we have that
$\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$ indeed makes sense as a
probability measure. We note that it suffices to prove Lemma 6.2 under
$\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$ in place of
$\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$ since $\mu_{h}(D)$ has finite
negative moments of all orders [1, Theorem 1.2]. Given that $(D,h)$ has law
$\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$, it is natural to
consider $(D,h)$ marked by a point $z$ chosen from $\mu_{h}$ and we will also
write $(D,h,z)\sim\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$ in this
case.

######  Lemma 6.3.

There exists $\alpha,\beta,c_{0}>0$ so that the following is true. Suppose
that $(\mathbf{D},h,0)\sim\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$.
Then

$\mathbf{P}[\exists\theta\in[0,2\pi):\nu_{h}([e^{i\theta},e^{i(\theta+\epsilon)}]_{\partial\mathbf{D}}^{\bm{\circlearrowleft}})\geq\epsilon^{\beta}]\leq
c_{0}\epsilon^{\alpha}\quad\text{for all}\quad\epsilon>0.$

###### Proof.

In the case that
$(\mathbf{D},h,-i,i)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$, this
follows from the same argument used to prove Lemma A.6 except here we use the
moment upper bound for the boundary length measure for a free boundary GFF. We
will thus omit the details in this case. The result in the case of
$\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$ follows since [1, Theorem
1.2] implies that $\mu_{h}(D)$ has a finite $p$th moment for each
$p<4/\gamma^{2}$ [1, Theorem 1.2] so we can get the result from the case of
$\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$ and apply Hölder’s
inequality. ∎

###### Proof of Lemma 6.2.

Let $\alpha$, $\beta$ be as in Lemma 6.3 and let $\epsilon,\delta>0$. We
assume that $(D,h,z)\sim\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$. As
remarked after the lemma statement, it suffices to give the proof in this
setting. Note that $z$ is a.s. surrounded by a loop $\mathcal{L}_{z}$ of
$\Gamma$. Let $D_{z}$ be the component of $\mathbf{C}\setminus\mathcal{L}_{z}$
which contains $z$. Let $\varphi_{z}\colon D_{z}\to\mathbf{D}$ be the unique
conformal map with $\varphi_{z}(z)=0$, $\varphi_{z}^{\prime}(z)>0$, and let
$h_{z}=h\circ\varphi_{z}^{-1}+Q\log|(\varphi_{z}^{-1})^{\prime}|$. Let $F$ be
the event that $\mathcal{L}_{z}$ has quantum length at least $\epsilon$ and
there exists $\theta\in[0,2\pi)$ so that
$\nu_{h_{z}}([e^{i\theta},e^{i(\theta+\delta)}]_{\partial\mathbf{D}}^{\bm{\circlearrowleft}})\geq\delta^{\beta}$.
Letting $\ell>0$ be the quantum length of $\mathcal{L}_{z}$, we have that the
conditional law of $(\mathbf{D},h_{z},0)$ given $\ell$ is equal to
$\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$. Consequently, Lemma 6.3
implies that $\mathbf{P}[F]=O((\delta/\epsilon)^{\alpha})$.

Fix $\xi>1$ and $\sigma>0$. Let $G$ be the event that

  1. (i)

$E_{1}$ occurs

  2. (ii)

$\mu_{h}(D)\leq\epsilon^{-\sigma}$ and

  3. (iii)

there is $\mathcal{L}\in\Gamma$ with quantum length at least $2\epsilon$ which
satisfies (i) in the definition of an $(\epsilon,\alpha_{\mathrm{PP}})$-pinch
point and the domain surrounded by $\mathcal{L}$ has quantum area at least
$\epsilon^{2\xi}$.

Let us now make a few observations. First, it follows from Lemma B.4 that for
each $a>0$ there exists $b>0$ so that the probability that the number of loops
of $\Gamma$ in $D$ with quantum length at least $\epsilon$ is
$O(\epsilon^{-4/\kappa-1/2-a})$ is $1-O(\epsilon^{b})$. By [1, Theorem 1.2],
we have that the total quantum area associated with a sample from
$\mathsf{QDiskW}_{\bm{\gamma}=\gamma}^{\bm{L}=\ell}$ for $\ell\geq\epsilon$ is
at least $\epsilon^{2\xi}$ off an event whose probability decays to $0$ as
$\epsilon\to 0$ faster than any power of $\epsilon$. Combining these two facts
implies that off an event whose probability decays to $0$ faster than any
power of $\epsilon$ as $\epsilon\to 0$, the quantum area of the region
surrounded by each loop of $\Gamma$ with quantum length at least $\epsilon$ is
at least $\epsilon^{2\xi}$. Moreover, by [1, Theorem 1.2] we have that
$\mathbf{P}[\mu_{h}(D)\geq\epsilon^{-\sigma}]=O(\epsilon^{(4/\gamma^{2}-1)\sigma})$.
It thus follows that $\mathbf{P}[E_{1}\cap
E_{2}]\leq\mathbf{P}[G]+O(\epsilon^{(4/\gamma^{2}-1)\sigma})$.

We will argue that there exists a constant $c_{0}>0$ so that

(6.1) $\displaystyle\mathbf{P}[F\,|\,G]\geq c_{0}\epsilon^{2\xi+\sigma}$

which, in turn, implies that

(6.2)
$\displaystyle\mathbf{P}[G]\leq\frac{\mathbf{P}[F]}{\mathbf{P}[F\,|\,G]}=\frac{O((\delta/\epsilon)^{\alpha})}{c_{0}\epsilon^{2\xi+\sigma}}=O(\delta^{\alpha}\epsilon^{-\alpha-2\xi-\sigma}).$

Combining, this will prove the result provided we take $\delta$ to be a
sufficiently large power of $\epsilon$.

Suppose that we are working on $G$, that $\mathcal{L}$ is a loop of $\Gamma$
with quantum length at least $2\epsilon$ and suppose that $\mathcal{L}$
satisfies (i) in the definition of an $(\epsilon,\alpha_{\mathrm{PP}})$-pinch
point. We assume further that the domain $U$ surrounded by $\mathcal{L}$
satisfies $\mu_{h}(U)\geq\epsilon^{2\xi}$ and that among all such loops
$\mathcal{L}$ is leftmost (smallest real part) and among all such leftmost
loops is bottommost (smallest imaginary part). We let $w$ be the
$(\epsilon,\alpha_{\mathrm{PP}})$-pinch point of $\mathcal{L}$ which is
leftmost and among all leftmost $(\epsilon,\alpha_{\mathrm{PP}})$-pinch points
of $\mathcal{L}$ is bottommost. We note that if $d\in(0,\epsilon_{0})$ and
$\mathrm{diam}(U)\leq d$ then on $E_{1}$ we have that $\mu_{h}(U)\leq
d^{\alpha_{\mathrm{UBD}}}$. That is, we have that $\epsilon^{2\xi}\leq
d^{\alpha_{\mathrm{UBD}}}$. Therefore
$\mathrm{diam}(U)\geq\min(\epsilon^{2\xi/\alpha_{\mathrm{UBD}}},\epsilon_{0})$.
Since $\mu_{h}(D)\leq\epsilon^{-\sigma}$ and $\mu_{h}(U)\geq\epsilon^{2\xi}$,
it follows that $\mathbf{P}[z\in U\,|\,G]\geq\epsilon^{2\xi+\sigma}$.
Moreover, $\mathbf{P}[z\in B(w,\delta)\,|\,G,\ z\in
U]=O(\delta^{\alpha_{\mathrm{UBD}}}\epsilon^{-2\xi})$. Suppose that we are
working on $G$ and $z\in U\setminus B(w,\delta)$. Let $\varphi\colon
U\to\mathbf{D}$ be the unique conformal transformation with $\varphi(z)=0$ and
$\varphi^{\prime}(z)>0$. Then there exists a component $L$ of
$\mathcal{L}\setminus B(z,\epsilon^{\alpha_{\mathrm{PP}}})$ which has quantum
length at least $\epsilon$ and is not part of the boundary of the component of
$U\setminus B(w,\delta)$ which contains $z$. The Beurling estimate implies
that $\varphi(L)$ is mapped to an interval of $\partial\mathbf{D}$ of
(Euclidean) length $O((\epsilon^{\alpha_{\mathrm{PP}}}/\delta)^{1/2})$.
Altogether, this implies that

$\displaystyle\mathbf{P}[F\,|\,G]$ $\displaystyle\geq\mathbf{P}[z\in
U\setminus B(w,\delta)\,|\,G]\geq\mathbf{P}[z\in U\,|\,G](1-\mathbf{P}[z\in
B(w,\delta)\,|\,G,\ z\in U])$
$\displaystyle\geq\epsilon^{2\xi+\sigma}(1-O(\delta^{\alpha_{\mathrm{UBD}}}\epsilon^{-2\xi})).$

We assume that $\delta$ is a sufficiently large power of $\epsilon$ so that
$O(\delta^{\alpha_{\mathrm{UBD}}}\epsilon^{-2\xi})=o(1)$ as $\epsilon\to 0$.
This implies (6.1). Taking $\delta$ to be an even larger power of $\epsilon$
so that the power of $\epsilon$ in the right side of (6.2) is positive and
then $\alpha_{\mathrm{PP}}>0$ large enough so that
$(\epsilon^{\alpha_{\mathrm{PP}}}/\delta)^{1/2}=O(\delta^{\beta})$ completes
the proof in the case of $(\epsilon,\alpha_{\mathrm{PP}})$ pinch points
satisfying (i). The proof is analogous for $(\epsilon,\alpha_{\mathrm{PP}})$
pinch points satisfying (ii). ∎

###  6.2. Proof of tightness in the interior

The first step to completing the proof of Proposition 6.1 is to give the
definition of a good chunk for the adaptive ${\rm SLE}_{\kappa}^{0}(\kappa-6)$
exploration of a quantum half-plane $\mathcal{H}=(\mathbf{H},h,0,\infty)$ (see
Figure 6.1 for an illustration). The definition that we will give below will
be different than in Section 5 because the percolation exploration will serve
a different purpose. Suppose that $\eta$ is an ${\rm
SLE}_{\kappa}^{0}(\kappa-6)$ from $0$ to $\infty$ sampled independently of $h$
and then subsequently parameterized according to the quantum natural time of
its trunk. As in the beginning of Section 5, for the quantum surface
$\mathcal{N}_{t}$ disconnected from $\infty$ by $\eta|_{[0,t]}$, we let
$z_{L}(\mathcal{N}_{t})$ (resp. $z_{R}(\mathcal{N}_{t})$) denote the leftmost
(resp. rightmost) point on the bottom of $\mathcal{N}_{t}$. We also let
$z_{C}(\mathcal{N}_{t})=\eta(0)$.

Figure 6.1.  Top: Illustration of part (ii) of Definition 6.4. Shown in blue
is the exploration up until $\mathcal{N}_{i,M}^{\uparrow}$ is discovered and
the images of $z_{L}(\mathcal{N}_{i,M}^{\uparrow})$,
$z_{C}(\mathcal{N}_{i,M}^{\uparrow})$, and
$z_{R}(\mathcal{N}_{i,M}^{\uparrow})$ under $\varphi$ are respectively given
by $-1$, $-i$, and $1$. Bottom: Illustration of part (iii) of Definition 6.4.
The chunk $\mathcal{N}_{\ell,L}^{\uparrow}$ is shown in light yellow even
though it is “good” in order to differentiate it from
$\mathcal{N}_{i,M}^{\uparrow}$. The images of
$z_{L}(\mathcal{N}_{\ell,L}^{\uparrow})$,
$z_{C}(\mathcal{N}_{\ell,L}^{\uparrow})$, and
$z_{R}(\mathcal{N}_{\ell,L}^{\uparrow})$ under $\psi$ are $-1$, $-i$, and $1$,
respectively.

######  Definition 6.4.

We suppose that we have the setup described in Section 4.1. Fix
$R_{1},R_{2},R_{3}\geq 1$. For each $i,M$, we define $E_{i,M}^{\uparrow}$ to
be the event for $\mathcal{N}_{i,M}^{\uparrow}$ and $\eta_{i,M}^{\uparrow}$
that $\eta_{i,M}^{\uparrow}(\sigma_{i,M}^{\uparrow})\in\partial\mathbf{H}$ and
the following hold.

  1. (i)

The quantum lengths of the top, the left side of the bottom, and the right
side of the bottom of $\mathcal{N}_{i,M}^{\uparrow}$ are in
$[R_{1}^{-1}2^{-(\kappa/4)M},R_{1}2^{-(\kappa/4)M}]$.

  2. (ii)

Let $\varphi\colon\mathcal{N}_{i,M}^{\uparrow}\to\mathbf{D}$ be the unique
conformal transformation with
$\varphi(z_{L}(\mathcal{N}_{i,M}^{\uparrow}))=-1$,
$\varphi(z_{C}(\mathcal{N}_{i,M}^{\uparrow}))=-i$, and
$\varphi(z_{R}(\mathcal{N}_{i,M}^{\uparrow}))=1$. Let
$\mathcal{B}_{i,M}^{\uparrow}=\varphi^{-1}(B(0,1/2))$. Then we have that

$R_{2}^{-1}2^{-M}\leq\mu_{h}(\mathcal{B}_{i,M}^{\uparrow})\leq\mu_{h}(\mathcal{N}_{i,M}^{\uparrow})\leq
R_{2}2^{-M}.$

  3. (iii)

Let $\mathcal{N}_{\ell,L}^{\uparrow}$ be the chunk whose top contains
$\eta_{i,M}^{\uparrow}(0)$ (so that the left side of the bottom of
$\mathcal{N}_{i,M}^{\uparrow}$ is glued to the top of
$\mathcal{N}_{\ell,L}^{\uparrow}$) if it exists, otherwise we set
$\mathcal{N}_{\ell,L}^{\uparrow}=\emptyset$. If
$\mathcal{N}_{\ell,L}^{\uparrow}\neq\emptyset$, let $\widetilde{\mathcal{N}}$
be the quantum surface parameterized by
$\mathrm{int}(\mathrm{cl}(\mathcal{N}_{\ell,L}^{\uparrow}\cup\mathcal{N}_{i,M}^{\uparrow}))$
and let $\psi\colon\widetilde{\mathcal{N}}\to\mathbf{D}$ be the unique
conformal map so that $\psi(z_{L}(\mathcal{N}_{\ell,L}^{\uparrow}))=-1$,
$\psi(z_{C}(\mathcal{N}_{\ell,L}^{\uparrow}))=-i$, and
$\psi(z_{R}(\mathcal{N}_{\ell,L}^{\uparrow}))=1$. Then

    1. (I)

$R_{2}^{-1}2^{-M}\leq\mu_{h}(\psi^{-1}(B(0,1-R_{2}^{-1})))\leq\mu_{h}(\widetilde{\mathcal{N}})\leq
4R_{2}2^{-M}$ and

    2. (II)

$\psi^{-1}(B(0,1-R_{2}^{-1}))$ contains both $\mathcal{B}_{i,M}^{\uparrow}$
and $\mathcal{B}_{\ell,L}^{\uparrow}$.

  4. (iv)

If $\widehat{\mathcal{N}}$ is another ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ chunk
defined using the time-interval $[\delta_{0}2^{-j},2^{-j}]$ for
$j\in\\{M-1,M,M+1\\}$ which starts from a point on the top of
$\mathcal{N}_{i,M}^{\uparrow}$ with clockwise boundary length distance from
$\partial\mathcal{H}_{i,M}^{\uparrow}$ given by an integer multiple of
$a_{0}2^{-(\kappa/4)M}$ chosen in an $\mathcal{F}_{i,M}^{\uparrow}$-measurable
manner and with boundary length distance from a point in $\Upsilon$ at most
$a_{0}2^{-(\kappa/4)M}$, then the conditional probability given
$\mathcal{F}_{i,M}^{\uparrow}$ that the previous item holds for the pair
$(\mathcal{N}_{i,M}^{\uparrow},\widehat{N})$ in place of
$(\mathcal{N}_{\ell,L}^{\uparrow},\mathcal{N}_{i,M}^{\uparrow})$ is at least
$1-R_{3}^{-1}$.

We define $E_{i,M}^{\downarrow}$ analogously to $E_{i,M}^{\uparrow}$.

######  Lemma 6.5.

Suppose that we have the setup described in Section 4.1 and the events
$E_{i,M}^{\uparrow}$, $E_{i,M}^{\downarrow}$ are as in Definition 6.4. For
every $p_{0}\in(0,1)$ there exists $R_{1},R_{2},R_{3}\geq 1$, $\delta_{0}>0$
so that on the event that the exploration has not failed before exploring
$\mathcal{N}_{i,M}^{\uparrow}$ (resp. $\mathcal{N}_{i,M}^{\downarrow}$), we
have that
$\mathbf{P}[E_{i,M}^{\uparrow}\,|\,\mathcal{F}_{i-1,M}^{\uparrow}]\geq p_{0}$
(resp.
$\mathbf{P}[E_{i,M}^{\downarrow}\,|\,\mathcal{F}_{i-1,M}^{\downarrow}]\geq
p_{0}$) for all $i,M\in\mathbf{N}$.

###### Proof.

Lemmas C.2 and C.4 imply that we can take $\delta_{0}\in(0,1)$ sufficiently
small so that the probability that
$\eta_{i,M}^{\uparrow}(\sigma_{i,M}^{\uparrow})\in\partial\mathbf{H}$ is at
least $1-q_{0}/3$ where $q_{0}=1-p_{0}$. The scaling properties of
$4/\kappa$-stable Lévy processes imply that (i) holds with probability tending
to $1$ as $R_{1}\to\infty$. Likewise, it is also clear that (ii) holds with
probability tending to $1$ as $R_{2}\to\infty$. By the definition of
$E_{\ell,L}^{\uparrow}$, we have that (iii) holds with probability at least
$1-R_{3}^{-1}$. We choose $R_{3}\geq 1$ sufficiently large so that
$1-R_{3}^{-1}\geq 1-q_{0}/3$. To see that (iv) holds with probability tending
to $1$ as $R_{2}\to\infty$ (when $R_{3}$ is fixed), we note that we can
construct the gluing of $\mathcal{N}_{i,M}^{\uparrow}$ and
$\widehat{\mathcal{N}}$ as follows. Recall that $\mathcal{H}_{i,M}^{\uparrow}$
is the quantum half-plane which corresponds to the unexplored region when
generating $\eta_{i,M}^{\uparrow}$. We suppose that we are working on the
event that
$\eta_{i,M}^{\uparrow}(\sigma_{i,M}^{\uparrow})\in\partial\mathbf{H}$ and
properties (i) and (ii) of Definition 6.4 hold. Then there are at most
$R_{1}a_{0}^{-1}$ points on the top of $\mathcal{N}_{i,M}^{\uparrow}$ with
clockwise boundary length distance to $\partial\mathcal{H}_{i,M}^{\uparrow}$
given by an integer multiple of $a_{0}2^{-(\kappa/4)M}$ and with boundary
length distance from a point in $\Upsilon$ at most $a_{0}2^{-(\kappa/4)M}$.
For a given such point chosen in an $\mathcal{F}_{i,M}^{\uparrow}$-measurable
manner, we run another ${\rm SLE}_{\kappa}^{0}(\kappa-6)$ process
$\widehat{\eta}$ in
$\mathcal{H}_{i+1,M}^{\uparrow}=\mathcal{H}_{i,M}^{\uparrow}\setminus\mathcal{N}_{i,M}^{\uparrow}$.
Suppose that we have fixed $j\in\\{M-1,M,M+1\\}$. On the event that
$\widehat{\sigma}=\inf\\{t\geq\delta_{0}2^{-j}:\widehat{\eta}(t)\in\partial\mathcal{H}_{i+1,M}^{\uparrow}\\}\leq
2^{-j}$ and property (i) of Definition 6.4 holds for the surface
$\widehat{\mathcal{N}}$ disconnected from $\infty$ by
$\widehat{\eta}|_{[0,\widehat{\sigma}]}$, the surface
$\widetilde{\mathcal{N}}$ parameterized by
$\mathrm{int}(\mathrm{cl}(\mathcal{N}_{i,M}^{\uparrow}\cup\widehat{\mathcal{N}}))$
is homeomorphic to $\mathbf{D}$. Therefore it is clear that (iv) holds with
probability tending to $1$ as $R_{2}\to\infty$ (with $R_{1},R_{3}\geq 1$
fixed) as well. ∎

######  Lemma 6.6.

Suppose that $D$ is a bounded, simply connected domain,
$(D,h)\sim\mathsf{QDisk}_{\bm{\gamma}=\gamma}^{\bm{L}=1}$, and $\Gamma$ is a
${\rm CLE}_{\kappa}$ on $D$ which is independent of $h$. Fix
$0<\alpha_{\mathrm{UBD}}<2<\alpha_{\mathrm{LBD}}$ and
$\alpha_{\mathrm{PP}},\epsilon_{0}>0$. Let $E$ be the event that for every
$z\in D$ and $\epsilon\in(0,\epsilon_{0})$ we have that
$\epsilon^{\alpha_{\mathrm{LBD}}}\leq\mu_{h}(B(z,\epsilon))\leq\epsilon^{\alpha_{\mathrm{UBD}}}$
and no loop of $\Gamma$ has an $(\epsilon,\alpha_{\mathrm{PP}})$-pinch point
for $\epsilon\in(0,\epsilon_{0})$. For each $\beta>0$ there exists $\xi>0$ so
that the following is true. Fix $\delta\in(0,\epsilon)$ and let $F$ be the
event that for every $z\in D$ with $\mathrm{dist}(z,\partial D)\leq\epsilon$
there exists a curve $\eta$ in $\Upsilon$ so that

  1. (i)

$\mathfrak{d}_{\delta}(u,v;\Gamma)/{\mathfrak{m}}_{\delta}^{\mathrm{HP}}\leq\epsilon^{\xi}$
for all $u,v\in\eta$ and

  2. (ii)

the component $U$ of $D\setminus\eta$ which contains $z$ satisfies
$\mathrm{diam}(U)\leq\epsilon^{\xi}$ and $\partial U\cap\partial
D\neq\emptyset$.

Uniformly in $\delta\in(0,\epsilon)$ we have that $\mathbf{P}[E\cap
F^{c}]=O(\epsilon^{\beta})$.

###### Proof.

Step 0: Setup. Suppose that $x\in\partial D$ is picked from $\nu_{h}$. For
each $k\in\mathbf{N}$, we let $N_{k}=2^{k}$ and let $I_{1},\ldots,I_{N_{k}}$
be arcs of $\partial D$ of equal quantum length $2^{-k}$ with disjoint
interior whose union is $\partial D$. We assume that the $I_{j}$ are given in
counterclockwise order and the left endpoint of $I_{1}$ is $x$. Fix $1\leq
j\leq N_{k}$ and let $I=I_{j}$. Fix $\zeta>0$. We will adjust the value of
$\zeta$ in the proof. The role of $\epsilon>0$ in the statement of the lemma
will be played by $2^{-\zeta k}$. Let $U=U_{j}$ be the $2^{-\zeta
k}$-neighborhood of $I$ with respect to the internal (Euclidean) metric in
$D$. Suppose that $\delta\in(0,2^{-\zeta k})$. We are going to show that there
exists $\xi>0$ (which does not depend on $k$) so that there exists a path
$\eta$ in $\Upsilon$ which satisfies (i) and such that
$\mathrm{diam}(U)\leq\epsilon^{\xi}$ off an event of probability
$O(2^{-(1+\beta)k})$. The result will then follow by performing a union bound
over $1\leq j\leq N_{k}$ and then over $k\in\mathbf{N}$.

Let $x_{0}$ be the left endpoint of $I_{j-1}$ and let $x_{1}$ be the right
endpoint of $I_{j+1}$, where we take the convention that $I_{0}=I_{N_{k}}$ and
$I_{N_{k}+1}=I_{1}$. Then $\nu_{h}([x_{0},x_{1}]_{\partial
D}^{\bm{\circlearrowleft}})=3\cdot 2^{-k}$ and
$I\subseteq[x_{0},x_{1}]_{\partial D}^{\bm{\circlearrowleft}}$. We let
$h_{0}=h+\tfrac{2}{\gamma}\log(2^{k}/3)$ so that with $x_{0}^{0}=x_{0}$ and
$x_{1}^{0}=x_{1}$ we have that $\nu_{h_{0}}([x_{0}^{0},x_{1}^{0}]_{\partial
D}^{\bm{\circlearrowleft}})=1$. We let $\mathcal{D}_{0}=(D,h_{0})$ be the
resulting (rescaled) quantum disk. We then perform the exploration as in Lemma
4.4 from $x_{0}^{0}$ to $x_{1}^{0}$ where the definition of a good chunk is as
in Definition 6.4 and where we take $J=\xi_{0}k$ and $\xi_{0}>0$ is a fixed
small constant whose value we will adjust later in the proof.

Step 1. Exploration succeeds. We first suppose that we are on the event that
the exploration does not fail and we will explain why there exists a path
$\eta$ as claimed. See Figure 6.2 for an illustration of the setup.
Afterwards, we will explain how to iterate the exploration to construct a path
$\eta$ in the case that the exploration fails. Let
$\mathcal{N}_{1},\ldots,\mathcal{N}_{n}$ be the (good) chunks in the
exploration which have (the maximal) quantum natural time in
$[\delta_{0}2^{-\xi_{0}k},2^{-\xi_{0}k}]$ (as measured using $h_{0}$). Note
that if we rescale the boundary length back by the factor $3/2^{k}$ (to
transform $h_{0}$ back to $h$) then quantum natural time gets scaled by the
factor $(3/2^{k})^{4/\kappa}$. Therefore the (maximal) quantum natural time of
a good chunk under the exploration as measured by $h$ is equal to a constant
times $2^{-\xi_{1}k}$ where $\xi_{1}=\xi_{0}+4/\kappa$. For each
$1\leq\ell\leq n-1$, let $\psi_{\ell}$ be the unique conformal map from
$\mathrm{int}(\mathrm{cl}(\mathcal{N}_{\ell}\cup\mathcal{N}_{\ell+1}))$ to
$\mathbf{D}$ which takes $z_{L}(\mathcal{N}_{\ell})$,
$z_{C}(\mathcal{N}_{\ell})$, and $z_{R}(\mathcal{N}_{\ell})$ to $-1$, $-i$,
and $1$, respectively. We then let
$\mathcal{A}_{\ell}=\psi_{\ell}^{-1}(B(0,1-R_{2}^{-1}))$.

Figure 6.2.  Illustration of the definition of $\omega_{\ell}$ in the case
that the exploration succeeds. Shown is part of $\partial D$ which contains
$I_{j-1}$, $I_{j}=I$, and $I_{j+1}$. The blue regions are the initial and
terminal parts of the exploration before and after the chunks with the maximal
size are discovered. Among the chunks with maximal size, those in green are
good (with the exception of $\mathcal{N}_{\ell}$ to differentiate it from
$\mathcal{N}_{\ell+1}$) and those in red are bad. Recall that $\psi_{\ell}$ is
the unique conformal map from
$\mathrm{int}(\mathrm{cl}(\mathcal{N}_{\ell}\cup\mathcal{N}_{\ell+1}))$ to
$\mathbf{D}$ which takes $z_{L}(\mathcal{N}_{\ell})$,
$z_{C}(\mathcal{N}_{\ell})$, $z_{R}(\mathcal{N}_{\ell})$ respectively to $-1$,
$-i$, and $1$ and $\varphi_{\ell}$ is the unique conformal map from
$\mathcal{N}_{\ell}$ to $\mathbf{D}$ which takes $z_{L}(\mathcal{N}_{\ell})$,
$z_{C}(\mathcal{N}_{\ell})$, $z_{R}(\mathcal{N}_{\ell})$ respectively to $-1$,
$-i$, and $1$. Shown also in red are the regions
$\mathcal{B}_{\ell}=\varphi_{\ell}^{-1}(B(0,1/2))$ and
$\mathcal{B}_{\ell+1}=\varphi_{\ell+1}^{-1}(B(0,1/2))$. The dashed region
which contains $\mathcal{B}_{\ell}$, $\mathcal{B}_{\ell+1}$ is
$\mathcal{A}_{\ell}=\psi_{\ell}^{-1}(B(0,1-R_{2}^{-1}))$. The path
$\omega_{\ell}$ (orange) is the image under $\psi_{\ell}^{-1}$ of the
concatenation of the line segment from
$z_{\ell}^{1}=\psi_{\ell}(\varphi_{\ell}^{-1}(0))$ to $0$ and from $0$ to
$z_{\ell}^{2}=\psi_{\ell}(\varphi_{\ell+1}^{-1}(0))$.

We are now going to show that we can adjust the value of $\zeta>0$ so that
$\mathcal{T}=\mathcal{N}_{1}\cup\mathcal{N}_{n}\cup(\cup_{\ell=1}^{n-1}\mathcal{A}_{\ell})$
contains a path $\omega$ which connects $I_{j-1}$ to $I_{j+1}$ and has
(Euclidean) distance at least $2^{-\zeta k}$ from $I_{j}$. This, in turn, will
imply that there is a component of $D\setminus\omega$ which contains $U$. To
see this, for each $1\leq\ell\leq n-1$ we let
$\varphi_{\ell}\colon\mathcal{N}_{\ell}\to\mathbf{D}$ be the unique conformal
map which takes $z_{L}(\mathcal{N}_{\ell})$, $z_{C}(\mathcal{N}_{\ell})$, and
$z_{R}(\mathcal{N}_{\ell})$ to $-1$, $-i$, and $1$, respectively. Let
$\mathcal{B}_{\ell}=\varphi_{\ell}^{-1}(B(0,1/2))$. Then we know from property
(iii) of Definition 6.4 that $\mathcal{A}_{\ell}$ contains
$\mathcal{B}_{\ell}$ and $\mathcal{B}_{\ell+1}$. By property (ii) of
Definition 6.4,

(6.3)
$\mu_{h}(\mathcal{B}_{\ell})=\left(\frac{3}{2^{k}}\right)^{2}\mu_{h_{0}}(\mathcal{B}_{\ell})\geq
R_{2}^{-1}2^{-(\xi_{0}+2)k}=R_{2}^{-1}2^{-\xi_{2}k}\quad\text{where}\quad\xi_{2}=\xi_{0}+2.$

From the definition of $E$, we have that

$\mathrm{diam}(\mathcal{B}_{\ell})^{\alpha_{\mathrm{UBD}}}\geq\mu_{h}(\mathcal{B}_{\ell})\geq
R_{2}^{-1}2^{-\xi_{2}k}.$

That is, on $E$ we have that

(6.4) $\mathrm{diam}(\mathcal{B}_{\ell})\geq
R_{2}^{-1/\alpha_{\mathrm{UBD}}}2^{-\xi_{2}k/\alpha_{\mathrm{UBD}}}.$

Assume that $1\leq\ell\leq n-1$. We now define a path $\omega_{\ell}$ as
follows. We let $z_{\ell}^{1}=\psi_{\ell}(\varphi_{\ell}^{-1}(0))$ and
$z_{\ell}^{2}=\psi_{\ell}(\varphi_{\ell+1}^{-1}(0))$. We then let
$\omega_{\ell}$ be the image under $\psi_{\ell}^{-1}$ of the concatenation of
$[z_{\ell}^{1},0]$ with $[0,z_{\ell}^{2}]$. As $B(0,1-R_{2}^{-1})$ contains
$\psi_{\ell}(\mathcal{B}_{\ell})$ and $\mathcal{B}_{\ell}$ has diameter at
least $R_{2}^{-1/\alpha_{\mathrm{UBD}}}2^{-\xi_{2}k/\alpha_{\mathrm{UBD}}}$,
standard distortion estimates for conformal maps imply that

$|(\psi_{\ell}^{-1})^{\prime}(z)|\gtrsim
2^{-\xi_{2}k/\alpha_{\mathrm{UBD}}}\quad\text{for all}\quad z\in
B(0,1-R_{2}^{-1})$

where the implicit constant in $\gtrsim$ depends only on $R_{2}$. Therefore
there exists a ($R_{2}$-dependent) constant $c_{0}>0$ so that $\mathcal{T}$
contains the (Euclidean) $c_{0}2^{-\xi_{2}k/\alpha_{\mathrm{UBD}}}$
neighborhood of $\omega_{\ell}$. We take
$\zeta>\xi_{2}/\alpha_{\mathrm{UBD}}$.

In order to finish proving the claim, it is left to explain why:

  * •

$\mathcal{N}_{1}$ (resp. $\mathcal{N}_{n}$) intersects $I_{j-1}$ (resp.
$I_{j+1}$) and

  * •

We can define a path $\omega_{0}$ in $\mathcal{N}_{1}\cup\mathcal{A}_{1}$
which connects $I_{j-1}$ to the starting point of $\omega_{1}$ and a path
$\omega_{n}$ in $\mathcal{A}_{n-1}\cup\mathcal{N}_{n}$ which connects the
ending point of $\omega_{n-1}$ to $I_{j+1}$ so that both $\omega_{0}$ and
$\omega_{n}$ have distance at least $2^{-\zeta k}$ from $I_{j}$.

The first point is in fact a consequence of the definition of the exploration
succeeding. This leaves us to address the second point.

Let us first explain how to construct the path $\omega_{0}$. Consider the set
$\mathcal{B}_{1}=\varphi_{1}^{-1}(B(0,1/2))$. As explained above, we know that
$\mathrm{diam}(\mathcal{B}_{1})\geq
R_{2}^{-1/\alpha_{\mathrm{UBD}}}2^{-\xi_{2}k/\alpha_{\mathrm{UBD}}}$. Let
$L_{1}$ (resp. $L_{2}$) be the line segment in $\mathbf{D}$ from $-1/4$ (resp.
$1/4$) to the point $z_{1}=-i$ (resp. the point $z_{2}$ on the bottom of
$\partial\mathbf{D}$ with real part $1/4$). We claim that there exists a
constant $c>0$ so that
$\mathrm{dist}(\varphi_{1}^{-1}(L_{1}),\varphi_{1}^{-1}(L_{2}))\geq
c2^{-\xi_{2}k/\alpha_{\mathrm{UBD}}}$. To see this, let $B$ be a Brownian
motion starting from $0$. Then $B$ has a positive chance of exiting
$\partial\mathbf{D}$ in
$[z_{1},z_{2}]_{\partial\mathbf{D}}^{\bm{\circlearrowleft}}$ before hitting
$L_{1}$, $L_{2}$. Therefore the same is also true for the Brownian motion
$\varphi_{1}^{-1}(B)$. The claim therefore follows from the Beurling estimate.
We can therefore take $\omega_{0}$ to be the image under $\varphi_{1}^{-1}$ of
the line segment from the point on the bottom of $\partial\mathbf{D}$ with
real part $1/8$ to $0$ and we see that the distance of $\omega_{0}$ to $I_{j}$
is at least $c2^{-\xi_{2}k/\alpha_{\mathrm{UBD}}}$ (possibly decreasing
$c>0$). We define $\omega_{n}$ in a similar manner.

We now define the path $\eta$. Let $x\in\partial D\setminus(I_{j-1}\cup
I_{j}\cup I_{j+1})$ be any fixed point. On the event that the exploration
succeeds, its outer boundary as viewed from $x$ does not depend on the choice
of $x$. We take $\eta$ to be the curve which corresponds to the outer boundary
of the exploration as viewed from $x$ and assume that $\eta$ is parameterized
by quantum length. We note that in this case the quantum length of $\eta$ is
at most a constant times $2^{-k}$. On $E$, (5.2) from Proposition 5.1 thus
gives an upper bound on $\mathrm{diam}(I_{j-1}\cup I_{j}\cup I_{j+1})$ and
$\mathrm{diam}(\eta)$ which is a power of $2^{-k}$ off an event which occurs
with probability $O(2^{-(1+\beta)k})$, which completes the proof in the case
that the exploration does not fail.

Step 2. Exploration fails. We now consider the possibilities which can occur
if the exploration described above fails. Measuring quantities using $h_{0}$,
we recall that the exploration fails if it makes either a large upward jump
(i.e., of size at least $2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$ when
exploring chunks with quantum natural time in $[\delta_{0}2^{-M},2^{-M}]$), a
large downward jump (i.e., of size at least
$2^{-\alpha_{\mathrm{PERC}}(\kappa/4)M}$ when exploring chunks with quantum
natural time in $[\delta_{0}2^{-M},2^{-M}]$), requires too many chunks at a
particular scale (i.e., at least
$c_{F}2^{(1-\alpha_{\mathrm{PERC}})(\kappa/4)M}$ when exploring chunks of size
in $[\delta_{0}2^{-M},2^{-M}]$ for $M>J$ or at least $c_{F}2^{(\kappa/4)J}$
when exploring chunks of size in $[\delta_{0}2^{-J},2^{-J}]$), or the
deviations in the boundary length process are too large (i.e., the absolute
value of the quantum length of the top minus the bottom exceeds
$c_{F}2^{(1-4/\kappa)M}$ when exploring chunks of size in
$[\delta_{0}2^{-M},2^{-M}]$). We stop the exploration immediately if either of
these possibilities happens, and then proceed as described below.

Figure 6.3.  Illustration of the possibility that the exploration fails by
making an upward jump whose size measured using $h_{0}$ is at least $3$. Shown
in blue is the exploration up until it makes the large upward jump and the
corresponding ${\rm CLE}_{\kappa}$ loop is shown in red. The two prime ends
which correspond to where the loop is rooted on the previous exploration are
$\widetilde{z}_{0}^{0}$, $\widetilde{z}_{1}^{0}$. The point $z_{0}^{0}$ (resp.
$z_{0}^{1}$) is the one on the loop so that the clockwise (resp.
counterclockwise) boundary length from $\widetilde{z}_{0}^{0}$ to $z_{0}^{0}$
(resp. $\widetilde{z}_{1}^{0}$ to $z_{1}^{0}$) measured using $h_{0}$ is equal
to $1$. Since we are working on the event that $\Gamma$ does not have any
$(\epsilon,\alpha_{\mathrm{PP}})$-pinch points for
$\epsilon\in(0,\epsilon_{0})$, by choosing $\zeta>0$ sufficiently large
(depending only on $\alpha_{\mathrm{PP}}$) we have that the clockwise arc of
the loop from $z_{0}^{0}$ to $z_{1}^{0}$ does not hit the ball of radius
$2^{-\zeta k}$ centered at $\widetilde{z}_{0}^{0}$. The point $w_{0}^{0}$
(resp. $w_{1}^{0}$) is the one which is $1$ unit of boundary length distance
in the clockwise (resp. counterclockwise) direction from $x_{0}^{0}$ (resp.
$x_{1}^{0}$) measured using $h_{0}$. The exploration is then continued from
$w_{0}^{0}$ to $z_{0}^{0}$ and from $z_{1}^{0}$ to $w_{1}^{0}$.

Case 1: large upward jump. We let $\widetilde{\mathcal{D}}_{0}$ be the quantum
disk in the complement of the exploration with $x_{0}^{0}$ and $x_{1}^{0}$ on
its boundary. We let $\widetilde{z}_{0}^{0}$, $\widetilde{z}_{1}^{0}$ be the
two prime ends on $\partial\widetilde{\mathcal{D}}_{0}$ which correspond to
where the discovered ${\rm CLE}_{\kappa}$ loop $\mathcal{L}$ is rooted to the
boundary. There are two possibilities. Either the size of the upward jump
(measured using $h_{0}$) is smaller or larger than $3$.

We first suppose that the upward jump (measured using $h_{0}$) has size at
most $3$. We let $z_{0}^{0}$ (resp. $z_{1}^{0}$) be the point on
$\partial\widetilde{\mathcal{D}}_{0}$ so that
$\nu_{h_{0}}([z_{0}^{0},x_{0}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})=1$
(resp.
$\nu_{h_{0}}([x_{1}^{0},z_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})=1$).
We let $\mathcal{D}_{00}$ be given by $\widetilde{\mathcal{D}}_{0}$ after
adding $\tfrac{2}{\gamma}\log L_{00}^{-1}$ to the field where
$L_{00}=\nu_{h_{0}}([z_{0}^{0},z_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{00}$, $x_{1}^{00}$ for the points which correspond to
$z_{0}^{0}$, $z_{1}^{0}$. We refer to
$(\mathcal{D}_{00},x_{0}^{00},x_{1}^{00})$ as the _child_ of
$(\mathcal{D}_{0},x_{0}^{0},x_{1}^{0})$. We then perform the exploration
inside of $\mathcal{D}_{00}$ from $x_{0}^{00}$ to $x_{1}^{00}$ with the
definition of the good chunk as in Definition 6.4. We note in this case that

(6.5) $3-c_{F}2^{(1-4/\kappa)J}\leq L_{00}\leq 6+c_{F}2^{(1-4/\kappa)J}.$

We now suppose that the upward jump (measured using $h_{0}$) has size at least
$3$. See Figure 6.3 for an illustration in this case. We let $z_{0}^{0}$
(resp. $z_{1}^{0}$) be the point on $\mathcal{L}$ so that
$\nu_{h_{0}}([\widetilde{z}_{0}^{0},z_{0}^{0}]_{\mathcal{L}}^{\bm{\circlearrowright}})=1$
(resp.
$\nu_{h_{0}}([z_{1}^{0},\widetilde{z}_{1}^{0}]_{\mathcal{L}}^{\bm{\circlearrowright}})=1$).
This means that

$\nu_{h}([\widetilde{z}_{0}^{0},z_{0}^{0}]_{\mathcal{L}}^{\bm{\circlearrowright}})\geq
2^{-k}\quad\text{and}\quad\nu_{h}([z_{1}^{0},\widetilde{z}_{1}^{0}]_{\mathcal{L}}^{\bm{\circlearrowright}})\geq
2^{-k}.$

It thus follows from the definition of $E$ that if we assume
$\zeta>1/\alpha_{\mathrm{PP}}$ then
$[z_{0}^{0},z_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}}$
does not intersect $B(\widetilde{z}_{0}^{0},2^{-\zeta k})$.

We also let $w_{0}^{0}$ (resp. $w_{1}^{0}$) be the point on
$\partial\widetilde{\mathcal{D}}_{0}$ so that
$\nu_{h_{0}}([w_{0}^{0},x_{0}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})=1$
(resp.
$\nu_{h_{0}}([x_{1}^{0},w_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})=1$.
We let $\mathcal{D}_{00}$ be given by $\widetilde{\mathcal{D}}_{0}$ after
adding $\tfrac{2}{\gamma}\log L_{00}^{-1}$ to the field where
$L_{00}=\nu_{h_{0}}([w_{0}^{0},z_{0}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{00}$, $x_{1}^{00}$ for the points corresponding to
$w_{0}^{0}$, $z_{0}^{0}$. We also let $\mathcal{D}_{01}$ be given by
$\widetilde{\mathcal{D}}_{0}$ after adding $\tfrac{2}{\gamma}\log L_{01}^{-1}$
where
$L_{01}=\nu_{h_{0}}([z_{1}^{0},w_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{01}$, $x_{1}^{01}$ for the points which correspond to
$z_{1}^{0}$, $w_{1}^{0}$. We refer to
$(\mathcal{D}_{0i},x_{0}^{0i},x_{1}^{0i})$ for $i=0,1$ as the _children_ of
$(\mathcal{D}_{0},x_{0}^{0},x_{1}^{0})$. We then perform the exploration
inside of $\mathcal{D}_{0i}$ from $x_{0}^{0i}$ to $x_{1}^{0i}$ for $i=0,1$
with the definition of the good chunk as in Definition 6.4. We note in this
case that

(6.6) $2\leq L_{0i}\leq 3+c_{F}2^{(1-4/\kappa)J}\quad\text{for}\quad i=0,1.$

We will now explain how to construct the path $\omega$ in the case that both
of the explorations succeed. As mentioned just above, we have that the
distance between
$[z_{0}^{0},z_{0}^{1}]_{\mathcal{L}}^{\bm{\circlearrowright}}$ and
$\widetilde{z}_{0}^{0}$ is at least $2^{-\zeta k}$. Let
$\mathcal{N}_{1},\ldots,\mathcal{N}_{\ell}$ be the good chunks from the
exploration in $\mathcal{D}_{00}$ from $x_{0}^{00}$ to $x_{1}^{00}$ which have
the maximal chunk size. Arguing as in the proof that the exploration succeeds
to begin with, we can construct a path inside of
$\mathrm{cl}(\cup_{j=1}^{\ell}\mathcal{N}_{j})$ which has distance at least
$2^{-\zeta k}$ (possibly increasing $\zeta>0$) from $I_{j}$ and connects
$I_{j-1}$ to $x_{1}^{00}=z_{0}^{0}$. We then take the concatenation of this
path with
$[z_{0}^{0},\widetilde{z}_{0}^{0}]_{\mathcal{L}}^{\bm{\circlearrowleft}}$.
Finally, we concatenate this path with
$[\widetilde{z}_{1}^{0},z_{1}^{0}]_{\mathcal{L}}^{\bm{\circlearrowleft}}$ and
the analogous path constructed from the exploration in $\mathcal{D}_{01}$ from
$x_{0}^{01}=z_{1}^{0}$ to $x_{1}^{01}=x_{1}^{0}$.

If one of the two explorations fail, then we iterate the procedure depending
on the type of failure in the same manner that we continue the iteration
procedure in the case that the initial exploration has failed with the
following exception. If the exploration in $\mathcal{D}_{00}$ from
$x_{0}^{00}$ to $x_{1}^{00}$ has a large downward jump whose terminal point
disconnects $x_{0}^{01}$ and $x_{1}^{01}$, then we do not perform the
exploration in $\mathcal{D}_{01}$ from $x_{0}^{01}$ to $x_{1}^{01}$ and only
perform an exploration from $x_{0}^{00}$ to $x_{1}^{01}$ in the component
which contains these two points (as in the first type of large downward jump
considered just below).

Figure 6.4.  Illustration of the possibility that the exploration fails by
making a large downward jump. Top: The terminal point of the downward jump is
within $2$ units of boundary length distance (measured using $h_{0}$) from
$I_{j-1}\cup I_{j}\cup I_{j+1}$. Shown is the case that the terminal point is
not in $I_{j-1}\cup I_{j}\cup I_{j+1}$ but it is also possible for it to be in
$I_{j-1}\cup I_{j}\cup I_{j+1}$. Bottom: The terminal point of the downward
jump is more than $2$ units of boundary length distance (measured using
$h_{0}$) from $I_{j-1}\cup I_{j}\cup I_{j+1}$.

Case 2: large downward jump. There are two possibilities in the case that the
exploration makes a large downward jump: either the terminal point of the
downward jump is within boundary length distance $2$ (measured using $h_{0}$)
of $I_{j-1}\cup I_{j}\cup I_{j+1}$ or it is not. See Figure 6.4 for an
illustration of the setup.

Suppose that the terminal point of the downward jump is within boundary length
distance $2$ (measured using $h_{0}$) of $I_{j-1}\cup I_{j}\cup I_{j+1}$. Let
$w_{0}^{0}$ (resp. $w_{1}^{0}$) be such that
$\nu_{h_{0}}([w_{0}^{0},x_{0}^{0}]_{\partial\mathcal{D}}^{\bm{\circlearrowright}})=3$
(resp.
$\nu_{h_{0}}([x_{1}^{0},w_{1}^{0}]_{\partial\mathcal{D}}^{\bm{\circlearrowleft}})=3$)
and let $\widetilde{\mathcal{D}}_{0}$ be the quantum disk in the complement of
the exploration whose boundary contains
$[w_{0}^{0},w_{1}^{0}]_{\partial\mathcal{D}}^{\bm{\circlearrowright}}$. We
then let $\mathcal{D}_{00}$ be given by $\widetilde{\mathcal{D}}_{0}$ after
adding $\tfrac{2}{\gamma}\log L_{00}^{-1}$ to the field where
$L_{00}=\nu_{h_{0}}([z_{0}^{0},z_{1}^{0}]_{\partial\widetilde{\mathcal{D}}_{0}}^{\bm{\circlearrowleft}})$
and we write $x_{0}^{00}$, $x_{1}^{00}$ for the points on
$\partial\mathcal{D}_{00}$ which correspond to $w_{0}^{0}$, $w_{1}^{0}$. We