# Crossing exponent in the Brownian loop soup

Antoine Jego EPFL<EMAIL_ADDRESS>Titus Lupu Sorbonne Université and
Université Paris Cité, CNRS, Laboratoire de Probabilités, Statistique et
Modélisation, F-75005 Paris, France<EMAIL_ADDRESS>Wei
Qian City University of Hong Kong<EMAIL_ADDRESS>
###### Abstract

We study the clusters of loops in a Brownian loop soup in some bounded two-
dimensional domain with subcritical intensity $\theta\in(0,1/2]$. We obtain an
exact expression for the asymptotic probability of the existence of a cluster
crossing a given annulus of radii $r$ and $r^{s}$ as $r\to 0$ ($s>1$ fixed).
Relying on this result, we then show that the probability for a macroscopic
cluster to hit a given disc of radius $r$ decays like $|\log
r|^{-1+\theta+o(1)}$ as $r\to 0$. Finally, we characterise the polar sets of
clusters, i.e. sets that are not hit by the closure of any cluster, in terms
of $\log^{\alpha}$-capacity.

This paper reveals a connection between the 1D and 2D Brownian loop soups.
This connection in turn implies the existence of a second critical intensity
$\theta=1$ that describes a phase transition in the percolative behaviour of
large loops on a logarithmic scale targeting an interior point of the domain.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Outline of the proof and further results
    3. 1.3 Two critical values: $\theta=1/2$ and $\theta=1$
  2. 2 Preliminaries
    1. 2.1 FKG inequality in the loop soup
    2. 2.2 Crossing an annulus with loops
    3. 2.3 Boundary Poisson kernel in an annulus
    4. 2.4 Surrounding a disk with loops
    5. 2.5 A priori upper bound on the crossing probability of an annulus by a cluster
    6. 2.6 Generalisation of Theorem 1.2
  3. 3 One-arm event in the Brownian loop soup
    1. 3.1 Tightness
    2. 3.2 Integral equation
    3. 3.3 Fixed-points of the integral equation and Proof of Theorem 3.1
    4. 3.4 $f_{\infty}$ in terms of Bessel processes
    5. 3.5 Probability of large crossings – Proof of Theorem 1.2
    6. 3.6 Percolation of large loops
  4. 4 Polar sets and capacity
    1. 4.1 Crossing two annuli with loops
    2. 4.2 Proof of Proposition 4.1
    3. 4.3 Proof of Theorem 1.5
  5. A Lower bound on $f_{\infty}$
  6. B Hypergeometric function

##  1 Introduction

For a planar domain $D\subset\mathbb{C}$, the Brownian loop soup
$\mathcal{L}_{D}^{\theta}$ in $D$ with intensity $\theta>0$ is a random
collection of Brownian loops lying in $D$ that is distributed according to a
Poisson point process with intensity $\theta\mu^{\rm loop}_{D}$. The measure
$\mu^{\rm loop}_{D}$ on Brownian loops is defined by

$\mu^{\rm
loop}_{D}(\mathrm{d}\wp)=\int_{D}\mathrm{d}z\int_{0}^{\infty}\frac{\mathrm{d}t}{t}\,p_{D}(t,z,z)\,\mathbb{P}_{D}^{t,z,z}(\mathrm{d}\wp)$
(1.1)

where $p_{D}$ is the heat kernel in $D$ and $\mathbb{P}_{D}^{t,z,z}$ denotes
the law of a Brownian bridge in $D$ from $z$ to $z$ of duration $t$. Because
the total mass of $\mu^{\rm loop}_{D}$ is infinite, $\mathcal{L}_{D}^{\theta}$
contains countably infinitely many loops a.s.

First introduced by [28], the Brownian loop soup became a central object of
study in 2D random conformal geometry, in particular due to its connections
with the Gaussian free field (GFF) and with Conformal loop ensemble
(CLE)/Schramm-Loewner evolution (SLE). Note that a different choice of the
intensity parameter $c=2\theta$ was used in [28, 46] and many other works,
where $c$ corresponds to the “central charge” in conformal field theory. As
established in the seminal work [46], the percolative property of the loop
soup in the unit disk $\mathbb{D}$ (this can be easily extended to any domain
$D$ such that $\mathbb{C}\setminus D$ is non polar for the Brownian motion)
undergoes a phase transition:

  * •

Supercritical ($c>1,\theta>1/2$): there is a.s. a unique cluster of loops;

  * •

Critical and subcritical ($c\leq 1,\theta\leq 1/2$): there are a.s. infinitely
many clusters. In this case, the outermost boundaries of the outermost
clusters are distributed as a CLEκ where $\kappa\in(8/3,4]$ and $\theta$ are
related by

$\displaystyle 2\theta=c=(6-\kappa)(3\kappa-8)/(2\kappa).$ (1.2)

As a consequence of the relation to CLE, one can compute many exact quantities
about the critical and subcritical loop soup. For example, for
$\theta\in(0,1/2]$, the probability $p(\theta,n,r,R)$ of having $n\geq 2$
different clusters in $\mathcal{L}_{\mathbb{D}}^{\theta}$ that cross the
annulus $R\mathbb{D}\setminus r\mathbb{D}$ for $R>r>0$ is the same as the
probability that the (outer or inner) boundaries of these $n$ clusters cross
this annulus. Even though [46] only looked at the outer boundaries of the
outermost clusters, both the inner and outer boundaries of any cluster can be
transformed into outer boundaries of outermost clusters after some
conditioning and conformal mapping (see e.g. [16, Section 7]). We believe that
with some additional work, it should be possible to show that

$p(\theta,n,r,R)=r^{\alpha_{2n}+o(1)}\text{ as }r\to 0,\quad\text{ for
}\alpha_{2n}=(16n^{2}-(\kappa-4)^{2})/(8\kappa)$

where $\theta$ and $\kappa$ are related through (1.2), and $\alpha_{2n}$ is
the $2n$-arm exponent of SLE (following from [49, 24, 25, 27], see also [55]).

The main purpose of the current paper is to obtain the crossing probability
for one cluster, i.e. $n=1$, for all intensities $\theta\in(0,1/2]$. This case
is fundamentally different from the $n\geq 2$ case, because as a cluster
crosses the annulus $R\mathbb{D}\setminus r\mathbb{D}$, its inner or outer
boundaries typically do not cross this annulus. In particular, we do not
expect to be able to express the crossing probability for $n=1$ naturally in
terms of SLE or CLE. As we will reveal below, the crossing probability for
$n=1$ will decay as

$|\log r|^{-1+\theta+o(1)},$

much slower than the polynomial rate for $n\geq 2$. Heuristically, as a
macroscopic cluster approaches a given point, it will contain infinitely many
small loops that surround this point. In contrast, in the $n\geq 2$ case, we
require each cluster not to contain small loops that surround the origin, in
order to leave some space for the other cluster(s) to make the crossing(s),
which is a much rarer event.

The value $c=1$ ($\theta=1/2$) is known to be critical for various reasons,
and is the central charge associated with the Gaussian free field. As shown in
[46], the outer boundaries of the critical loop soup $(\theta=1/2)$ is
distributed as an CLE4. It is also known that SLE4 or CLE4 can be interpreted
as the level lines of the Gaussian free field (see e.g. [43, 44, 15, 37, 6]).
A more direct connection with the Gaussian free field was established by Le
Jan [29, 30] via the occupation time field. Thanks to these connections to the
GFF, the structure of the clusters at $\theta=1/2$ is very well understood
[32, 35, 42, 4, 5]. However, the understanding of the clusters when
$\theta<1/2$ is still limited to their outer boundaries [46, 50] and to the
loops touching them [40, 41, 16].

In this paper, we study in great details the clusters of the loop soup when
$\theta\in(0,1/2]$, considerably refining the picture when $\theta\in(0,1/2)$.
In a nutshell, we compute the crossing exponent of these clusters (Theorem
1.2) and give a characterisation of the polar sets of the clusters (Theorem
1.5). We also reveal a connection between the 1D and 2D Brownian loop soups
(Section 1.2) and show that it implies the existence of a second critical
point $\theta=1$ (Section 1.3).

###  1.1 Main results

######  Notation 1.1.

For any family $\mathcal{L}$ of loops and any sets $A,B\subset\mathbb{C}$, we
will denote by $\\{A\overset{\mathcal{L}}{\leftrightarrow}B\\}$ the event that
there is a cluster of loops in $\mathcal{L}$ that intersects both $A$ and $B$.

The first main result of this paper is:

######  Theorem 1.2.

Let $\theta\in(0,1/2]$ and let $R_{0}\in(0,1)$ be a macroscopic radius. Then

$\mathbb{P}\Big{(}R_{0}\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}r\partial\mathbb{D}\Big{)}=|\log
r|^{-1+\theta+o(1)}\quad\quad\text{as~{}}\quad r\to 0.$

In the companion paper [21], we build and study the properties of a field
naturally associated to a subcritical loop soup. This field can be thought of
as the generalisation of the Gaussian free field to subcritical intensities
$\theta<1/2$. Theorem 1.2 above is a crucial input in [21] to show that the
covariance of this field at points $x$ and $y$ blows up like
$|\log|x-y||^{2(1-\theta)+o(1)}$ as $x-y\to 0$.

Theorem 1.2 can be seen as an interpolation of the two known cases
$\theta=1/2$ and $\theta\to 0^{+}$. Indeed, when the intensity $\theta=1/2$,
the connection with the Gaussian free field allows one to show that the
crossing probability is asymptotic to a constant times $|\log r|^{-1/2}$ (see
Lemma 2.6). When $\theta\to 0^{+}$, the crossing probability can be informally
compared with the probability that a Brownian path starting somewhere in the
unit disc (not at the origin) hits $r\mathbb{D}$ before reaching the unit
circle. This probability decays like a constant times $|\log r|^{-1}$. We
believe that the probability in Theorem 1.2 should also be asymptotic to a
constant times $|\log r|^{-1+\theta}$ (see also [21, Section 9.5] for strong
heuristics in this direction), but we do not achieve this precision in this
work. However, remarkably, we are able to obtain the following exact
expression of the “scaling limit” of a certain crossing probability (see
Theorem 3.1 for a stronger statement). As the logarithmic decay suggests, the
relevant events to consider here are scaled versions of one another only on a
logarithmic scale.

######  Theorem 1.3.

Let $\theta\in(0,1/2]$. For all $s>1$, the following crossing probability
converges

$\displaystyle\mathbb{P}\Big{(}\delta\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{s}\partial\mathbb{D}\Big{)}\xrightarrow[\delta\to
0]{}f_{\infty}(s)$ (1.3)

where

$\displaystyle
f_{\infty}(s)=\frac{\sin(\pi\theta)}{\pi}\int_{s-1}^{\infty}t^{\theta-1}(t+1)^{-1}\mathrm{d}t.$
(1.4)

See (1.11) for the link with the generalised arcsine law with parameter
$1-\theta$ and a discussion concerning regenerative sets.

Theorem 1.3 is another demonstration of some form of “exact solvability” of
the loop soup (in addition to the “solvability” coming from its connection
with CLE). To prove Theorem 1.3, we first work out a functional equation that
is satisfied by $f_{\infty}$, and then identify $f_{\infty}$ as the unique
fixed point of the functional equation. The outline of the proof of Theorem
1.3 will be further explained in Section 1.2, where we also point out the
intimate connection between $f_{\infty}$ and the 1D loop soup.

One can observe from (1.4) that

$f_{\infty}(s)\sim\frac{\sin(\pi\theta)}{\pi(1-\theta)}s^{\theta-1},\quad\quad\text{as}\quad
s\to\infty.$ (1.5)

We will first focus on showing Theorem 1.3, and then show (in Section 3.5)
that the exponent $1-\theta$ indeed describes the decay of the probability of
a macroscopic crossing, thus proving Theorem 1.2.

The crossing probability is an important quantity of interest in many other
models in statistical mechanics. In the well-known model of Bernoulli site
percolation on the triangular lattice, at criticality, the probability of one
cluster crossing an annulus $\mathbb{D}\setminus\varepsilon\mathbb{D}$ decays
like $\varepsilon^{5/48+o(1)}$ [26]. One can also look at the probability of
having $k\geq 2$ clusters, that is, $2k$ disjoint “arms” with alternating
colors (or more generally any given number of arms with prescribed colors).
Many of these exponents have been computed [49], thanks to the relation to SLE
in the scaling limit [48]. In the subcritical regime, the crossing probability
for a percolation cluster decays exponentially fast (instead of polynomially),
marking a sharp phase transition [36, 3].

In contrast, in our setting of the Brownian loop soup, the crossing
probability decays like a power of $|\log\varepsilon|$, for all intensities
$\theta\leq 1/2$. This is closer to the setting of a system depending on two
parameters (e.g. temperature and pressure) that is critical on a one-
dimensional curve. The random cluster model and long range percolation on
$\mathbb{Z}^{d}$ are two such examples. The latter has the interesting feature
that there are two points of non-analyticity on the critical line for the one-
arm exponent (see the introduction of [17] for background on critical long
range percolation). This shares some similarities with the Brownian loop soup:
we show in Section 1.3 that $\theta=1/2$ and $\theta=1$ can be seen as two
critical intensities for the percolation behaviour of the Brownian loop soup.

We now discuss a consequence of Theorem 1.2 on a characterisation of the sets
that can be hit by a cluster of the loop soup. Recall that a set $A$ is said
to be polar for planar Brownian motion $(B_{t})_{t\geq 0}$ if for all starting
point $x$, $\mathbb{P}_{x}\left(\exists t>0:B_{t}\in A\right)=0.$ Since there
are countably many loops in a Brownian loop soup, if a set $A$ is polar for 2D
Brownian motion, then a.s. it will not be visited by any cluster. However, we
are going to see that the _closures_ of the clusters can hit much thinner
sets. Identifying the polar sets of a random set provides a way of
understanding how large and spread out that set is.

######  Definition 1.4.

Let $D\subset\mathbb{C}$ be a bounded open domain and let $A\subset D$ be a
Borel set. We will say that $A$ is polar for the clusters of
$\mathcal{L}_{D}^{\theta}$ if

$\mathbb{P}\left(\exists~{}\mathcal{C}\text{~{}cluster~{}of~{}}\mathcal{L}_{D}^{\theta}:\overline{\mathcal{C}}\cap
A\neq\varnothing\right)=0.$

For any measurable function $K:\mathbb{C}\times\mathbb{C}\to[0,\infty]$, let

$\mathrm{Cap}_{K}(A):=\left(\inf\left\\{\int
K(x,y)\mu(\mathrm{d}x)\mu(\mathrm{d}y):\mu\text{~{}probability~{}measure~{}on~{}}A\right\\}\right)^{-1}$
(1.6)

be the capacity of $A$ with respect to the kernel $K$. These capacities can be
thought of as a tool measuring precisely the size of set. For instance, by
Frostman’s lemma, the Hausdorff dimension of $A$ is the supremum of
$\alpha\geq 0$ such that $\mathrm{Cap}_{K}(A)>0$ where
$K(x,y)=|x-y|^{-\alpha}$. Here we will be interested in measuring the size of
very thin sets and we will choose kernels $K$ of the form
$K(x,y)=|\log|x-y||^{\alpha}$ for $\alpha>0$. We will denote
$\mathrm{Cap}_{\log^{\alpha}}$ the capacity associated to that kernel.

######  Theorem 1.5.

Let $\theta\in(0,1/2]$, $D\subset\mathbb{C}$ be a bounded open domain and let
$A\subset D$ be a closed set.

  * •

If $A$ is polar for the clusters of $\mathcal{L}_{D}^{\theta}$, then
$\mathrm{Cap}_{\log^{\alpha}}(A)=0$ for all $\alpha>1-\theta$.

  * •

Conversely, if $A$ is not polar for the clusters of
$\mathcal{L}_{D}^{\theta}$, then $\mathrm{Cap}_{\log^{\alpha}}(A)>0$ for all
$\alpha<1-\theta$.

Recall that Kakutani’s theorem states that a closed set $A$ is polar for
planar Brownian motion if and only if $\mathrm{Cap}_{\log^{\alpha}}(A)=0$ for
$\alpha=1$ (see e.g. [38, Theorem 8.20]). Informally, this well-known result
is the limiting case $\theta\to 0$ of Theorem 1.5 above.

With this notion of polar sets, Theorem 1.2 immediately implies a more general
statement:

######  Corollary 1.6.

If $D$ is a bounded open domain, $x\in D$ and $A$ is a closed set which does
not contain $x$ and which is not polar for the clusters of
$\mathcal{L}_{D}^{\theta}$, then

$\mathbb{P}\Big{(}\exists~{}\mathcal{C}\text{~{}cluster~{}of~{}}\mathcal{L}_{D}^{\theta}:\overline{\mathcal{C}}\cap
A\neq\varnothing\text{~{}and~{}}\mathcal{C}\cap
D(x,r)\neq\varnothing\Big{)}=|\log
r|^{-1+\theta+o(1)}\quad\quad\text{as~{}}\quad r\to 0.$

#### Related works

Let us mention that the clusters in loop soups have been also studied in
dimensions $d\geq 3$, both on discrete lattices (random walk loop soups) and
on metric graphs. Particularly investigated was the case $\theta=1/2$ due to
the relation to the GFF [12, 13, 10, 8]. See also the conjectures of Wendelin
Werner about the scaling limits in continuum for $\theta=1/2$ in dimensions
$d\geq 3$ [53]. For studies of clusters of random walk loop soups for more
general $\theta$, we refer to [31, 11, 9, 51]. In dimensions $d\geq 3$ the
one-arm probabilities of clusters decay as a power rather than as a power of
$\log$ as in $d=2$. More precisely, it can be shown that for every
$\theta\in(0,1/2]$, the probability of a one-arm event is bounded between two
powers, both on discrete lattice and on metric graph. At the metric graph
level, the precise value of the exponent is known for $\theta=1/2$ in $d=3$
[12, 13] (this is a result on the metric graph GFF), and in high dimensions
$d>6$ [53, 8]. At the discrete lattice level, the value of the exponent is
known in $d\geq 5$ for $\theta$ small enough [11].

###  1.2 Outline of the proof and further results

We now explain our approach to Theorem 1.2. Let us recall that we cannot rely
on CLE tools since the event we consider is not a CLE-type event, meaning that
this is not an event concerning the outermost boundaries of the clusters.
Obtaining a lower bound on the crossing probability should be in principle the
easy direction. Indeed, this amounts to finding a good strategy to make the
crossing and to be able to control the probability of this strategy. One such
strategy could for instance consist in crossing successive annuli
$r_{i+1}\partial\mathbb{D}\setminus r_{i}\mathbb{D}$ with one loop at a time
($r_{0}=r<r_{1}<\dots<r_{n}=e^{-1}$) and welding these successive crossings to
ensure that a cluster has crossed the annuli $e^{-1}\mathbb{D}\setminus
r\mathbb{D}$. Summing over all possible intermediate scales
$r_{1},\dots,r_{n}$, this should lead to a close-to-optimal strategy. Since
the probability that there is a loop crossing a given annulus can be
accurately estimated (see Lemma 2.2), one can lower bound the probability of
this strategy (together with FKG inequality). Although this computation can be
explicitly written down, the analysis of the lengthy expressions one obtains
when doing so seems far from simple.

More importantly, the above line of argument would not provide any upper bound
on the crossing probability. This direction is indeed much more challenging
since we need to control the probability of _all_ possible scenarios where the
crossing event occurs. We therefore need to proceed differently. In our
approach, we will obtain the upper and lower bounds (almost) at the same time.
Our main intermediate step towards proving Theorem 1.2 is to prove a “scaling
limit” result for the crossing probabilities, namely to prove (1.3).

Now, let’s assume that the convergence (1.3) has been established, without
knowing yet that $f_{\infty}$ takes the simple form (1.4), and let’s see how
we pursue. A standard argument shows that $f_{\infty}$ is supermultiplicative:
for all $s,t\geq 1$, $f_{\infty}(s)f_{\infty}(t)\leq f_{\infty}(st)$. See
Lemma 3.7 and Corollary 3.8. In particular, Fekete’s subadditive lemma implies
the existence of a one-arm exponent $\xi_{1}\geq 0$ such that

$f_{\infty}(s)=s^{-\xi_{1}+o(1)}\quad\quad\text{as}\quad s\to\infty.$ (1.7)

We show in Section 3.5 that this exponent indeed describes the decay of the
probability of a macroscopic crossing, i.e. that

$\mathbb{P}\Big{(}r\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}e^{-1}\partial\mathbb{D}\Big{)}=|\log
r|^{-\xi_{1}+o(1)}\quad\quad\text{as~{}}\quad r\to 0.$

It then remains to compute the exponent $\xi_{1}$. In planar percolation, this
step is usually done using SLE computations. Here, we exploit some of the
exact solvability inherent to the Brownian loop soup by showing that
$f_{\infty}$ satisfies some integral equation:

######  Theorem 1.7.

The limiting function $f_{\infty}$ satisfies

$f_{\infty}(s)=1-\Big{(}1-\frac{1}{s}\Big{)}^{\theta}+\theta(s-1)^{\theta}\int_{1}^{\infty}(s+t-1)^{-\theta-1}f_{\infty}(t)\mathrm{d}t,\quad\quad
s\geq 1.$ (1.8)

Moreover, for all $\alpha\in(0,1-\theta)$, $f_{\infty}$ is the only function
belonging to $\mathcal{F}_{\alpha}$ (3.7) satisfying (1.8).

This integral equation is obtained by exploring the cluster from the outside
and checking how deep inside a first layer of loops can go. The first term
$1-(1-1/s)^{\theta}$ corresponds to the case where the crossing of
$\delta\mathbb{D}\setminus\delta^{s}\mathbb{D}$ has been made entirely by a
single loop (which is unlikely when $s$ is large). The second term with the
integral corresponds to the mixed case where the first layer of loops crosses
exactly $\delta\mathbb{D}\setminus\delta^{1+(s-1)/t}\mathbb{D}$. The term
$f_{\infty}(t)$ in the integral accounts for the remaining crossing of
$\delta^{1+(s-1)/t}\mathbb{D}\setminus\delta^{s}\mathbb{D}$ by a cluster of
loops staying in $\delta\mathbb{D}$. This reasoning is made precise in Section
3.2.

Theorem 1.7 provides a uniqueness result for the subsequential limits of the
crossing probabilities appearing in Theorem 1.3. These two theorems will
therefore be proved at the same time.

#### Identification of $f_{\infty}$ and connection to Bessel processes

The Brownian loop soup can be also defined on the half line $(0,\infty)$ (or
on more general intervals). In this one-dimensional setting, the total local
time of the loop soup is well defined pointwise and finite a.s. For any
intensity $\theta$, the local time profile has the law of a squared Bessel
process $(R_{t}^{(\theta)})_{t\geq 0}$ of dimension $2\theta$, reflected at
the origin, with $R_{0}^{(\theta)}=0$ a.s. (the point 0 is the boundary of
$(0,\infty)$). See [34] for more details and precise statements. In dimension
1, the probability that a cluster of loops connects two points is simply equal
to the probability that the local time stays positive on the interval joining
these two points. The one-dimensional analogue of the function $f_{\infty}$ is
therefore given by

$\lim_{\delta\to 0}\mathbb{P}\left(\forall
t\in[|\log\delta|,s|\log\delta|],R_{t}^{(\theta)}>0\right)=\mathbb{P}\left(\forall
t\in[1,s],R_{t}^{(\theta)}>0\right),\quad\quad s\geq 1,$

by Brownian scaling. With Theorem 1.7 in hand, we can actually show that the
function $f_{\infty}$ agrees with the above function:

######  Theorem 1.8.

For all $s\geq 1$,

$\displaystyle f_{\infty}(s)$ $\displaystyle=\mathbb{P}\big{(}\forall
t\in[1,s],R_{t}^{(\theta)}>0\big{)}.$

In particular,

$f_{\infty}(s)=\frac{\sin(\pi\theta)}{\pi}\int_{s-1}^{\infty}t^{\theta-1}(t+1)^{-1}\mathrm{d}t,\quad\quad
s\geq 1.$ (1.9)

Theorem 1.8 in particular shows that $\xi_{1}=1-\theta$ concluding the proof
of Theorem 1.2. Notice that $1-\theta$ is also the exponent describing the
decay of $\mathbb{P}\big{(}\forall t\in[1,s],R_{t}^{(\theta)}>0\big{)}$ as
$s\to\infty$.

In several instances, there is a strong relationship between a 2D random
conformally invariant object and its one-dimensional analogue. Particularly
relevant to us, the circle average of the 2D Gaussian free field with fixed
centre, seen as a function of the radius of the circle, has the law of a 1D
Brownian motion (which is the GFF in dimension 1). See [45, Section 2.8] for a
precise statement. Another important example concerns the local time of
concentric circles of planar Brownian motion. As the radius of the circle
becomes small, these local times become asymptotically like the local time
time of 1D Brownian motion (which turns out to be a Bessel process). See [19,
20].

We believe that such a relationship between the 1D and 2D local times of the
Brownian loop soup should also exist. In a planar Brownian loop soup, the
local time of a circle is infinite, so this relation is more subtle than in
the case of a single Brownian trajectory. The usual way to circumvent this
issue is to define the local time via an approximation procedure where one
subtracts a deterministic diverging quantity (Wick normalisation). However,
this resulting local time is signed, i.e. takes negative values. An
alternative possibility is to restrict ourselves to loops that are not too
small (specifically, when looking at the local time of a circle of radius
$\varepsilon$, we only keep loops with duration at least
$\varepsilon^{O(1)}$). As we show in this article (see Section 1.3), removing
these small loops does not change the percolative properties of the loop soup
when $\theta\in(0,1/2]$. We believe that with a cutoff of this type, the local
time of concentric circles becomes asymptotically like the local time of a
one-dimensional Brownian loop soup. Theorem 1.8 is a strong indication that
such a relationship should hold. We now give some further heuristics in this
direction.

#### Relation between 1D and 2D loop soups

The punctured unit disc $\mathbb{D}\setminus\\{0\\}$ and the half infinite
cylinder $\mathtt{C}=(0,+\infty)\times\mathbb{S}^{1}$ are conformally
equivalent via the map $re^{i\theta}\mapsto(|\log r|,e^{i\theta})$. Because
the Brownian loop soup is conformally invariant (up to time reparametrization,
see [22, Proposition 5.27]) and because $\\{0\\}$ is polar for Brownian
motion, we may work on $\mathtt{C}$ instead of $\mathbb{D}$. The loop measure
in $\mathtt{C}$ is then given by

$\mu^{\rm
loop}_{\mathtt{C}}(\mathrm{d}\wp)=\int_{0}^{\infty}\mathrm{d}t\int_{\mathtt{C}}\mathrm{d}z~{}\dfrac{p_{\mathtt{C}}(t,z,z)}{t}\mathbb{P}_{\mathtt{C}}^{t,z,z}(\mathrm{d}\wp)$

where $p_{\mathtt{C}}$ and $\mathbb{P}_{\mathtt{C}}^{t,z,z}$ denote the heat
kernel in $\mathtt{C}$ and the Brownian bridge probability measure of duration
$t$ from $z$ to $z$ in $\mathtt{C}$. We will denote by
$\boldsymbol{\pi}:(x,e^{i\theta})\in\mathtt{C}\mapsto x\in(0,+\infty)$ the
longitudinal projection that induces a projection on paths in $\mathtt{C}$
onto paths in $(0,+\infty)$. In the cylinder, the two components of a Brownian
bridge of duration $t$ from $(x,e^{i\theta})$ to $(x,e^{i\theta})$ are simply
two independent Brownian bridges of duration $t$: one from $x$ to $x$ in
$\mathbb{R}^{+}$ and the other one from $e^{i\theta}$ to $e^{i\theta}$ in
$\mathbb{S}^{1}$. Similarly, the heat kernel $p_{\mathtt{C}}$ factorises: for
$z=(x,e^{i\theta})$,
$p_{\mathtt{C}}(t,z,z)=p_{\mathbb{S}^{1}}(t,e^{i\theta},e^{i\theta})p_{\mathbb{R}^{+}}(t,x,x)$.
The push forward $\boldsymbol{\pi}_{*}\mu^{\rm loop}_{\mathtt{C}}$ of
$\mu^{\rm loop}_{\mathtt{C}}$ by $\boldsymbol{\pi}$ is therefore equal to

$\boldsymbol{\pi}_{*}\mu^{\rm loop}_{\mathtt{C}}(\mathrm{d}\wp^{\rm
1D})=\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}x\left\\{2\pi
p_{\mathbb{S}^{1}}(t,1,1)\right\\}\frac{p_{\mathbb{R}^{+}}(t,x,x)}{t}\mathbb{P}_{\mathbb{R}^{+}}^{t,x,x}(\mathrm{d}\wp^{\rm
1D}),$ (1.10)

with the factor $2\pi$ coming from the integration over $\mathbb{S}^{1}$. The
upshot is that, since $2\pi p_{\mathbb{S}^{1}}(t,1,1)\to 1$ as $t\to\infty$,
if we restrict ourselves to long loops, then we approximately recover the
Brownian loop measure in $(0,+\infty)$:

$\mu^{\rm loop}_{\mathbb{R}^{+}}(\mathrm{d}\wp^{\rm
1D})=\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}x~{}\frac{p_{\mathbb{R}^{+}}(t,x,x)}{t}\mathbb{P}_{\mathbb{R}^{+}}^{t,x,x}(\mathrm{d}\wp^{\rm
1D}).$

In particular, heuristically, the long crossings on $\mathtt{C}$, as in
Theorems 1.2 and 1.3, are achieved by combining a small number of large loops
rather than a large number of small loops. So essentially, we are in a regime
where the 2D Brownian loop soup on $\mathtt{C}$ is well approximated by the 1D
Brownian loop soup on $(0,+\infty)$. Moreover, if two large Brownian loops
$\wp_{1}$ and $\wp_{2}$ on $\mathtt{C}$ are such that their 1D projections
under $\boldsymbol{\pi}$ intersect, then with high probability $\wp_{1}$ and
$\wp_{2}$ themselves do intersect, the winding around the cylinder creating
many opportunities for that.

#### Random coverings of $\mathbb{R}^{+}$ and regenerative sets

By forgetting the non trivial local time that the 1D Brownian loops carry, we
can view the 1D Brownian loop soup as a random collection of intervals. For
any one-dimensional loop $\wp$, let
$\boldsymbol{\tilde{\pi}}(\wp)=(\min\wp,\max\wp)$. By rooting the loops at
their minimum (in a similar manner as in (2.6); see [34, Proposition 3.18]),
one can show that the push forward $\boldsymbol{\tilde{\pi}}_{*}\mu^{\rm
loop}_{\mathbb{R}^{+}}$ of $\mu^{\rm loop}_{\mathbb{R}^{+}}$ by
$\boldsymbol{\tilde{\pi}}$ is equal to

$\boldsymbol{\tilde{\pi}}_{*}\mu^{\rm
loop}_{\mathbb{R}^{+}}(\mathrm{d}a,\mathrm{d}b)=(b-a)^{-2}\mathrm{d}a~{}\mathrm{d}b\mathbf{1}_{\left\\{0<a<b\right\\}}.$

In other words, one can sample a Poisson point process $\\{(a_{i},b_{i}),i\geq
1\\}$ on $(\mathbb{R}^{+})^{2}$ with intensity measure
$\theta(b-a)^{-2}\mathrm{d}a~{}\mathrm{d}b\mathbf{1}_{\left\\{0<a<b\right\\}}$
and look at the random set $\bigcup_{i\geq 1}[a_{i},b_{i}]$ which has the same
law as the points visited by a loop in
$\mathcal{L}_{\mathbb{R}^{+}}^{\theta}$.

This type of random coverings of $\mathbb{R}^{+}$ has been greatly studied;
see e.g. [7, Chapter 7] for an introduction to this topic. One could consider
intensity measures with more general forms $\mathrm{d}a~{}\mathrm{d}\nu(b-a)$
where $\nu$ is a measure on $\mathbb{R}^{+}$. The set of uncovered points of
$\mathbb{R}^{+}$ is then a regenerative set and one can characterise whether
this set is trivial or not in terms of $\nu$. In the concrete example
$\nu(\mathrm{d}x)=\theta x^{-2}\mathrm{d}x$ that is of interest to us, it is
known that the uncovered set is $\\{0\\}$ if and only if $\theta\geq 1$ (which
is consistent with the Bessel process approach). Let us however stress once
more that the connectivity properties on the cylinder $\mathtt{C}$ cannot be
immediately deduced from those in $\mathbb{R}^{+}$ (they are for instance
drastically different if $\theta\in(1/2,1)$).

Theorem 1.3 can also be interpreted through the lens of regenerative sets. To
this end, let us denote by

$g_{\delta,s}:=\inf\\{g\in(0,s):\delta^{g}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{s}\\},\quad\delta\in(0,1),\quad
s>1.$

Performing a change of variable in the integral in (1.9), Theorem 1.3 can be
rephrased as saying that $g_{\delta,s}/s$ converges in distribution as
$\delta\to 0$ to the generalised arcsine law with parameter $1-\theta$:

$\frac{\sin(\pi\theta)}{\pi}t^{-\theta}(1-t)^{\theta-1}\mathbf{1}_{\left\\{0<t<1\right\\}}\mathrm{d}t.$
(1.11)

This is reminiscent of the convergence of the last passage times in
regenerative sets towards generalised arcsine laws. See [7] for an
introduction to this theory, especially Theorem 3.2 therein.

###  1.3 Two critical values: $\theta=1/2$ and $\theta=1$

We finish this introduction by explaining why the exponent $1-\theta$ vanishes
at $\theta=1$ instead of vanishing at the critical point $\theta=1/2$. As we
will see, this due to the existence of two different critical values of
$\theta$ for the Brownian loop soup on the cylinder $\mathtt{C}$. By conformal
invariance of the Brownian loop soup, claims about infinite clusters in
$\mathtt{C}$ can be translated to claims concerning clusters targeting a given
point of a domain.

The fact that $\theta=1$ is a critical value boils down to the following
dichotomy (see Lemma 3.5 and Figure 3.1):

  * •

$\theta\in(0,1)$: the functions $s\mapsto 1$ and $s\mapsto\mathbb{P}(\forall
t\in[1,s],R_{t}^{(\theta)}>0)$ are two distinct solutions to the integral
equation (1.8);

  * •

$\theta>1$: the function $s\mapsto 1$ is the unique measurable function in
$[0,1]^{[1,\infty)}$ solution to (1.8).

For $\theta>0$ and $\tau>0$, let $\mathcal{L}_{\mathtt{C},\tau}^{\theta}$ be
the subset $\mathcal{L}_{\mathtt{C}}^{\theta}$ consisting of all loops of
duration at least $\tau$. Recall that $\tau\in(0,\infty)\mapsto 2\pi
p_{\mathbb{S}^{1}}(\tau,1,1)$ is decreasing and converges to 1 as
$\tau\to\infty$. For $\theta\in(1/2,1)$, let

$\tau_{\theta}:=\inf\\{\tau>0:2\pi p_{\mathbb{S}^{1}}(\tau,1,1)\leq
1/\theta\\}.$ (1.12)

######  Theorem 1.9.

Let $\theta\in(0,1)$ and let $\tau\geq 0$. Assume that $\tau>\tau_{\theta}$ if
$\theta\in(1/2,1)$. For all $s>1$,

$\mathbb{P}\left(\\{T\\}\times\mathbb{S}^{1}\overset{\mathcal{L}_{\mathtt{C},\tau}^{\theta}}{\longleftrightarrow}\\{sT\\}\times\mathbb{S}^{1}\right)\to\frac{\sin(\pi\theta)}{\pi}\int_{s-1}^{\infty}t^{\theta-1}(t+1)^{-1}\mathrm{d}t\quad\quad\text{as}\quad
T\to\infty.$ (1.13)

In particular, there is almost surely no infinite cluster in
$\mathcal{L}_{\mathtt{C},\tau}^{\theta}$.

Let $\theta\geq 1$. For any $\tau\geq 0$ and $s>1$,

$\mathbb{P}\left(\\{T\\}\times\mathbb{S}^{1}\overset{\mathcal{L}_{\mathtt{C},\tau}^{\theta}}{\longleftrightarrow}\\{sT\\}\times\mathbb{S}^{1}\right)\to
1\quad\quad\text{as}\quad T\to\infty.$ (1.14)

To summarise,

  1. 1.

$\theta\in(0,1/2]$. It is known that all the clusters are bounded. In Theorem
1.2, we show that the crossing exponent is given by $1-\theta$. Furthermore,
if one removes the small loops (as described above), the value of the exponent
is unchanged (Theorem 1.9). This is reminiscent of [35].

  2. 2.

$\theta\in(1/2,1)$. It is known that the loops form a unique cluster, which is
thus unbounded. However, we show in Theorem 1.9 that if one removes the small
loops (at a sufficiently high threshold), each cluster formed by the remaining
loops is bounded and the crossing exponent is again given by $1-\theta$.

  3. 3.

$\theta>1$. Again, it is known that the loops form a unique cluster. We
further conjecture that if one removes the small loops, no matter the
threshold, the remaining loops will still form one unbounded cluster,
alongside infinitely many bounded clusters. Theorem 1.9 is a partial result in
this direction.

We do not state a conjecture for $\theta=1$ (although Theorem 1.9 also applies
to this case). The one-dimensional loop soup does percolate at this intensity
since a two-dimensional Bessel process does not vanish. However, such a Bessel
process goes very close to zero so it could be the case that the large loops
in $\mathcal{L}_{\mathtt{C}}^{\theta=1}$ does not possess any infinite cluster
but still percolates longitudinally. The situation is unclear to us.

So on the half-infinite cylinder, $\theta=1/2$ is the threshold for
percolation of small loops, and $\theta=1$ is the threshold for percolation of
large loops. Nevertheless, the appearance of the second critical point
$\theta=1$ depends strongly on the domain, since the size of the loops is not
conformally invariant. For instance on the half-plane, the two thresholds are
the same and equal $\theta=1/2$, as can be deduced from a percolation by
blocks construction similar to [33].

Finally, let us mention that the intensity $\theta=1$ was also known to be
special due to its connection to the uniform spanning tree (such as the
Wilson’s algorithm [54, 39]). This is closely related to the spatial Markov
property of the loop soup at $\theta=1$ when the loops are viewed as oriented
loops, as pointed out in [52].

#### Acknowledgements

The authors thank Juhan Aru, Nicolas Curien, Tom Hutchcroft, Pierre Nolin,
Avelio Sepúlveda, Xin Sun and Zijie Zhuang for discussions related to this
work. This material is based upon work supported by the National Science
Foundation under Grant No. DMS-1928930 while AJ and WQ participated in a
program hosted by the Mathematical Sciences Research Institute in Berkeley,
California, during the Spring 2022 semester. AJ is supported by Eccellenza
grant 194648 of the Swiss National Science Foundation and is a member of NCCR
SwissMAP. WQ was in CNRS and Laboratoire de Mathématiques d’Orsay, Université
Paris-Saclay, when this work was initiated.

##  2 Preliminaries

Let $D\subset\mathbb{C}$ be an open set whose boundary is a finite collection
of analytic curves. Let $x\in D$ and run a standard Brownian motion starting
from $x$ until it reaches the boundary of $D$. Denote by $B_{\tau}$ the exit
point. The law of $B_{\tau}$ has a density $H_{D}(x,\cdot)$, called Poisson
kernel, with respect to the one-dimensional Hausdorff measure on $\partial D$.
The boundary Poisson kernel is defined by

$H_{D}(w,z)=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}H_{D}(w+\varepsilon
n_{w},z),\quad\quad w,z\in\partial D,$ (2.1)

where $n_{w}$ denotes the inward normal at $w$. When $D$ is the unit disc,
$H_{D}$ has the following explicit expression:

$H_{\mathbb{D}}(x,z)=\left\\{\begin{array}[]{cc}\frac{1}{2\pi}\frac{1-|x|^{2}}{|x-z|^{2}},&x\in\mathbb{D},z\in\partial\mathbb{D},\\\
\frac{1}{\pi}\frac{1}{|x-z|^{2}},&x\in\partial\mathbb{D},z\in\partial\mathbb{D}.\end{array}\right.$
(2.2)

###  2.1 FKG inequality in the loop soup

We now recall the FKG inequality which is an import tool that we will use
frequently in the paper. A function $f$ is said to be increasing if for all
$\mathcal{L},\mathcal{L}^{\prime}$ with
$\mathcal{L}\subset\mathcal{L}^{\prime}$, $f(\mathcal{L})\leq
f(\mathcal{L}^{\prime})$.

######  Lemma 2.1 (FKG inequality [18, Lemma 2.1]).

For all increasing bounded functions $f,g$,

$\mathbb{E}\left[f(\mathcal{L}_{D}^{\theta})g(\mathcal{L}_{D}^{\theta})\right]\geq\mathbb{E}\left[f(\mathcal{L}_{D}^{\theta})\right]\mathbb{E}\left[g(\mathcal{L}_{D}^{\theta})\right].$

In most places we will apply this result to indicator functions of increasing
events.

###  2.2 Crossing an annulus with loops

The aim of this section is to prove the following estimates on the loop
measure of some crossing event.

######  Lemma 2.2.

Let $0<r_{1}<r_{2}<1$. We have

$\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}crossing~{}}r_{2}\mathbb{D}\setminus
r_{1}\mathbb{D}\\})=\left(1+O(1)\frac{r_{1}}{r_{2}}\right)\log\frac{\log(1/r_{1})}{\log(r_{2}/r_{1})}.$
(2.3)

Moreover, the left hand side of (2.3) is differentiable with respect to
$r_{1}$ and

$\frac{\mathrm{d}}{\mathrm{d}r_{1}}\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}crossing~{}}r_{2}\mathbb{D}\setminus
r_{1}\mathbb{D}\\})=\Big{(}1+\frac{O(1)}{\log(r_{2}/r_{1})}\Big{)}\frac{1}{r_{1}}\Big{(}\frac{1}{\log(r_{2}/r_{1})}-\frac{1}{\log(1/r_{1})}\Big{)}.$
(2.4)

In (2.3) and (2.4), $O(1)$ refers to a quantity uniformly bounded with respect
to $r_{1}$ and $r_{2}$.

We will see in (2.7) below that the left hand side of (2.3) can be expressed
in terms of a Poisson kernel in a disc and a boundary Poisson kernel in an
annulus. Contrary to the case of a disc (see (2.2)), the Poisson kernel in an
annulus does not have a simple explicit expression. Before proving Lemma 2.2,
we first state a result concerning this latter Poisson kernel. This result
will be useful in the proof of (2.4).

######  Lemma 2.3.

Let $q\in(0,1)$. For all $\theta\in[0,2\pi]$, $H_{\mathbb{D}\setminus
q\mathbb{D}}(qe^{i\theta},1)$ is differentiable with respect to $q$ and

$\frac{\mathrm{d}}{\mathrm{d}q}\left(qH_{\mathbb{D}\setminus
q\mathbb{D}}(qe^{i\theta},1)\right)=\left(1+\frac{O(1)}{\log
q}\right)\frac{1}{|\log q|}H_{\mathbb{D}\setminus
q\mathbb{D}}(qe^{i\theta},1).$ (2.5)

We now assume that Lemma 2.3 holds and we prove Lemma 2.2. The proof of Lemma
2.3 is contained in Section 2.3.

###### Proof of Lemma 2.2.

By rooting the loop at the unique point whose distance to the origin is
maximal, we find that

$\mu^{\rm
loop}_{\mathbb{D}}=\frac{1}{\pi}\int_{0}^{1}r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta_{2}~{}\mu^{\rm
bub}_{r\mathbb{D}}(re^{i\theta_{2}}).$ (2.6)

where $\mu^{\rm bub}_{r\mathbb{D}}(re^{i\theta_{2}})$ is the bubble measure
defined in [28]. This decomposition is similar to [28, Proposition 8] except
that in our normalisation the loop measure is twice larger than in [28]
(equivalently, the critical intensity is 1/2 for us whereas it is 1 for them).
Moreover, one can compute for all $r\geq r_{1}$ and $\theta_{2}\in[0,2\pi]$,

$\mu^{\rm
bub}_{r\mathbb{D}}(re^{i\theta_{2}})(\\{\wp\text{~{}hit~{}}r_{1}\mathbb{D}\\})=\pi\int_{0}^{2\pi}\mathrm{d}\theta_{1}r_{1}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})H_{r\mathbb{D}}(r_{1}e^{i\theta_{1}},re^{i\theta_{2}}),$

leading to

$\displaystyle\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}crossing~{}}r_{2}\mathbb{D}\setminus
r_{1}\mathbb{D}\\})=\int_{r_{2}}^{1}r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{0}^{2\pi}\mathrm{d}\theta_{1}r_{1}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})H_{r\mathbb{D}}(r_{1}e^{i\theta_{1}},re^{i\theta_{2}}).$
(2.7)

Using the explicit expression (2.2) of the Poisson kernel in a disc, we see
that for all $r\geq r_{2}$,

$H_{r\mathbb{D}}(r_{1}e^{i\theta_{1}},re^{i\theta_{2}})=\left(1+O(1)\frac{r_{1}}{r_{2}}\right)\frac{1}{2\pi
r}$ (2.8)

and

$\frac{\mathrm{d}}{\mathrm{d}r_{1}}H_{r\mathbb{D}}(r_{1}e^{i\theta_{1}},re^{i\theta_{2}})=-\frac{2\cos(\theta_{1}-\theta_{2})}{2\pi
r^{2}}+O(1)\frac{r_{1}}{r^{3}}.$ (2.9)

Plugging the estimate (2.8) in (2.7), we obtain that

$\displaystyle\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}crossing~{}}r_{2}\mathbb{D}\setminus
r_{1}\mathbb{D}\\})=\Big{(}1+O(1)\frac{r_{1}}{r_{2}}\Big{)}\frac{1}{2\pi}\int_{r_{2}}^{1}\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{0}^{2\pi}\mathrm{d}\theta_{1}~{}r_{1}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}}).$

Since for all $\theta_{1}\in[0,2\pi]$ and $r\in[r_{2},1]$ (see (2.1))

$\displaystyle\int_{r\partial\mathbb{D}}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(z,r_{1}e^{i\theta})\mathrm{d}z$
$\displaystyle=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\int_{r\partial\mathbb{D}}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(z,(r_{1}+\varepsilon)e^{i\theta})\mathrm{d}z$ (2.10)
$\displaystyle=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\mathbb{P}_{(r_{1}+\varepsilon)e^{i\theta}}\left(\tau_{r\partial\mathbb{D}}<\tau_{r_{1}\partial\mathbb{D}}\right)=\frac{1}{r_{1}\log(r/r_{1})},$

we deduce that

$\displaystyle\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}crossing~{}}r_{2}\mathbb{D}\setminus
r_{1}\mathbb{D}\\})=\left(1+O(1)\frac{r_{1}}{r_{2}}\right)\int_{r_{2}}^{1}\frac{\mathrm{d}r}{r\log(r/r_{1})}=\left(1+O(1)\frac{r_{1}}{r_{2}}\right)\log\frac{\log(1/r_{1})}{\log(r_{2}/r_{1})}.$

This proves (2.3).

We now move on to the proof of (2.4). In Lemma 2.3 we studied the boundary
Poisson kernel in an annulus of the form $\mathbb{D}\setminus q\mathbb{D}$
which is related to the Poisson kernel we are interested in by

$H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})=\frac{1}{r^{2}}H_{\mathbb{D}\setminus\frac{r_{1}}{r}\mathbb{D}}\Big{(}1,\frac{r_{1}}{r}e^{i(\theta_{1}-\theta_{2})}\Big{)}.$

By Lemma 2.3, the above left hand side is therefore differentiable with
respect to $r_{1}$ and

$\frac{\mathrm{d}}{\mathrm{d}r_{1}}\left(r_{1}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})\right)=\left(1+\frac{O(1)}{\log(r/r_{1})}\right)\frac{1}{\log(r/r_{1})}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}}).$

Hence, $\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}crossing~{}}r_{2}\mathbb{D}\setminus
r_{1}\mathbb{D}\\})$ is differentiable with respect to $r_{1}$ and using (2.8)
and (2.9) we find that its derivative is equal to

$\displaystyle\int_{r_{2}}^{1}r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{0}^{2\pi}\mathrm{d}\theta_{1}~{}\Big{\\{}\frac{\mathrm{d}}{\mathrm{d}r_{1}}\left(r_{1}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})\right)H_{r\mathbb{D}}(r_{1}e^{i\theta_{1}},re^{i\theta_{2}})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+r_{1}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})\frac{\mathrm{d}}{\mathrm{d}r_{1}}H_{r\mathbb{D}}(r_{1}e^{i\theta_{1}},re^{i\theta_{2}})\Big{\\}}$
$\displaystyle=\left(1+\frac{O(1)}{\log(r_{2}/r_{1})}\right)\int_{r_{2}}^{1}r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{0}^{2\pi}\mathrm{d}\theta_{1}~{}\Big{\\{}\frac{1}{2\pi
r}\frac{1}{\log(r/r_{1})}+O(1)\frac{r_{1}}{r^{2}}\Big{\\}}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})$
$\displaystyle=\left(1+\frac{O(1)}{\log(r_{2}/r_{1})}\right)\int_{r_{2}}^{1}r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta_{2}\int_{0}^{2\pi}\mathrm{d}\theta_{1}~{}\frac{1}{2\pi
r}\frac{1}{\log(r/r_{1})}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}}).$

We again use the fact that (see (2.10))

$\int_{0}^{2\pi}H_{r\mathbb{D}\setminus
r_{1}\mathbb{D}}(re^{i\theta_{2}},r_{1}e^{i\theta_{1}})\mathrm{d}\theta_{2}=\frac{1}{rr_{1}\log(r/r_{1})}$

to obtain that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}r_{1}}\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}crossing~{}}r_{2}\mathbb{D}\setminus
r_{1}\mathbb{D}\\})$
$\displaystyle=\left(1+\frac{O(1)}{\log(r_{2}/r_{1})}\right)\int_{r_{2}}^{1}\frac{\mathrm{d}r}{rr_{1}\log(r/r_{1})^{2}}$
$\displaystyle=\left(1+\frac{O(1)}{\log(r_{2}/r_{1})}\right)\frac{1}{r_{1}}\left(\frac{1}{\log(r_{2}/r_{1})}-\frac{1}{\log(1/r_{1})}\right).$

This concludes the proof of (2.4), assuming Lemma 2.3. ∎

###  2.3 Boundary Poisson kernel in an annulus

The goal of this section is to prove Lemma 2.3. We start by stating and
proving an intermediate result.

######  Lemma 2.4.

Let $f(\theta):=H_{{\mathbb{D}}\setminus q{\mathbb{D}}}(qe^{i\theta},1)$. For
all $\theta\in[0,2\pi]$, we have

$\displaystyle f(\theta)=(1+o(1))f(0)\,\text{ as }\,q\to 0.$ (2.11)

For all $\alpha,\theta\in[0,2\pi]$, we have

$\displaystyle
f(\alpha)-f(\theta)=f^{\prime}(\theta)(\alpha-\theta)+\frac{1}{2}(\alpha-\theta)^{2}f^{\prime\prime}(0)\left(1+o(1)\right)\,\text{
as }\,q\to 0.$ (2.12)

Moreover,

$\displaystyle f^{\prime\prime}(0)=\frac{O(1)}{(\log q)^{2}}f(0)\,\text{ as
}\,q\to 0.$ (2.13)

###### Proof.

The multivalued conformal map $z\mapsto\log z/|\log q|$ sends the region
${\mathbb{D}}\setminus q{\mathbb{D}}$ to the vertical strip
$V:=\\{z=x+iy:-1\leq x\leq 0,y\in\mathbb{R}\\}$. We have

$\displaystyle H_{{\mathbb{D}}\setminus
q{\mathbb{D}}}(1,qe^{i\theta})=\frac{1}{q|\log
q|^{2}}\sum_{n\in{\mathbb{Z}}}H_{V}(0,-1+i(\theta+2n\pi)/|\log q|).$ (2.14)

We can map the strip $V$ onto the upper half-plane ${\mathbb{H}}$ by the
conformal map $\varphi:z\mapsto\exp(iz\pi)$, so that

$\displaystyle|\varphi^{\prime}(0)||\varphi^{\prime}(-1+i(\theta+2n\pi)/|\log
q|)|H_{V}(0,-1+i(\theta+2n\pi)/|\log q|)=$ $\displaystyle
H_{\mathbb{H}}(1,-\exp((\theta+2n\pi)\pi/\log q)),$

which tends to $H_{\mathbb{H}}(1,-1)$ as $q\to 0$. Note that

$\displaystyle|\varphi^{\prime}(0)|=\pi,\quad|\varphi^{\prime}(-1+i(\theta+2n\pi)/|\log
q|)|=\pi\exp((\theta+2n\pi)\pi/\log q).$

This implies that

$\displaystyle H_{V}(0,-1+i(\theta+2n\pi)/|\log
q|)=\pi\exp((\theta+2n\pi)\pi/\log q)(1+\exp((\theta+2n\pi)\pi/\log q))^{-2}.$

Therefore

$\displaystyle f(\theta)=$ $\displaystyle\frac{\pi}{q|\log
q|^{2}}\sum_{n\in{\mathbb{Z}}}\exp((\theta+2n\pi)\pi/\log
q)(1+\exp((\theta+2n\pi)\pi/\log q))^{-2}$ (2.15) $\displaystyle=$
$\displaystyle\frac{\pi}{q|\log
q|^{2}}\sum_{n\in{\mathbb{Z}}}\cosh\left(\frac{(\theta+2n\pi)\pi}{2\log
q}\right)^{-2}.$ (2.16)

Therefore

$\displaystyle f^{\prime}(\theta)=-\frac{\pi}{q(\log
q)^{3}}\sum_{n\in{\mathbb{Z}}}\tanh\left(\frac{(\theta+2n\pi)\pi}{2\log
q}\right)\cosh\left(\frac{(\theta+2n\pi)\pi}{2\log q}\right)^{-2},$ (2.17)
$\displaystyle f^{\prime\prime}(\theta)=-\frac{\pi}{q(\log
q)^{4}}\sum_{n\in{\mathbb{Z}}}\left(1-3\tanh\left(\frac{(\theta+2n\pi)\pi}{2\log
q}\right)^{2}\right)\cosh\left(\frac{(\theta+2n\pi)\pi}{2\log q}\right)^{-2}$
(2.18)

Using that

$\displaystyle\exp(\theta\pi/(2\log q))=1+o(1)\text{ as }q\to 0,$

we deduce that for all $\theta\in[0,2\pi]$,

$\displaystyle f(\theta)=f(0)\left(1+o(1)\right),\quad
f^{\prime\prime}(\theta)=f^{\prime\prime}(0)\left(1+o(1)\right).$

This proves (2.11). Noting that $\tanh$ takes values in $(-1,1)$, we can also
deduce that

$\displaystyle f^{\prime\prime}(0)=\frac{O(1)}{(\log q)^{2}}f(0),$

proving (2.13). Therefore for all $\alpha,\theta\in[0,2\pi]$,

$\displaystyle f(\alpha)-f(\theta)=$
$\displaystyle\int_{\theta}^{\alpha}f^{\prime}(\beta)d\beta=\int_{\theta}^{\alpha}(f^{\prime}(\beta)-f^{\prime}(\theta))d\beta+(\alpha-\theta)f^{\prime}(\theta)$
$\displaystyle=$
$\displaystyle\int_{\theta}^{\alpha}\int_{\theta}^{\beta}f^{\prime\prime}(\gamma)d\gamma
d\beta+(\alpha-\theta)f^{\prime}(\theta)$ $\displaystyle=$
$\displaystyle\int_{\theta}^{\alpha}(\beta-\theta)d\beta
f^{\prime\prime}(0)\left(1+\frac{O(1)}{\log
q}\right)+(\alpha-\theta)f^{\prime}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\alpha-\theta)^{2}f^{\prime\prime}(0)\left(1+\frac{O(1)}{\log
q}\right)+(\alpha-\theta)f^{\prime}(\theta).$

This proves (2.12), and completes the proof of the lemma. ∎

We are now ready to prove Lemma 2.3.

###### Proof of Lemma 2.3.

Fix $q_{1}\in(0,1)$ and $q_{2}=q_{1}-\varepsilon$, where $\varepsilon>0$ is
small and will tend to $0$. We have

$\displaystyle H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(1,q_{2}e^{i\theta})=$ $\displaystyle
q_{1}\int_{0}^{2\pi}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\alpha})H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{2}e^{i\theta})d\alpha.$

Therefore

$\displaystyle\frac{1}{q_{1}}H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(1,q_{2}e^{i\theta})$
$\displaystyle=\int_{0}^{2\pi}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{2}e^{i\theta})d\alpha$ (2.19)
$\displaystyle+\int_{0}^{2\pi}(H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\alpha})-H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta}))H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{2}e^{i\theta})d\alpha.$

The first term on the right hand side of (2.19) is equal to

$\displaystyle H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})\int_{0}^{2\pi}H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{2}e^{i\theta})d\alpha=H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})\int_{0}^{2\pi}H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1},q_{2}e^{i\alpha})d\alpha$
$\displaystyle=\frac{1}{q_{2}}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})\frac{|\log q_{1}|}{|\log
q_{2}|}=\frac{1}{q_{2}}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})\left(1+\frac{1}{q_{1}\log
q_{1}}\varepsilon+O_{\varepsilon}(\varepsilon^{2})\right).$ (2.20)

Let $f(\theta):=H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})$. By Lemma 2.4, the second term on the
right hand side of (2.19) is equal to

$\displaystyle\int_{\theta-\pi}^{\theta+\pi}f^{\prime}(\theta)(\alpha-\theta)H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{2}e^{i\theta})d\alpha+\frac{1}{2}\int_{\theta-\pi}^{\theta+\pi}(\alpha-\theta)^{2}f^{\prime\prime}(0)(1+o(1))H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{2}e^{i\theta})d\alpha.$

By symmetry, the first term in the above line is $0$, hence the second term on
the right hand side of (2.19) is equal to

$\displaystyle\frac{1}{2}\int_{\theta-\pi}^{\theta+\pi}(\alpha-\theta)^{2}f^{\prime\prime}(0)(1+o(1))H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{2}e^{i\theta})d\alpha.$ (2.21)

Dividing the term in (2.21) by $\varepsilon$ and take the limit as
$\varepsilon\to 0$, we get

$\displaystyle\frac{1}{2}f^{\prime\prime}(0)(1+o(1))\int_{\theta-\pi}^{\theta+\pi}(\alpha-\theta)^{2}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(q_{1}e^{i\alpha},q_{1}e^{i\theta})d\alpha$
$\displaystyle=\frac{1}{2}(1+o(1))f^{\prime\prime}(0)\int_{-\pi}^{\pi}\beta^{2}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(q_{1},q_{1}e^{i\beta})d\beta.$ (2.22)

Note that

$\displaystyle H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(q_{1},q_{1}e^{i\beta})\leq H_{\mathbb{C}\setminus
q_{1}{\mathbb{D}}}(q_{1},q_{1}e^{i\beta})=\frac{1}{\pi
q^{2}}\sin(\theta/2)^{-2}.$

Therefore,

$\displaystyle\int_{-\pi}^{\pi}\beta^{2}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(q_{1},q_{1}e^{i\beta})d\beta=O(1)q^{-2}.$

Therefore, combined with (2.13), (2.22) is equal to
$\frac{O(1)}{q_{1}^{2}(\log q_{1})^{2}}f(0),$ which is, by (2.11), further
equal to

$\displaystyle\frac{O(1)}{q_{1}^{2}(\log q_{1})^{2}}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta}).$

Combined with (2.19) and (2.20), we get that

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon^{-1}\left(q_{2}H_{{\mathbb{D}}\setminus
q_{2}{\mathbb{D}}}(1,q_{2}e^{i\theta})-q_{1}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})\right)=\frac{1}{\log
q_{1}}H_{{\mathbb{D}}\setminus
q_{1}{\mathbb{D}}}(1,q_{1}e^{i\theta})\left(1+\frac{O(1)}{\log q_{1}}\right).$

This completes the proof. ∎

###  2.4 Surrounding a disk with loops

We will say that a loop $\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}$ surrounds
the disc $r\mathbb{D}$ if $\wp$ does not intersect $r\mathbb{D}$ but
disconnects it from $\partial\mathbb{D}$.

######  Lemma 2.5.

There exists $c=c(\theta)>0$ such that for all $r\in(0,1/10)$,

$\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\mathrm{~{}surrounding~{}}r\mathbb{D}\right)\geq
1-r^{c}.$ (2.23)

###### Proof.

By monotonicity, it suffices to prove the lemma for $r$ of the form $r=e^{-n}$
for some integer $n\geq 1$. By independence of the collections of loops
$\\{\wp\in\mathcal{L}_{\mathbb{D}}^{\theta},\wp\subset
e^{-k}\mathbb{D}\setminus e^{-k-1}\mathbb{D}\\}$, $k=0\dots n-1$, and by scale
invariance, the probability that no loop in
$\mathcal{L}_{\mathbb{D}}^{\theta}$ surrounds $r\mathbb{D}$ is at most

$\displaystyle\mathbb{P}\left(\forall k=0\dots
n-1,\nexists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\mathrm{~{}included~{}in~{}}e^{-k}\mathbb{D}\mathrm{~{}which~{}surrounds~{}}e^{-k-1}\mathbb{D}\right)=p^{n}$

where
$p=\mathbb{P}\left(\nexists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\mathrm{~{}surrounding~{}}e^{-1}\mathbb{D}\right)$.
The fact that $p<1$ concludes the proof. ∎

###  2.5 A priori upper bound on the crossing probability of an annulus by a
cluster

We continue this preliminary section by recalling that when $\theta=1/2$,

######  Lemma 2.6.

When $\theta=1/2$, the following limit exists and is nontrivial:

$\lim_{r\to 0}|\log
r|^{1/2}\mathbb{P}\Big{(}e^{-1}\overset{\mathcal{L}_{\mathbb{D}}^{1/2}}{\longleftrightarrow}r\Big{)}\in(0,\infty).$

###### Proof.

First consider the event that there is a cluster $\mathcal{C}$ of
$\mathcal{L}_{\mathbb{D}}^{1/2}$ crossing from $e^{-1}\partial\mathbb{D}$ to
$r\partial\mathbb{D}$ without surrounding the point $0$. Such a cluster cannot
exist if $\mathcal{L}_{\mathbb{D}}^{1/2}$ contains a Brownian loop $\wp$
contained in the annulus $e^{-1}\mathbb{D}\setminus(r\overline{\mathbb{D}})$
and disconnecting $e^{-1}\partial\mathbb{D}$ from $r\partial\mathbb{D}$. So,
by Lemma 2.5,

$\mathbb{P}\left(\exists\mathcal{C}\text{ cluster of
}\mathcal{L}_{\mathbb{D}}^{1/2},e^{-1}\overset{\mathcal{C}}{\longleftrightarrow}r,~{}\mathcal{C}\text{
does not surround }0\right)\leq(er)^{c},$

for some constant $c>0$ and $r$ small enough. So with the additional condition
of not surrounding $0$, the crossing probability decays much faster than
logarithmically. With this out of the way, let us concentrate on clusters of
$\mathcal{L}_{\mathbb{D}}^{1/2}$ that surround $0$.

Let $(\mathcal{C}_{n})_{n\geq 1}$ be the infinite sequence of clusters of
$\mathcal{L}_{\mathbb{D}}^{1/2}$ that surround $0$, where the clusters are
ordered from the most exterior in the nesting order ($\mathcal{C}_{n}$
surrounds $\mathcal{C}_{n+1}$). We will denote by
$\partial_{o}\mathcal{C}_{n}$ the outer boundary of $\mathcal{C}_{n}$ and by
$\partial_{i}\mathcal{C}_{n}$ the inner boundary component that surrounds $0$.
Both $\partial_{o}\mathcal{C}_{n}$ and $\partial_{i}\mathcal{C}_{n}$ are known
to be CLE4-type loops. We will denote by
$\operatorname{Int}(\partial_{o}\mathcal{C}_{n})$, resp.
$\operatorname{Int}(\partial_{i}\mathcal{C}_{n})$, the connected component of
$0$ in $\mathbb{D}\setminus\partial_{o}\mathcal{C}_{n}$, resp.
$\mathbb{D}\setminus\partial_{i}\mathcal{C}_{n}$. These are simply connected
subdomains containing $0$. We will denote

$\operatorname{CR}_{o}(n)=\operatorname{CR}(0,\operatorname{Int}(\partial_{o}\mathcal{C}_{n})),\qquad\operatorname{CR}_{i}(n)=\operatorname{CR}(0,\operatorname{Int}(\partial_{i}\mathcal{C}_{n})).$

For every $n\geq 1$, the r.v.
$\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n)$ is independent from
$(\mathcal{C}_{1},\dots,\mathcal{C}_{n-1},\partial_{o}\mathcal{C}_{n})$.
Moreover, the family of r.v.s
$(\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n))_{n\geq 1}$ is i.i.d. The
common law is known thanks to the relation between the clusters of
$\mathcal{L}_{\mathbb{D}}^{1/2}$ and some local sets of a continuum GFF on
$\mathbb{D}$ [5, 4, 42]. We refer in particular to [5, Corollary 5.4] and [4,
Proposition 4.8]. To express this law, consider $(B_{t})_{t\geq 0}$ a standard
one-dimensional Brownian motion starting at $0$. For $a>0$, denote

$T_{-a}=\inf\\{t\geq 0:B_{t}=-a\\}.$

Then $\log\big{(}\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n)\big{)}$ has
the same distribution as $T_{-\pi}$. Note that

$\mathbb{P}\left(T_{-\pi}\geq
t\right)=\mathbb{P}\left(|B_{t}|\leq\pi\right)\sim\sqrt{2\pi}t^{-1/2},\quad\quad\text{as}\quad
t\to\infty.$

We will also need the following comparison between conformal radius and the
Euclidean distance. For every open simple connected domain
$D\subset\mathbb{C}$, with $D\neq\mathbb{C}$, and for every $z\in D$,

$d(z,\partial D)\leq\operatorname{CR}(z,D)\leq 4d(z,\partial D).$

The lower bound follows simply from the monotonicity of the conformal radius.
The upper bound is the Koebe quarter theorem (Theorem 5-3 in [2] and the
subsequent corollary).

Denote

$n_{\rm cross}(r)=\inf\\{n\geq
1:e^{-1}\overset{\mathcal{C}_{n}}{\longleftrightarrow}r\\}.$

If the crossing does not occur, we set $n_{\rm cross}(r)=+\infty$. Let $A$ be
the annulus

$A=\mathbb{D}\setminus(e^{-1}\overline{\mathbb{D}})=\\{z\in\mathbb{C}:e^{-1}<|z|<1\\}.$

Denote

$\Sigma=\sum_{n\geq 1}\mathbb{P}\left(\mathcal{C}_{n}\cap
A\neq\emptyset\right).$

We claim that $\Sigma<+\infty$ and that

$\mathbb{P}\left(n_{\rm cross}(r)<+\infty\right)\sim\sqrt{2\pi}\Sigma|\log
r|^{-1/2}\quad\quad\text{as}\quad r\to 0.$ (2.24)

This implies our lemma.

First, let us check that $\Sigma<+\infty$. We have that

$\log(d(0,\mathcal{C}_{n-1})^{-1})\geq\log(\operatorname{CR}_{i}(n-1)^{-1})-\log
4\geq\Big{(}\sum_{k=1}^{n-1}\log(\operatorname{CR}_{o}(k)/\operatorname{CR}_{i}(k))\Big{)}-\log
4.$

Since the r.v.s $\log(\operatorname{CR}_{o}(k)/\operatorname{CR}_{i}(k))$ are
i.i.d. and

$\mathbb{E}\Big{[}\log(\operatorname{CR}_{o}(k)/\operatorname{CR}_{i}(k))\Big{]}=\mathbb{E}[T_{-\pi}]=+\infty;$

the large deviation principle ensures that there is $u\in(0,1)$ such that for
every $n\geq 2$,

$\mathbb{P}\left(\sum_{k=1}^{n-1}\log(\operatorname{CR}_{o}(k)/\operatorname{CR}_{i}(k))\leq
n-1+\log 4\right)\leq u^{n-1}.$

Thus,

$\mathbb{P}\left(d(0,\mathcal{C}_{n-1})\geq e^{-(n-1)}\right)\leq u^{n-1}.$

Further,

$\displaystyle\mathbb{P}\left(\mathcal{C}_{n}\cap A\neq\emptyset\right)$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\left(\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})<e^{-(n-1)}\right)+\mathbb{P}\left(d(0,\mathcal{C}_{n-1})\geq
e^{-(n-1)}\right)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\left(\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})<e^{-(n-1)}\right)+u^{n-1}.$

For $n\geq 3$, on the event $\\{\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})<e^{-(n-1)}\\}$, the loop soup
$\mathcal{L}^{1/2}_{\mathbb{D}}$ cannot contain a Brownian loop $\wp$ with
$\operatorname{Range}(\wp)\subset
e^{-1}\mathbb{D}\setminus(e^{-(n-1)}\overline{\mathbb{D}})$ that surrounds
$e^{-(n-1)}\overline{\mathbb{D}}$. Indeed, such a loop would have to intersect
$\mathcal{C}_{n-1}$, thus be contained in $\mathcal{C}_{n-1}$, thus surround
$\mathcal{C}_{n}$, and thus prevent $\mathcal{C}_{n}$ from intersecting $A$,
which is a contradiction. So, by Lemma 2.5, there is a constant $c>0$, such
that for every $n\geq 3$, $\mathbb{P}\left(\mathcal{C}_{n}\cap
A\neq\emptyset\right)\leq e^{-c(n-2)}+u^{n-1}.$ This ensures that
$\Sigma<+\infty$.

We now turn to the proof of (2.24). We will first establish the lower bound.
Let $n\geq 1$. The conjunction of the following conditions is sufficient to
ensure that $n_{\rm cross}(r)=n$:

$d(0,\mathcal{C}_{n-1})>r,\quad\partial_{o}\mathcal{C}_{n}\cap
A\neq\emptyset\quad\text{and}\quad\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n)\geq
r^{-1}.$

For $n=1$ the first condition is irrelevant. By using the independence of
$\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n)$ from
$(\mathcal{C}_{1},\dots,\mathcal{C}_{n-1},\partial_{o}\mathcal{C}_{n})$, we
get

$\displaystyle\mathbb{P}\left(n_{\rm cross}(r)=n\right)$ $\displaystyle\geq$
$\displaystyle\mathbb{P}\left(\partial_{o}\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})>r\right)\mathbb{P}\left(\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n)\geq
r^{-1}\right)$ $\displaystyle=$
$\displaystyle\mathbb{P}\left(\partial_{o}\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})>r\right)\mathbb{P}\left(T_{-\pi}\geq|\log
r|\right)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left(\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})>r\right)\mathbb{P}\left(T_{-\pi}\geq|\log
r|\right).$

Thus,

$\mathbb{P}\left(n_{\rm
cross}(r)<+\infty\right)\geq\mathbb{P}\left(T_{-\pi}\geq|\log
r|\right)\sum_{n\geq 1}\mathbb{P}\left(\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})>r\right).$

Further, as $r\to 0$,

$\mathbb{P}\left(T_{-\pi}\geq|\log r|\right)\sim\sqrt{2\pi}|\log
r|^{-1/2},\qquad\lim_{r\to 0}\sum_{n\geq 1}\mathbb{P}\left(\mathcal{C}_{n}\cap
A\neq\emptyset,d(0,\mathcal{C}_{n-1})>r\right)=\Sigma,$

concluding the proof of the lower bound of (2.24).

Let us now derive the upper bound of (2.24). Fix $v\in(0,1)$.
$\mathbb{P}\left(n_{\rm cross}(r)<+\infty\right)$ is equal to

$\displaystyle\mathbb{P}\left(n_{\rm
cross}(r)<+\infty,\operatorname{CR}_{i}(n_{\rm
cross}(r))^{-1}\geq(4r)^{-1}\right)$ $\displaystyle\leq\mathbb{P}\left(n_{\rm
cross}(r)<+\infty,\operatorname{CR}_{o}(n_{\rm
cross}(r))^{-1}\geq(4r)^{-(1-v)}\right)$ $\displaystyle+\mathbb{P}\left(n_{\rm
cross}(r)<+\infty,\operatorname{CR}_{o}(n_{\rm
cross}(r))/\operatorname{CR}_{i}(n_{\rm cross}(r))\geq(4r)^{-v}\right)$
$\displaystyle\leq\mathbb{P}\left(n_{\rm
cross}(r)<+\infty,d(0,\partial_{o}\mathcal{C}_{n_{\rm
cross}(r)})\leq(4r)^{1-v}\right)$ (2.25) $\displaystyle+\mathbb{P}\left(n_{\rm
cross}(r)<+\infty,\operatorname{CR}_{o}(n_{\rm
cross}(r))/\operatorname{CR}_{i}(n_{\rm cross}(r))\geq(4r)^{-v}\right)$ (2.26)

On the event $\\{n_{\rm cross}(r)<+\infty,d(0,\partial_{o}\mathcal{C}_{n_{\rm
cross}(r)})\leq(4r)^{1-v}\\}$, the loop soup $\mathcal{L}^{1/2}_{\mathbb{D}}$
cannot contain a Brownian loop $\wp$ with $\operatorname{Range}(\wp)\subset
e^{-1}\mathbb{D}\setminus((4r)^{1-v}\overline{\mathbb{D}})$ that surrounds
$(4r)^{1-v}\overline{\mathbb{D}}$. Indeed, then $\wp$ would have to intersect
$\partial_{o}\mathcal{C}_{n_{\rm cross}(r)}$, which is impossible by
construction. Thus, by Lemma 2.5, there is a constant $c>0$ such that for
every $r$ small enough, (2.25) is at most $e^{c}(4r)^{c(1-v)}$. On the other
hand, (2.26) is at most

$\displaystyle\sum_{n\geq 1}\mathbb{P}\left(\partial_{o}\mathcal{C}_{n}\cap
A\neq\emptyset,\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n)\geq(4r)^{-v}\right),$
$\displaystyle=\sum_{n\geq 1}\mathbb{P}\left(\partial_{o}\mathcal{C}_{n}\cap
A\neq\emptyset\right)\mathbb{P}\left(\operatorname{CR}_{o}(n)/\operatorname{CR}_{i}(n)\geq(4r)^{-v}\right)=\Sigma\mathbb{P}\left(T_{-\pi}\geq
v\log((4r)^{-1})\right).$

Therefore,

$\limsup_{r\to 0}|\log r|^{1/2}\mathbb{P}\left(n_{\rm
cross}(r)<+\infty\right)\leq\sqrt{2\pi}\Sigma v^{-1/2}.$

By letting $v\to 1$, we get the desired result. ∎

In particular, this provides a first rough upper bound on the probability of
crossing for any $\theta\leq 1/2$:

######  Corollary 2.7 (A priori upper bound).

For all $\theta\leq 1/2$ and $s\geq 1$,

$\limsup_{\varepsilon\to
0}\mathbb{P}\Big{(}\varepsilon\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s}\Big{)}\leq
s^{-1/2}.$ (2.27)

###### Proof.

By monotonicity of the left hand side of (2.27) w.r.t. $\theta$, we can assume
that $\theta=1/2$. Let $\eta>0$ be a small parameter and let $s>1$ (the case
$s=1$ is trivial). Conditioned on
$\\{e^{-1}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\eta\varepsilon\\}$
and
$\\{\varepsilon\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s}\\}$,
to have a cluster crossing the annulus
$e^{-1}\mathbb{D}\setminus\varepsilon^{s}\mathbb{D}$, it is enough to have a
loop included in $\varepsilon\mathbb{D}$ that surrounds the disc
$\eta\varepsilon\mathbb{D}$. By FKG inequality and scale invariance, we obtain
that

$\displaystyle\mathbb{P}\Big{(}e^{-1}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s}\Big{)}\geq\mathbb{P}\Big{(}e^{-1}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\eta\varepsilon\Big{)}\mathbb{P}\Big{(}\varepsilon\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s}\Big{)}\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}surrounding~{}}\eta\mathbb{D}\right).$

By Lemma 2.6, this leads to

$\limsup_{\varepsilon\to
0}\mathbb{P}\Big{(}\varepsilon\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s}\Big{)}\leq
s^{-1/2}\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}surrounding~{}}\eta\mathbb{D}\right)^{-1}.$

As $\eta\to 0$, the probability on the right hand side converges to 1. This
concludes the proof. ∎

###  2.6 Generalisation of Theorem 1.2

We conclude this preliminary section by showing that Corollary 1.6 is a quick
consequence of Theorem 1.2 and FKG inequality.

###### Proof of Corollary 1.6, assuming Theorem 1.2.

Let $R_{1},R_{2}>0$ be such that $D(x,R_{1})\subset D\subset D(x,R_{2})$ and
let $d$ be the distance between $x$ and $A$. The probability that there is a
cluster $\mathcal{C}$ of $\mathcal{L}_{D}^{\theta}$ such that
$\overline{\mathcal{C}}\cap A\neq\varnothing$ and $\mathcal{C}\cap
D(x,r)\neq\varnothing$ is at most

$\displaystyle\mathbb{P}\Big{(}\partial
D(x,r)\overset{\mathcal{L}_{D(x,R_{2})}^{\theta}}{\longleftrightarrow}\partial
D(x,d)\Big{)}=\mathbb{P}\Big{(}\frac{r}{R_{2}}\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\frac{d}{R_{2}}\partial\mathbb{D}\Big{)}$

by invariance under scaling and translation. Since $A$ is closed, $d$ is
positive and Theorem 1.2 yields the upper bound stated in Corollary 1.6. A
similar argument shows that if $r_{0}\in(0,R_{1})$ is fixed, then

$\mathbb{P}\Big{(}\partial
D(x,r)\overset{\mathcal{L}_{D}^{\theta}}{\longleftrightarrow}\partial
D(x,r_{0})\Big{)}=|\log r|^{-1+\theta+o(1)}.$

Moreover, because $A$ is not polar, the probability that there is a cluster in
$\mathcal{L}_{D}^{\theta}$ whose closure hits $A$ and that disconnects
$\partial D(x,r_{0}/2)$ and $\partial D(x,r_{0})$ is positive. On the
intersection of this event and the event $\\{\partial
D(x,r)\overset{\mathcal{L}_{D}^{\theta}}{\longleftrightarrow}\partial
D(x,r_{0})\\}$, there must be a cluster whose closure joins $D(x,r)$ to $A$.
By FKG inequality (see Lemma 2.1), the probability of the intersection of
these two events is at least the product of the probabilities. This shows that

$\mathbb{P}\Big{(}\exists~{}\mathcal{C}\text{~{}cluster~{}of~{}}\mathcal{L}_{D}^{\theta}:\overline{\mathcal{C}}\cap
A\neq\varnothing\text{~{}and~{}}\mathcal{C}\cap
D(x,r)\neq\varnothing\Big{)}\geq|\log r|^{-1+\theta+o(1)}$

as claimed. ∎

##  3 One-arm event in the Brownian loop soup

The goal of this section is to prove Theorem 1.2. We start by introducing a
few notations and state a stronger form of Theorem 1.3. We will denote by

$p_{\varepsilon,\delta}=\mathbb{P}\left(\varepsilon\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta\partial\mathbb{D}\right).$

Let

$I_{\leq}^{*}:=\\{(s_{1},s_{2})\in(0,\infty)^{2}:s_{1}\leq s_{2}\\}$ (3.1)

and for all $\delta>0$, let $F_{\delta}$ be the following function

$F_{\delta}:(s_{1},s_{2})\in I_{\leq}^{*}\mapsto
p_{\delta^{s_{1}},\delta^{s_{2}}}\in[0,1].$

We will show in this section that:

######  Theorem 3.1.

$F_{\delta}$ converges as $\delta\to 0$ to some function $F_{\infty}$. The
convergence is uniform on each compact subset of $I_{\leq}^{*}$.

As a consequence of this convergence, the limiting function $F_{\infty}$ is
actually only a function of the ratio $s_{2}/s_{1}$, i.e. for all
$(s_{1},s_{2})\in I_{\leq}$,
$F_{\infty}(s_{1},s_{2})=F_{\infty}(1,s_{2}/s_{1})$. The function $f_{\infty}$
appearing in Theorem 1.3 is then simply given by:
$f_{\infty}(s)=F_{\infty}(1,s)$ for all $s\geq 1$.

The proofs of Theorems 1.2 and 3.1 are then divided as follows:

  * •

Section 3.1: the sequence $(F_{\delta})_{\delta>0}$ is tight;

  * •

Section 3.2: any subsequential limit of $(F_{\delta})_{\delta>0}$ satisfies
some integral equation;

  * •

Section 3.3: uniqueness of the fixed-points of this integral equation. This
will finish the proof of Theorem 3.1;

  * •

Section 3.4: identification of $f_{\infty}$ in terms of Bessel processes as
stated in Theorem 1.8. This will in particular nail down the decay of
$f_{\infty}(s)$ as $s\to\infty$;

  * •

Section 3.5: proof of Theorem 1.2 by relating the decay of the probability of
large crossings to the decay of $f_{\infty}$ at infinity.

###  3.1 Tightness

Recall the definition (3.1) of $I_{\leq}^{*}$.

######  Lemma 3.2 (Tightness).

For every decreasing sequence $(\delta_{n})_{n\geq 1}$ converging to zero, we
can extract a subsequence $(\delta_{n_{k}})_{k\geq 1}$ such that
$(F_{\delta_{n_{k}}})_{k\geq 1}$ converges uniformly on every compact subsets
of $I_{\leq}^{*}$ towards a continuous function.

###### Proof of Lemma 3.2.

It is enough to prove the following uniform equicontinuity property: there
exist $C,c>0$ such that for all $\delta>0$, for all
$(s_{1},t_{1}),(s_{2},t_{2})\in I_{\leq}^{*}$,

$|F_{\delta}(s_{1},t_{1})-F_{\delta}(s_{2},t_{2})|\leq
C\frac{|s_{2}-s_{1}|^{c}+|t_{2}-t_{1}|^{c}}{\max(s_{1},s_{2})^{\theta}}+C\frac{1}{|\log\delta|^{c}}.$
(3.2)

Indeed, let $(\delta_{n})_{n\geq 1}$ be a decreasing sequence converging to
zero. By a slight variant of the Arzelà–Ascoli theorem (see e.g. [14, Theorem
6.2]), the above equicontinuity property and the fact that
$(F_{\delta_{n}})_{n\geq 1}$ is bounded ($0\leq F_{\delta}\leq 1$ for all
$\delta$) imply that for every compact subset of $I_{\leq}^{+}$, we can
extract a subsequence that converges uniformly on that compact set to a
continuous function. We then obtain the result stated in Lemma 3.2 by a
diagonalisation argument (so that the convergence is uniform on every compact
subset simultaneously).

It remains to prove the equicontinuity property (3.2). Let $\delta>0$ and
$(s_{1},t_{1}),(s_{2},t_{2})\in I_{\leq}^{*}$. By symmetry, we can assume that
$s_{1}\leq s_{2}$. By the triangle inequality,

$|F_{\delta}(s_{1},t_{1})-F_{\delta}(s_{2},t_{2})|\leq|F_{\delta}(s_{1},t_{1})-F_{\delta}(s_{1},t_{2})|+|F_{\delta}(s_{1},t_{2})-F_{\delta}(s_{2},t_{2})|.$
(3.3)

Let us start by bounding the last term on the right hand side. Because
$s_{1}\leq s_{2}$, $F_{\delta}(s_{1},t_{2})\leq F_{\delta}(s_{2},t_{2})$.
Obtaining an inequality in the other direction amounts to bounding from below
the probability to cross
$\delta^{s_{1}}\mathbb{D}\setminus\delta^{t_{2}}\mathbb{D}$ conditioned on the
fact that there is a crossing of
$\delta^{s_{2}}\mathbb{D}\setminus\delta^{t_{2}}\mathbb{D}$. Let $\eta>0$ be a
small parameter that we fix later. Conditionally on the smaller crossing,
there will be a larger crossing as soon as the following two events hold:

$\\{\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\delta^{s_{1}}\mathbb{D}\setminus\delta^{s_{2}+\eta}\mathbb{D}\\}\quad\text{and}\quad\\{\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}surrounding~{}}\delta^{s_{2}+\eta}\mathbb{D}\text{~{}while~{}staying~{}in~{}}\delta^{s_{2}}\mathbb{D}\\}.$

By FKG inequality we obtain that
$F_{\delta}(s_{1},t_{2})/F_{\delta}(s_{2},t_{2})$ is at least the product of
the probabilities of each of the two above events. By (2.3), there exists
$C>0$ such that the probability of the first event is at least
$1-C(s_{2}-s_{1}+\eta)^{\theta}/s_{2}^{\theta}$. By Lemma 2.5, the probability
of the second event is at least $1-\delta^{c\eta}$ for some $c>0$, giving that

$\displaystyle F_{\delta}(s_{1},t_{2})\geq
F_{\delta}(s_{2},t_{2})(1-\delta^{c\eta})\left(1-C\frac{(s_{2}-s_{1}+\eta)^{\theta}}{s_{2}^{\theta}}\right).$

We choose

$\eta=\left\\{\begin{array}[]{ll}\frac{|\log(s_{2}-s_{1})|}{|\log\delta|}&\text{if}~{}\frac{s_{2}-s_{1}}{|\log(s_{2}-s_{1})|}>|\log\delta|,\\\
\frac{\log|\log\delta|}{|\log\delta|}&\text{if}~{}\frac{s_{2}-s_{1}}{|\log(s_{2}-s_{1})|}\leq|\log\delta|.\end{array}\right.$

This choice leads to the following estimate:

$\left|F_{\delta}(s_{1},t_{2})-F_{\delta}(s_{2},t_{2})\right|\leq\left\\{\begin{array}[]{ll}C\frac{(s_{2}-s_{1})^{c}}{s_{2}^{\theta}}&\text{if}~{}\frac{s_{2}-s_{1}}{|\log(s_{2}-s_{1})|}>|\log\delta|,\\\
C|\log\delta|^{-c}&\text{if}~{}\frac{s_{2}-s_{1}}{|\log(s_{2}-s_{1})|}\leq|\log\delta|,\end{array}\right.$

for some $C,c>0$. In particular,

$\left|F_{\delta}(s_{1},t_{2})-F_{\delta}(s_{2},t_{2})\right|\leq
C\frac{(s_{2}-s_{1})^{c}}{s_{2}^{\theta}}+C|\log\delta|^{-c}$

for all $s_{1}\leq s_{2}$. The first term on the right hand side of (3.3) can
be bounded in a similar fashion:

$|F_{\delta}(s_{1},t_{1})-F_{\delta}(s_{1},t_{2})|\leq
C\frac{|t_{2}-t_{1}|^{c}}{\max(t_{1},t_{2})^{\eta}}+C|\log\delta|^{-c}.$

Putting things together, we obtain (3.2). This concludes the proof of Lemma
3.2. ∎

###  3.2 Integral equation

######  Lemma 3.3.

Let $u\in(0,1)$. For any $\delta\in(0,1)$ and $s\geq 1$,

$\displaystyle p_{\delta,\delta^{s}}$
$\displaystyle=1-\left(\frac{s-1+u}{s}\right)^{\theta}+\frac{O(1)}{s\log\delta}+\left(u^{\theta}+\frac{O(1)}{\log\delta}\right)p_{\delta^{u},\delta^{s-1+u}}+\left(1+O(1)\frac{(\log|\log\delta|)^{C}}{|\log\delta|^{\theta}}\right)\times$
$\displaystyle\times\theta(1-u)(s-1+u)^{\theta}\int_{1}^{1+\frac{s-1}{u}}(s+(1-u)(t-1))^{-\theta-1}p_{(\delta^{s-1+u})^{1/t},\delta^{s-1+u}}\mathrm{d}t.$

The $O(1)$ appearing above stands for a quantity uniformly bounded with
respect to $\delta$ and $s$ (it may depend on $u$).

###### Proof of Lemma 3.3.

Let $u\in(0,1)$, $s>1$ and $\delta>0$. By scaling,

$p_{\delta,\delta^{s}}=\mathbb{P}\left(\delta^{u}\overset{\mathcal{L}_{\delta^{-1+u}\mathbb{D}}}{\longleftrightarrow}\delta^{s-1+u}\right).$

Let

$t_{*}=\min\\{t\geq
1:\exists\wp\in\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}\text{~{}crossing~{}}\mathbb{D}\setminus(\delta^{s-1+u})^{1/t}\mathbb{D}\\}.$
(3.4)

If $t_{*}=1$, then it means that the crossing that we are interested in is
entirely made by a single loop which visits $\mathbb{D}^{c}$. If $t_{*}>1$,
then we need loops included in $\mathbb{D}$ to finish the crossing. Notice
further that if $t\geq 1+\frac{s-1}{u}$, then $(\delta^{s-1+u})^{1/t}$ is
larger than $\delta^{u}$ and the crossing of the annulus
$\delta^{u}\mathbb{D}\setminus\delta^{s-1+u}\mathbb{D}$ is entirely made by
loops staying in $\mathbb{D}$. Putting these remarks together, we obtain that

$\displaystyle p_{\delta,\delta^{s}}$
$\displaystyle=\mathbb{P}\left(t_{*}=1\right)+\mathbb{P}\left(t_{*}\geq
1+\frac{s-1}{u}\right)p_{\delta^{u},\delta^{s-1+u}}$
$\displaystyle+\int_{1}^{1+\frac{s-1}{u}}\mathbb{P}\left(t_{*}\in\mathrm{d}t\right)\mathbb{P}\left(\delta^{u}\overset{\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{s-1+u}\Big{|}t_{*}=t\right).$

We are now going to estimate precisely each term appearing in the above
display. These terms are divided into two different types of probabilities:
probabilities that either a _loop_ or a _cluster_ crosses some annulus. Terms
of the first type can be explicitly computed thanks to Lemma 2.2, whereas
terms of the second type will expressed in terms of $F_{\delta}$. In the
sequel, the $O(1)$ terms might depend on $u$. In particular, the fact that $u$
is positive will be important for the estimates we are going to write.

By scaling and by Lemma 2.2,

$\displaystyle\mathbb{P}\left(t_{*}=1\right)=\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}\text{~{}crossing~{}}\mathbb{D}\setminus\delta^{s-1+u}\mathbb{D}\right)=\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\delta^{1-u}\mathbb{D}\setminus\delta^{s}\mathbb{D}\right)$
$\displaystyle=1-\exp\left(-\theta\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\delta^{1-u}\mathbb{D}\setminus\delta^{s}\mathbb{D}\\})\right)=1-\left(\frac{s-1+u}{s}\right)^{\theta}+\frac{O(1)}{s\log\delta}.$

Similarly,

$\displaystyle\mathbb{P}\left(t_{*}\geq
1+\frac{s-1}{u}\right)=\mathbb{P}\left(\nexists\wp\in\mathcal{L}_{\delta^{-1+u}}^{\theta}\text{~{}crossing~{}}\mathbb{D}\setminus\delta^{u}\mathbb{D}\right)=u^{\theta}+\frac{O(1)}{\log\delta}.$

Finally, writing $r_{t}=\delta^{1-u+\frac{s-1+u}{t}}$,

$\displaystyle\mathbb{P}\left(t_{*}\in\mathrm{d}t\right)=\frac{\mathrm{d}}{\mathrm{d}t}\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}\text{~{}crossing~{}}\mathbb{D}\setminus(\delta^{s-1+u})^{1/t}\mathbb{D}\right)\mathrm{d}t$
$\displaystyle=\theta\left(\frac{\mathrm{d}}{\mathrm{d}t}\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\delta^{1-u}\mathbb{D}\setminus
r_{t}\mathbb{D}\\})\right)e^{-\theta\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\delta^{1-u}\mathbb{D}\setminus
r_{t}\mathbb{D}\\})}\mathrm{d}t.$

By Lemma 2.2,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\delta^{1-u}\mathbb{D}\setminus
r_{t}\mathbb{D}\\})=\left(\frac{\mathrm{d}r_{t}}{\mathrm{d}t}\right)\frac{\mathrm{d}}{\mathrm{d}r}\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\delta^{1-u}\mathbb{D}\setminus
r\mathbb{D}\\})\Big{|}_{r=r_{t}}$
$\displaystyle=\left(1+\frac{O(1)}{\frac{s-1+u}{t}\log\delta}\right)\left(\frac{s-1+u}{t^{2}}\log(\delta)r_{t}\right)\frac{1}{r_{t}}\frac{1}{\log\delta}\left(\frac{1}{\frac{s-1+u}{t}}-\frac{1}{1-u+\frac{s-1+u}{t}}\right)$
$\displaystyle=\left(1+\frac{O(1)}{\log\delta}\right)\frac{1-u}{t(1-u)+s-1+u}.$

Together with (2.3), it gives that

$\displaystyle\mathbb{P}\left(t_{*}\in\mathrm{d}t\right)=\left(1+\frac{O(1)}{\log\delta}\right)\theta\frac{1-u}{t(1-u)+s-1+u}\left(\frac{(1-u)t+s-1+u}{s-1+u}\right)^{-\theta}\mathrm{d}t$
$\displaystyle=\left(1+\frac{O(1)}{\log\delta}\right)\theta(1-u)(s-1+u)^{\theta}(s+(1-u)(t-1))^{-\theta-1}\mathrm{d}t.$

Putting things together, we have obtained that

$\displaystyle p_{\delta,\delta^{s}}$
$\displaystyle=1-\left(\frac{s-1+u}{s}\right)^{\theta}+\frac{O(1)}{s\log\delta}+\left(u^{\theta}+\frac{O(1)}{\log\delta}\right)p_{\delta^{u},\delta^{s-1+u}}+\left(1+\frac{O(1)}{\log\delta}\right)\times$
$\displaystyle\times\theta(1-u)(s-1+u)^{\theta}\int_{1}^{1+\frac{s-1}{u}}(s+(1-u)(t-1))^{-\theta-1}\mathbb{P}\left(\delta^{u}\overset{\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{s-1+u}\Big{|}t_{*}=t\right)\mathrm{d}t.$

It remains to estimate the conditional probability appearing in the integral
above. Let $t\in[1,1+\frac{s-1}{u}]$. We are going to conclude the proof by
showing that

$\mathbb{P}\left(\delta^{u}\overset{\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{s-1+u}\Big{|}t_{*}=t\right)=\left(1-O(1)\frac{\log|\log\delta|^{C}}{|\log\delta|^{\theta}}\right)p_{(\delta^{s-1+u})^{1/t},\delta^{s-1+u}}.$
(3.5)

The upper bound is clear. Indeed, conditionally on $t_{*}=t$, to make the
crossing from $\delta^{s-1+u}$ to $\delta^{u}$, the loops which stay in
$\mathbb{D}$ have to cross at least the annulus
$(\delta^{s-1+u})^{1/t}\mathbb{D}\setminus\delta^{s-1+u}\mathbb{D}$. This is
saying that

$\mathbb{P}\left(\delta^{u}\overset{\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{s-1+u}\Big{|}t_{*}=t\right)\leq
p_{(\delta^{s-1+u})^{1/t},\delta^{s-1+u}}.$

We will now provide a lower bound. The idea is to “weld” the crossing from
$\delta^{u}\mathbb{D}$ to $(\delta^{s-1+u})^{1/t}\mathbb{D}$, and the crossing
from $(\delta^{s-1+u})^{1/t}\mathbb{D}$ to $(\delta^{s-1+u})\mathbb{D}$. Let
$t_{+}$ and $t_{-}$ be close to $t$ and be such that $t_{-}<t<t_{+}$.
Conditionally on $t_{*}=t$, to create a crossing of the annulus
$\delta^{u}\mathbb{D}\setminus\delta^{s-1+u}\mathbb{D}$, it is enough for the
loops staying in $\mathbb{D}$ to do the following four events:

  * •

Event $E_{1}$: There is a cluster crossing the annulus
$(\delta^{s-1+u})^{1/t}\mathbb{D}\setminus\delta^{s-1+u}\mathbb{D}$;

  * •

Event $E_{2}$: There is a loop staying in $(\delta^{s-1+u})^{1/t}\mathbb{D}$
which surrounds $(\delta^{s-1+u})^{1/t_{-}}\mathbb{D}$;

  * •

Event $E_{3}$: There is a loop staying in
$(\delta^{s-1+u})^{1/t_{+}}\mathbb{D}$ which surrounds
$(\delta^{s-1+u})^{1/t}\mathbb{D}$;

  * •

Event $E_{4}$: There is a loop crossing the annulus
$(\delta^{s-1+u})^{1/t_{+}}\mathbb{D}\setminus(\delta^{s-1+u})^{1/t_{-}}\mathbb{D}$

By FKG inequality, the probability of the intersection of these four events is
at least the product of each probability. This gives

$\displaystyle\mathbb{P}\left(\delta^{s-1+u}\overset{\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{u}\Big{|}t_{*}=t\right)\geq
p_{(\delta^{s-1+u})^{1/t},\delta^{s-1+u}}\mathbb{P}\left(E_{2}\right)\mathbb{P}\left(E_{3}\right)\mathbb{P}\left(E_{4}\right).$

Let us denote $\varepsilon=\delta^{s-1+u}$. By Lemma 2.5, there exists $c>0$
such that $\mathbb{P}\left(E_{2}\right)\geq
1-\varepsilon^{c(\frac{1}{t_{-}}-\frac{1}{t})}.$ We choose $t_{+}$ and $t_{-}$
precisely so that

$\frac{1}{t_{-}}-\frac{1}{t}=\frac{1}{t}-\frac{1}{t_{+}}=\frac{\theta}{c}\frac{\log\log\frac{1}{\varepsilon}}{\log\frac{1}{\varepsilon}}.$

This choice leads to:
$\mathbb{P}\left(E_{2}\right)=\mathbb{P}\left(E_{3}\right)\geq
1-|\log\varepsilon|^{-\theta}.$ On the other hand, by Lemma 2.2,

$\mathbb{P}\left(E_{4}\right)=1-\left(\frac{\frac{1}{t_{-}}-\frac{1}{t_{+}}}{\frac{1}{t_{-}}}\right)^{\theta+o(1/\log\delta)}=1-O(1)t^{\theta}|\log\varepsilon|^{-\theta}\log|\log\delta|^{C}=1-O(1)\frac{\log|\log\delta|^{C}}{|\log\delta|^{\theta}}.$

Overall, we have obtained

$\mathbb{P}\left(\delta^{u}\overset{\mathcal{L}_{\delta^{-1+u}\mathbb{D}}^{\theta}}{\longleftrightarrow}\delta^{s-1+u}\Big{|}t_{*}=t\right)\geq\left(1-O(1)\frac{\log|\log\delta|^{C}}{|\log\delta|^{\theta}}\right)p_{(\delta^{s-1+u})^{1/t},\delta^{s-1+u}}.$

This concludes the proof of (3.5) and the proof of Lemma 3.3. ∎

######  Corollary 3.4.

Let $(\delta_{n})_{n\geq 1}$ be a decreasing sequence converging to zero such
that $(F_{\delta_{n}})_{n\geq 1}$ converges uniformly on every compact subsets
of $I_{\leq}^{*}$ to some continuous function $F_{\infty}$. $F_{\infty}$
satisfies for all $(s_{1},s_{2})\in I_{\leq}^{*}$,

$\displaystyle
F_{\infty}(s_{1},s_{2})=1-\left(1-\frac{s_{1}}{s_{2}}\right)^{\theta}+\theta\left(\frac{s_{2}}{s_{1}}-1\right)^{\theta}\int_{1}^{\infty}\left(\frac{s_{2}}{s_{1}}+t-1\right)^{-\theta-1}F_{\infty}\left(\frac{s_{2}-s_{1}}{t},s_{2}-s_{1}\right)\mathrm{d}t.$
(3.6)

###### Proof of Corollary 3.4.

Because the convergence is uniform on each compact set and by Lemma 3.3, we
have for all $(s_{1},s_{2})\in I_{\leq}^{*}$, for all $u\in(0,1]$,

$\displaystyle F_{\infty}(s_{1},s_{2})$
$\displaystyle=1-\left(1-(1-u)\frac{s_{1}}{s_{2}}\right)^{\theta}+u^{\theta}F_{\infty}(us_{1},s_{2}-(1-u)s_{1})+\theta(1-u)(s_{2}/s_{1}-1+u)^{\theta}\times$
$\displaystyle\times\int_{1}^{1+\frac{s_{2}/s_{1}-1}{u}}(s_{2}/s_{1}+(1-u)(t-1))^{-\theta-1}F_{\infty}\left(\frac{s_{2}-(1-u)s_{1}}{t},s_{2}-(1-u)s_{1}\right)\mathrm{d}t.$

We will get (3.6) by sending $u\to 0$. Indeed, the term
$u^{\theta}F_{\infty}(us_{1},s_{2}-(1-u)s_{1})$ goes to zero simply because
$F_{\infty}$ is bounded by 1. The integral is handled by continuity of
$F_{\infty}$ and dominated convergence theorem. ∎

###  3.3 Fixed-points of the integral equation and Proof of Theorem 3.1

In this section we study the solutions of the integral equation that appeared
in Corollary 3.4. We start by introducing the functional spaces we will be
working with. For $\alpha\in\mathbb{R}$, we define

$\mathcal{F}_{\alpha}:=\Big{\\{}f:[1,\infty)\to[0,1]~{}\text{measurable},~{}\left\|f\right\|_{\alpha}:=\sup_{s\geq
1}s^{\alpha}|f(s)|<\infty\Big{\\}}$ (3.7)

and for all measurable function $f:[1,\infty)\to[0,1]$, let

$T(f):s\in[1,\infty)\mapsto
1-\left(1-\frac{1}{s}\right)^{\theta}+\theta(s-1)^{\theta}\int_{1}^{\infty}\left(s+t-1\right)^{-\theta-1}f(t)\mathrm{d}t.$
(3.8)

We now introduce the two-point versions (recall the definition (3.1) of
$I_{\leq}^{*}$): for $\alpha\in\mathbb{R}0$, let

$\mathcal{G}_{\alpha}:=\Big{\\{}F:I_{\leq}^{*}\to[0,1]\text{~{}measurable},~{}\left\|F\right\|_{\mathcal{G}_{\alpha}}:=\sup_{(s_{1},s_{2})\in
I_{\leq}^{*}}(s_{2}/s_{1})^{\alpha}F(s_{1},s_{2})<\infty\Big{\\}}$ (3.9)

and for all measurable function $F:I_{\leq}^{*}\to[0,1]$, let

$S(F):(s_{1},s_{2})\in I_{\leq}^{*}\mapsto
1-\Big{(}1-\frac{s_{1}}{s_{2}}\Big{)}^{\theta}+\theta\Big{(}\frac{s_{2}}{s_{1}}-1\Big{)}^{\theta}\int_{1}^{\infty}\Big{(}\frac{s_{2}}{s_{1}}+t-1\Big{)}^{-\theta-1}F\Big{(}\frac{s_{2}-s_{1}}{t},s_{2}-s_{1}\Big{)}\mathrm{d}t.$
(3.10)

######  Lemma 3.5.

Let $\theta>0$. For all $\alpha\in(-\theta,1)$, $\mathcal{F}_{\alpha}$ is
stable under $T$ and $T:\mathcal{F}_{\alpha}\to\mathcal{F}_{\alpha}$ is a
Lipschitz map with Lipschitz constant $c(\alpha,\theta)$ where

$c(\alpha,\theta)=\frac{\Gamma(1-\alpha)\Gamma(\alpha+\theta)}{\Gamma(\theta)}.$
(3.11)

In particular,

  * •

If $\theta\in(0,1)$ and $\alpha\in(0,1-\theta)$, then $c(\alpha,\theta)<1$ and
$T:\mathcal{F}_{\alpha}\to\mathcal{F}_{\alpha}$ is a contraction mapping.
Therefore, for all $\alpha\in(0,1-\theta)$, there exists a unique function
$f_{\alpha}\in\mathcal{F}_{\alpha}$ satisfying $f_{\alpha}=T(f_{\alpha})$;

  * •

If $\theta>1$ and $\alpha\in(1-\theta,0)$, then $c(\alpha,\theta)<1$ and
$s\mapsto 1$ is the unique fixed point of $T$ in $\mathcal{F}_{\alpha}$.

The same statements hold with the map $T$ and the spaces
$\mathcal{F}_{\alpha}$ replaced by $S$ and $\mathcal{G}_{\alpha}$
respectively. For $\theta\in(0,1)$ and $\alpha\in(0,1-\theta)$, let
$F_{\alpha}$ denote the unique fixed-point of $S$ in $\mathcal{G}_{\alpha}$.
For all $\theta\in(0,1)$ and $\alpha,\beta\in(0,1-\theta)$,

$F_{\alpha}(s_{1},s_{2})=f_{\beta}(s_{2}/s_{1}),\quad\quad(s_{1},s_{2})\in
I_{\leq}^{*}.$ (3.12)

$c(\alpha,\theta)$
---
$\alpha$
---
$(0,1)$
---
$1-\theta$
---
(a) $\theta=0.8$

$c(\alpha,\theta)$
---
$\alpha$
---
$(0,1)$
---
(b) $\theta=1$

$c(\alpha,\theta)$
---
$\alpha$
---
$(0,1)$
---
$1-\theta$
---
(c) $\theta=1.2$

Figure 3.1: Plot of the function $\alpha\mapsto c(\alpha,\theta)$ from Lemma
3.5 for three values of $\theta$. The origin of the axes is fixed at $(0,1)$.
If $\theta<1$ (resp. $\theta>1$), $c(\alpha,\theta)$ takes values less than 1
for some positive (resp. negative) values of $\alpha$. This reflects a phase
transition at $\theta=1$ in the uniqueness of the fixed-points of $T$ (3.8),
which in turn reflects a phase transition of the percolative behaviour of
large loops in the loop soup (see Section 1.3).

The difference of behaviour of $\alpha\mapsto c(\alpha,\theta)$ depending on
whether $\theta<1$, $\theta=1$ or $\theta>1$ is depicted in Figure 3.1.

The transition at $\theta=1$ is another manifestation of the phase transition
of the percolative behaviour of large loops in the loop soup (see Section
1.3). At criticality $\theta=1$, the constant $c(\alpha,\theta)$ is equal to
$\pi\alpha/\sin(\pi\alpha)\geq 1$ by Euler’s reflection formula. Uniqueness of
fixed-points cannot be inferred in that case using this approach. In Theorem
1.9, we will nevertheless be able to cover the case $\theta=1$ by noticing
that the unique fixed point $f_{\alpha}^{\theta}\in\mathcal{F}_{\alpha}$ tends
to 1 as $\theta\to 1^{-}$.

Notice that it is not enough to work with the $L^{\infty}$-norm since it
corresponds to the limiting case where the Lipschitz constant $c(0,\theta)$
equals 1. And indeed, when $\theta\in(0,1)$, the function $s\mapsto 1$ is
another fixed point of $T$ that belongs to $\mathcal{F}_{0}$.

###### Proof.

We are going to prove the statements concerning $T$ and
$\mathcal{F}_{\alpha}$. The exact same arguments will also prove the
statements concerning $S$ and $\mathcal{G}_{\alpha}$. The equality (3.12) will
then follow by uniqueness.

Let $\theta>0$ and $\alpha\in(-\theta,1)$. We start by showing that
$T(\mathcal{F}_{\alpha})\subset\mathcal{F}_{\alpha}$. Let
$f\in\mathcal{F}_{\alpha}$. The fact that $T(f)\geq 0$ is clear. $T(f)$ does
not exceed 1 because for all $s\geq 1$,

$T(f)(s)\leq
1-\left(\frac{s-1}{s}\right)^{\theta}+\frac{\theta}{s-1}\int_{1}^{\infty}\left(1+\frac{t}{s-1}\right)^{-\theta-1}\mathrm{d}t=1.$

Concerning the $\alpha$-norm of $T(f)$, one can show that (see also below for
details)

$\displaystyle\frac{\theta}{s-1}\int_{1}^{\infty}\left(1+\frac{t}{s-1}\right)^{-\theta-1}\frac{1}{t^{\alpha}}\mathrm{d}t\sim\frac{c(\alpha,\theta)}{s^{\alpha}}\quad\quad\text{as~{}}s\to\infty$

where $c(\alpha,\theta)$ is as in (3.11). This proves that $T$ maps
$\mathcal{F}_{\alpha}$ to a subset of $\mathcal{F}_{\alpha}$.

We now study the Lipschitz property of $T$. Let
$f_{1},f_{2}\in\mathcal{F}_{\alpha}$. By definition of $T$,
$\left\|\cdot\right\|_{\alpha}$ and by the triangle inequality, we have

$\displaystyle\left\|T(f_{1})-T(f_{2})\right\|_{\alpha}$
$\displaystyle\leq\sup_{s\geq
1}s^{\alpha}\frac{\theta}{s-1}\int_{1}^{\infty}\left(1+\frac{t}{s-1}\right)^{-\theta-1}|f_{1}(t)-f_{2}(t)|\mathrm{d}t$
$\displaystyle\leq\left\|f_{1}-f_{2}\right\|_{\alpha}\sup_{s\geq
1}s^{\alpha}\frac{\theta}{s-1}\int_{1}^{\infty}\left(1+\frac{t}{s-1}\right)^{-\theta-1}t^{-\alpha}\mathrm{d}t.$

Doing the change of variable $x=1/t$ and then using the integral
representation (B.2) of the hypergeometric function ${}_{2}F_{1}(a,b,c,\cdot)$
whose definition is recalled in (B.1), we obtain that

$\displaystyle
s^{\alpha}\frac{\theta}{s-1}\int_{1}^{\infty}\left(1+\frac{t}{s-1}\right)^{-\theta-1}t^{-\alpha}\mathrm{d}t$
$\displaystyle=\theta
s^{\alpha}(s-1)^{\theta}\int_{0}^{1}x^{\theta+\alpha-1}(1+(s-1)x)^{-\theta-1}\mathrm{d}x$
$\displaystyle=\frac{\theta}{\theta+\alpha}s^{\alpha}(s-1)^{\theta}{}_{2}F_{1}(\theta+1,\theta+\alpha,\theta+\alpha+1,1-s).$

The Pfaff transformation (B.3) then gives that

$s^{\alpha}\frac{\theta}{s-1}\int_{1}^{\infty}\left(1+\frac{t}{s-1}\right)^{-\theta-1}t^{-\alpha}\mathrm{d}t=\frac{\theta}{\theta+\alpha}\left(\frac{s-1}{s}\right)^{\theta}{}_{2}F_{1}\left(\alpha+\theta,\alpha,1+\alpha+\theta,\frac{s-1}{s}\right).$
(3.13)

In particular, we see that seen as a function of $s\geq 1$, the above
expression is increasing and converges to (using the fact that $\alpha<1$ and
by (B.5))

$\frac{\theta}{\theta+\alpha}{}_{2}F_{1}\left(\alpha+\theta,\alpha,1+\alpha+\theta,1\right)=\frac{\theta}{\theta+\alpha}\frac{\Gamma(1+\alpha+\theta)\Gamma(1-\alpha)}{\Gamma(1+\theta)}=c(\alpha,\theta)$

where $c(\alpha,\theta)$ is the constant defined in (3.11). Wrapping things
up, we have proved that

$\left\|T(f_{1})-T(f_{2})\right\|_{\alpha}\leq
c(\alpha,\theta)\left\|f_{1}-f_{2}\right\|_{\alpha}.$

This concludes the proof that $T:\mathcal{F}_{\alpha}\to\mathcal{F}_{\alpha}$
is $c(\alpha,\theta)$-Lipschitz.

The existence of a unique fixed point of $T$ then follows from the Banach
fixed-point theorem and from the fact that $\mathcal{F}_{\alpha}$ is a
complete metric space (which in turn follows from the classical fact that
$L^{\infty}$ is a complete space). This proves the lemma. ∎

We now have all the ingredients to prove Theorem 3.1.

###### Proof of Theorem 3.1.

Lemma 3.2 proves tightness of $(F_{\delta})_{\delta>0}$. Corollary 3.4 proves
that any subsequential limit $F_{\infty}$ satisfies $S(F_{\infty})=F_{\infty}$
where we recall that $S$ is defined in (3.10). Moreover, any such
subsequential limit belongs to $\mathcal{G}_{1/2}$ since it can be bounded by
the analogous quantity with intensity $\theta=1/2$ for which the decay rate is
known (see Corollary 2.7). Finally, Lemma 3.5 shows that
$F_{\infty}=F_{\alpha}$ for any $\alpha<1-\theta$ where $F_{\alpha}$ is (as in
Lemma 3.5) the unique fixed-point of $S$ which belongs to
$\mathcal{G}_{\alpha}$. This concludes the proof. ∎

Let $f_{\infty}$ be the limiting function appearing in Theorem 1.3. As a
direct consequence of Lemma 3.5, $f_{\infty}$ belongs to
$\bigcap_{\alpha<1-\theta}\mathcal{F}_{\alpha}$, that is to say $f_{\infty}$
decays faster $s^{-\alpha}$ for all $\alpha<1-\theta$. In the next section, we
will identify precisely $f_{\infty}$ and we will in particular obtain that
$f_{\infty}$ decays exactly like a constant times $s^{\theta-1}$. In Appendix
A, we give an independent short proof of the fact that $f_{\infty}$ does not
decay much faster than $s^{-1+\theta}$ that does not rely on this explicit
expression.

###  3.4 $f_{\infty}$ in terms of Bessel processes

In this section we prove Theorem 1.8. Recall that we denote by
$(R_{t}^{(\theta)})_{t\geq 0}$ a squared Bessel process of dimension
$2\theta$, reflected at the origin, with $R_{0}^{(\theta)}=0$ a.s. Denote by

$f_{\rm Bes}:s\in[1,\infty)\mapsto\mathbb{P}\left(\forall
t\in[1,s],R_{t}^{(\theta)}>0\right).$

We first recall the explicit expression of the above probability and show that
$f_{\rm Bes}$ is also a fixed-point of $T$.

######  Lemma 3.6.

Let $\theta\in(0,1)$. For all $s\geq 1$,

$f_{\rm
Bes}(s)=\frac{\sin(\pi\theta)}{\pi}\int_{s-1}^{\infty}t^{\theta-1}(t+1)^{-1}\mathrm{d}t.$
(3.14)

Moreover, $Tf_{\rm Bes}=f_{\rm Bes}$.

###### Proof.

The last hitting time of 0 of $R^{(\theta)}$ before $s$, divided by $s$,
follows the generalised arcsine law (1.11) with parameter $1-\theta$; see e.g.
[7, Proposition 3.1]. The left hand side of (3.14) corresponds to the
probability that this last hitting time is smaller than 1. Integrating (1.11)
and then performing a change of variable concludes the proof of (3.14).

We now prove that $Tf_{\rm Bes}=f_{\rm Bes}$. It is likely that this can be
proved directly from the explicit expression of $f_{\rm Bes}$ and certain
identities between special functions. We prefer to take a different route that
explains what this integral equation means probabilistically for the Bessel
process. We can decompose (see [47])

$(R_{t}^{(\theta)})_{t\geq
1}\quad\overset{\mathrm{(d)}}{=}\quad(R_{t-1}^{(\theta)})_{t\geq
1}+(R^{(0)}_{t-1})_{t\geq 1}$

where on the right hand side $R^{(\theta)}$ and $R^{(0)}$ are independent and
are distributed as follows: $R^{(\theta)}$ is as before a squared Bessel
process of dimension $2\theta$ with $R_{0}^{(\theta)}=0$ a.s., and $R^{(0)}$
is a zero-dimensional squared Bessel process with $R_{0}^{(0)}\sim
R_{1}^{(\theta)}$. Elaborating on the discussion that follows Theorem 1.7, the
first term on the right hand side can be thought of as the local time of the
(large) loops that stay inside $\delta\mathbb{D}$, whereas the second term
corresponds to the local time of the loops that hit $\delta\partial\mathbb{D}$
(excursions). Let $\tau:=\inf\\{t>0:R^{(0)}_{t}=0\\}$ and fix $s\geq 1$. Using
the above decomposition, we can write

$f_{\rm Bes}(s)=\mathbb{P}\left(\tau\geq
s-1\right)+\int_{0}^{s-1}\mathbb{P}\left(\tau\in\mathrm{d}u\right)\mathbb{P}\left(\forall
v\in[u,s-1],R_{v}^{(\theta)}>0\right).$ (3.15)

By Brownian scaling, the probability $\mathbb{P}\left(\forall
v\in[u,s-1],R_{v}^{(\theta)}>0\right)$ is equal to $f_{\rm Bes}((s-1)/u)$. To
conclude that $Tf_{\rm Bes}=f_{\rm Bes}$, we need to compute the density of
$\tau$. Conditionally on $R_{0}^{(0)}=x^{2}$, the density of $\tau$ is given
by (see [23, Proposition 2.9])

$\mathbb{P}\left(\tau\in\mathrm{d}u|R_{0}^{(0)}=x^{2}\right)=\frac{1}{2}x^{2}u^{-2}e^{-x^{2}/(2u)}\mathbf{1}_{\left\\{u>0\right\\}}\mathrm{d}u.$

Using the explicit density of $R_{0}^{(0)}\sim R_{1}^{(\theta)}$ (see [23,
Equation (37)]), we obtain that

$\displaystyle\mathbb{P}\left(\tau\in\mathrm{d}u\right)$
$\displaystyle=\frac{1}{\Gamma(\theta)}2^{-\theta}u^{-2}\int_{0}^{\infty}\mathrm{d}x~{}x^{2\theta+1}e^{-\frac{u+1}{u}\frac{x^{2}}{2}}\mathbf{1}_{\left\\{u>0\right\\}}\mathrm{d}u$
$\displaystyle=\frac{1}{\Gamma(\theta)}u^{-1+\theta}(u+1)^{-1-\theta}\int_{0}^{\infty}\mathrm{d}y~{}y^{\theta}e^{-y}\mathbf{1}_{\left\\{u>0\right\\}}\mathrm{d}u=\theta
u^{-1+\theta}(u+1)^{-1-\theta}\mathbf{1}_{\left\\{u>0\right\\}}\mathrm{d}u$

thanks to the change of variable $y=\frac{u+1}{u}\frac{x^{2}}{2}$. In
particular,

$\mathbb{P}\left(\tau\geq
s-1\right)=\theta\int_{s-1}^{\infty}u^{-1+\theta}(u+1)^{-1-\theta}\mathrm{d}u=1-\Big{(}\frac{s-1}{s}\Big{)}^{\theta}.$

Going back to (3.15) and then performing the change of variable $t=(s-1)/u$,
we obtain that

$\displaystyle f_{\rm Bes}(s)$
$\displaystyle=1-\Big{(}\frac{s-1}{s}\Big{)}^{\theta}+\theta\int_{0}^{s-1}u^{-1+\theta}(u+1)^{-1-\theta}f_{\rm
Bes}\Big{(}\frac{s-1}{u}\Big{)}\mathrm{d}u$
$\displaystyle=1-\Big{(}\frac{s-1}{s}\Big{)}^{\theta}+\theta(s-1)^{\theta}\int_{1}^{\infty}(s+t-1)^{-1-\theta}f_{\rm
Bes}(t)\mathrm{d}t.$

This proves that $Tf_{\rm Bes}=f_{\rm Bes}$. ∎

Theorem 1.8 follows:

###### Proof of Theorem 1.8.

By Lemma 3.6, $f_{\rm Bes}$ is a fixed-point of $T$. The explicit expression
(3.14) of $f_{\rm Bes}$ also shows that it belongs to $\mathcal{F}_{\alpha}$
(3.7) for all $\alpha<1-\theta$. But by Lemma 3.5, $f_{\infty}$ is the only
fixed point of $T$ that belongs to these spaces. Hence $f_{\infty}=f_{\rm
Bes}$. ∎

###  3.5 Probability of large crossings – Proof of Theorem 1.2

This section is dedicated to the proof of Theorem 1.2. It will be divided into
two parts. The first part will prove the lower bound and the second part will
be focused on the upper bound.

#### Lower bound

We start by showing some supermultiplicativity-type estimate. This estimate is
a quick consequence of the FKG inequality.

######  Lemma 3.7.

There exists $c>0$ such that for all $\eta>0$, $\varepsilon$ small enough and
$s,t>1$,

$p_{\varepsilon,\varepsilon^{s+\eta}}(p_{\varepsilon^{s},\varepsilon^{s+t}}-\varepsilon^{c\eta})\leq
p_{\varepsilon,\varepsilon^{s+t}}.$

###### Proof.

Let us denote by $E$ the event that there exists a loop
$\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}$ which surrounds
$\varepsilon^{s+\eta}\mathbb{D}$ while staying in $\varepsilon^{s}\mathbb{D}$.
By scaling and by Lemma 2.5, there exists $c>0$ such that for all
$\varepsilon$ small enough $\mathbb{P}\left(E\right)\geq
1-\varepsilon^{c\eta}$. We have

$\displaystyle\frac{p_{\varepsilon,\varepsilon^{s+t}}}{p_{\varepsilon,\varepsilon^{s+\eta}}}$
$\displaystyle=\mathbb{P}\left(\varepsilon\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s+t}\partial\mathbb{D}\Big{|}\varepsilon\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s+\eta}\partial\mathbb{D}\right)\geq\mathbb{P}\left(E,\varepsilon^{s}\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s+t}\partial\mathbb{D}\Big{|}\varepsilon\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s+\eta}\partial\mathbb{D}\right).$

Now, by FKG inequality, the above probability is larger than the same one
without conditioning. Therefore,

$\frac{p_{\varepsilon,\varepsilon^{s+t}}}{p_{\varepsilon,\varepsilon^{s+\eta}}}\geq
p_{\varepsilon^{s},\varepsilon^{s+t}}-\mathbb{P}\left(E^{c}\right)$

which concludes the proof. ∎

######  Corollary 3.8.

The function $f_{\infty}:[1,\infty)\to(0,1]$ is supermultiplicative: for all
$s,t\geq 1$, $f_{\infty}(s)f_{\infty}(t)\leq f_{\infty}(st).$ Therefore, there
exists $\xi_{1}>0$ such that $f_{\infty}(s)=s^{-{\xi_{1}}+o(1)}$ as
$s\to\infty$.

###### Proof.

By Lemma 3.7, for all $s,t\geq 1$, $t^{\prime}>t$,
$f_{\infty}(s)f_{\infty}(t^{\prime})\leq f_{\infty}(st).$ We then obtain the
statement by continuity of $f_{\infty}$. The existence of the limit of $\log
f_{\infty}(s)/\log s$ as $s\to\infty$ is a consequence of Fekete’s subadditive
lemma. ∎

Note that thanks to Theorem 1.8 we already know that $\xi_{1}=1-\theta$. In
the remaining of this section we will still write $\xi_{1}$ instead of
$1-\theta$ to emphasise that the argument does not rely on the exact value of
the exponent $\xi_{1}$.

We are now ready to prove the lower bound part of Theorem 1.2.

###### Proof of Theorem 1.2 – Lower bound.

Let $\eta>0$ be a small parameter. Let $s>1$ be large enough so that for all
$t>s$, $f_{\infty}(t)\geq t^{-{\xi_{1}}-\eta}$. Let $c>0$ be the constant
appearing in Lemma 3.7. By Theorem 3.1,

$p_{\varepsilon,\varepsilon^{s(1+\eta)}}-\varepsilon^{c\eta}\xrightarrow[\varepsilon\to
0]{}f_{\infty}(s(1+\eta)).$

We can therefore pick $\varepsilon_{0}$ small enough so that for all
$\varepsilon\in(0,\varepsilon_{0})$,
$p_{\varepsilon,\varepsilon^{s(1+\eta)}}-\varepsilon^{\eta}\geq(1-\eta)f_{\infty}(s(1+\eta))$.
Let $\varepsilon>0$ be much smaller than $\varepsilon_{0}$. Let $k$ be the
largest integer so that
$\tilde{\varepsilon}_{0}:=\varepsilon^{1/s^{k}}\leq\varepsilon_{0}$, i.e. let

$k=\left\lfloor\frac{1}{\log
s}\log\frac{|\log\varepsilon|}{|\log\varepsilon_{0}|}\right\rfloor.$

We apply successively Lemma 3.7:

$\displaystyle p_{\varepsilon^{1/s^{k}},\varepsilon}$
$\displaystyle\geq\left(p_{\varepsilon^{1/s^{k}},\varepsilon^{(1+\eta)/s^{k-1}}}-\varepsilon^{\eta/s^{k-1}}\right)p_{\varepsilon^{1/s^{k-1}},\varepsilon}\geq\dots\geq\prod_{i=1}^{k}\left(p_{\varepsilon^{1/s^{i}},\varepsilon^{(1+\eta)/s^{i-1}}}-\varepsilon^{\eta/s^{i-1}}\right)$
$\displaystyle\geq\left((1-\eta)f_{\infty}((1+\eta)s)\right)^{k}\geq\left((1-\eta)\left\\{(1+\eta)s\right\\}^{-{\xi_{1}}-\eta}\right)^{k}.$

The main contribution comes from

$s^{-{\xi_{1}}k}\geq\left(\frac{|\log\varepsilon|}{|\log\varepsilon_{0}|}\right)^{-{\xi_{1}}}$

and we have shown that for some $\delta=\delta(\eta)$ going to zero as
$\eta\to 0$,
$p_{\tilde{\varepsilon}_{0},\varepsilon}\geq|\log\varepsilon|^{-{\xi_{1}}-\delta}.$
Since $\tilde{\varepsilon}_{0}\geq(\varepsilon_{0})^{s}$ is macroscopic, we
obtain by gluing a few extra loops and FKG inequality that

$p_{e^{-1},\varepsilon}\geq|\log\varepsilon|^{-{\xi_{1}}-\delta^{\prime}}$

where $\delta^{\prime}$ is arbitrarily small. It concludes the proof. ∎

#### Upper bound

To obtain an upper bound we need to establish a version of Lemma 3.7 in the
other direction. To achieve this, we need to find two independent collections
of loops whose clusters make non trivial crossings. This is very much similar
in spirit to the BK inequality in percolation and we state this reverse
inequality as:

######  Lemma 3.9.

Let $s>1$ and $k_{0}\geq 1$. For any $n\geq k_{0}+1$ and $\varepsilon>0$,

$\displaystyle p_{\varepsilon,\varepsilon^{s^{n}}}$
$\displaystyle\leq\sum_{k=1}^{k_{0}}p_{\varepsilon,\varepsilon^{s^{k}}}p_{\varepsilon^{s^{k+1}},\varepsilon^{s^{n}}}+\sum_{k=k_{0}+1}^{n-2}\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\varepsilon^{s}\mathbb{D}\setminus\varepsilon^{s^{k}}\right)p_{\varepsilon^{s^{k+1}},\varepsilon^{s^{n}}}$
(3.16)
$\displaystyle\quad+\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\varepsilon^{s}\mathbb{D}\setminus\varepsilon^{s^{n-1}}\mathbb{D}\right).$

###### Proof.

We explore the cluster of the circle $\varepsilon\partial\mathbb{D}$ as
follows. We start with $\mathcal{C}_{0}=\varepsilon\partial\mathbb{D}$ and for
all $i\geq 1$, we explore the loops that touch $\mathcal{C}_{i-1}$ and we let
$\mathcal{C}_{i}$ be the union of $\mathcal{C}_{i-1}$ and the newly revealed
loops. We stop the exploration at the first time $i_{*}$ that the cluster hits
$\varepsilon^{s}\partial\mathbb{D}$:

$i_{*}=\inf\\{i\geq
1:\mathcal{C}_{i}\cap\varepsilon^{s}\partial\mathbb{D}\neq\varnothing\\}.$

Let

$k_{*}=\max\\{k\geq
1:\mathcal{C}_{i_{*}}\cap\varepsilon^{s^{k}}\partial\mathbb{D}\neq\varnothing\\}.$

By definition, we have not revealed the loops that hit the disc of radius
$\varepsilon^{s^{k_{*}+1}}$. Therefore, for all $k=1\dots n-2$,

$\mathbb{P}\left(\varepsilon\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon^{s^{n}}|k_{*}=k\right)\leq
p_{\varepsilon^{s^{k+1}},\varepsilon^{s^{n}}}.$

With a union bound this gives,

$\displaystyle p_{\varepsilon,\varepsilon^{s^{n}}}$
$\displaystyle\leq\sum_{k=1}^{n-2}\mathbb{P}\left(k_{*}=k\right)p_{\varepsilon^{s^{k+1}},\varepsilon^{s^{n}}}+\mathbb{P}\left(k_{*}\geq
n-1\right).$

We bound $\mathbb{P}\left(k_{*}=k\right)\leq\mathbb{P}\left(k_{*}\geq
k\right)$ and when $k\leq k_{0}$ we bound this probability by
$p_{\varepsilon,\varepsilon^{s^{k}}}$. When $k\geq k_{0}+1$, we note that
$k_{*}$ can be larger or equal than $k$ only if there is a _loop_ crossing the
annulus $\varepsilon^{s}\mathbb{D}\setminus\varepsilon^{s^{k}}\mathbb{D}$.
This gives (3.16). ∎

We can now conclude this section with a proof of the upper bound in Theorem
1.2.

###### Proof of Theorem 1.2 – Upper bound.

Let $\delta>0$. Let $k_{0}$ be a large integer and $s>1$. Let
$\varepsilon_{0}$ be small enough so that for all $k=1\dots k_{0}$, for all
$\varepsilon\in(0,\varepsilon_{0})$,

$p_{\varepsilon,\varepsilon^{s^{k}}}\leq(s^{k})^{-{\xi_{1}}+\delta}.$ (3.17)

By Lemma 2.2,

$\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\varepsilon^{s}\mathbb{D}\setminus\varepsilon^{s^{k}}\mathbb{D}\right)=(1+o(1))\theta
s^{-(k-1)},$

where $o(1)\to 0$ as $\varepsilon\to 0$, uniformly in $k\geq 2$. We can assume
that $\varepsilon_{0}$ has been chosen small enough so that for all $k\geq 2$,
for all $\varepsilon\in(0,\varepsilon_{0})$,

$\mathbb{P}\left(\exists\wp\in\mathcal{L}_{\mathbb{D}}^{\theta}\text{~{}crossing~{}}\varepsilon^{s}\mathbb{D}\setminus\varepsilon^{s^{k}}\mathbb{D}\right)\leq
2\theta s^{-(k-1)}.$

We now apply Lemma 3.9 with $s$, $k_{0}$, $\varepsilon_{0}$ (instead of
$\varepsilon$) and some integer $n\geq k_{0}+2$:

$p_{\varepsilon_{0},\varepsilon_{0}^{s^{n}}}\leq\sum_{k=1}^{k_{0}}s^{-({\xi_{1}}-\delta)k}p_{\varepsilon_{0}^{s^{k+1}},\varepsilon_{0}^{s^{n}}}+2\theta\sum_{k=k_{0}+1}^{n-2}s^{-(k-1)}p_{\varepsilon_{0}^{s^{k+1}},\varepsilon_{0}^{s^{n}}}+2\theta
s^{-(n-2)}.$

We assume that $k_{0}$ is large enough to ensure that for all $k\geq k_{0}+1$,
$2\theta s^{-(k-1)}\leq s^{-({\xi_{1}}-\delta)k}$. The above inequality then
implies that for all $n\geq k_{0}+2$,

$p_{\varepsilon_{0},\varepsilon_{0}^{s^{n}}}+s^{-({\xi_{1}}-\delta)n}\leq\sum_{k=1}^{n-2}s^{-({\xi_{1}}-\delta)k}\left(p_{\varepsilon_{0}^{s^{k+1}},\varepsilon_{0}^{s^{n}}}+s^{-({\xi_{1}}-\delta)(n-k-1)}\right).$
(3.18)

We now iterate this inequality until only terms of the form
$p_{\varepsilon,\varepsilon^{s^{k}}}$ for some
$\varepsilon\in(0,\varepsilon_{0}]$ and $k\in\\{1,\dots k_{0}\\}$ remain.
Thanks to (3.17), we can bound these final terms by $s^{-({\xi_{1}}-\delta)k}$
and we obtain an estimate of the form

$p_{\varepsilon_{0},\varepsilon_{0}^{s^{n}}}+s^{-({\xi_{1}}-\delta)n}\leq
2\sum\prod_{i}s^{-({\xi_{1}}-\delta)k_{i}}$ (3.19)

Instead of describing precisely this summation, we make the following two key
observations:

  * •

The total number of terms involved in the sum is at most $C^{n}$ for some
$C>0$ independent of $\delta$, $s$ and $k_{0}$. This is due to the fact that
if a sequence $(S_{n})_{n\geq 1}$ of integers satisfies
$S_{n}\leq\sum_{k=1}^{n-1}S_{k}$ for all $n\geq 2$, then $(S_{n})$ grows at
most exponentially in $n$.

  * •

Each time that we use the inequality (3.18), the sum of the ratios of the
logarithms of the radii decreases by one. In (3.18), this is saying that
$k+(n-k-1)=n-1$. Because we do not use this inequality more than $n/k_{0}$
times, in each term of the sum in (3.19), we have $\sum_{i}k_{i}\geq
n-n/k_{0}$.

Putting things together, we obtain that

$p_{\varepsilon_{0},\varepsilon_{0}^{s^{n}}}\leq
2C^{n}s^{-({\xi_{1}}-\delta)(1-1/k_{0})n}=2s^{-\\{({\xi_{1}}-\delta)(1-1/k_{0})-\frac{\log
C}{\log s}\\}n}.$

The probability $p_{e^{-1},\varepsilon_{0}^{s^{n}}}$ is smaller than
$p_{\varepsilon_{0},\varepsilon_{0}^{s^{n}}}$ (if $\varepsilon_{0}\leq
e^{-1}$) and we have proven that for all $s>1$, $\delta>0$, $k_{0}\geq 1$
large enough, for all $\varepsilon_{0}$ small enough and $n\geq k_{0}+2$,
letting $r=\varepsilon_{0}^{s^{n}}$,

$p_{e^{-1},r}\leq 2\left(|\log\varepsilon_{0}||\log
r|\right)^{-({\xi_{1}}-\delta)(1-1/k_{0})+\frac{\log C}{\log s}}.$

It concludes the proof. ∎

###  3.6 Percolation of large loops

We now turn to the proof of the phase transition at $\theta=1$ of the
percolative behaviour of large loops, i.e. the proof of Theorem 1.9.

###### Proof of Theorem 1.9.

The proof of this result follows the exact same strategy as the proof of
Theorem 1.3. Let $\theta>0$ and $\tau\geq 0$. We consider the functions

$\hat{F}_{T}:(s,t)\in
I_{\leq}^{*}\mapsto\mathbb{P}\left(\\{sT\\}\times\mathbb{S}^{1}\overset{\mathcal{L}_{\mathtt{C},\tau}^{\theta}}{\longleftrightarrow}\\{tT\\}\times\mathbb{S}^{1}\right).$

We show that $(\hat{F}_{T})_{T\geq 1}$ is tight (in the same sense as Lemma
3.2) and that every subsequential limit $\hat{F}$ satisfies $S\hat{F}=\hat{F}$
where $S$ is defined in (3.10). These results are proved exactly like Lemma
3.2 and Corollary 3.4. Crucially, the proofs of Lemma 3.2 and Corollary 3.4
only use large loops to create additional connections. What remains to be
discussed is the identification of the subsequential limit $\hat{F}$. This is
where we need to distinguish different cases.

$\bullet~{}\theta\in(0,1/2]$: By Corollary 2.7, $\hat{F}$ belongs to
$\mathcal{G}_{1/2}$ (3.9). Lemma 3.5 states that there is a unique fixed point
of $S$ in $\mathcal{G}_{1/2}$. Lemma 3.6 then concludes the identification of
the limit with the right hand side of (1.13).

$\bullet~{}\theta\in(1/2,1)$: To use Lemma 3.5, we need to guarantee a certain
decay of $\hat{F}$ and this is where we need to assume that the threshold
$\tau$ is larger than $\tau_{\theta}$ (1.12). We cannot use the comparison
with the case $\theta=1/2$ any more. However, the restriction to large loops
allows us to use the comparison (1.10) with the 1D loop soup. Indeed, if there
is a crossing of $[sT,tT]\times\mathbb{S}^{1}$ by a cluster of
$\mathcal{L}_{\mathtt{C},\tau}^{\theta}$, the longitudinal projection of the
loops have to cross $[sT,tT]$. Let

$\lambda:=2\pi\theta p_{\mathbb{S}^{1}}(\tau,1,1).$

By definition of $\tau_{\theta}$, $\lambda<1$ and by (1.10),

$\theta\boldsymbol{\pi}_{*}\mu^{\rm loop}_{\mathtt{C}}(\mathrm{d}\wp^{\rm
1D})\mathbf{1}_{\left\\{T(\wp)>\tau\right\\}}\leq\lambda\mu^{\rm
loop}_{\mathbb{R}^{+}}(\mathrm{d}\wp^{\rm 1D}).$

Hence, the probability that there is a cluster of
$\mathcal{L}_{\mathtt{C},\tau}^{\theta}$ crossing
$[sT,tT]\times\mathbb{S}^{1}$ is bounded by the probability that a cluster of
$\mathcal{L}_{\mathbb{R}^{+}}^{\lambda}$ crosses $[sT,tT]$. The local time of
a 1D loop soup having the law of a Bessel process [34], the latter is equal to

$\displaystyle\mathbb{P}\left(\forall
x\in[sT,tT],R^{(\lambda)}_{x}>0\right)=\mathbb{P}\left(\forall
x\in[1,t/s],R^{(\lambda)}_{x}>0\right)=\frac{\sin(\pi\lambda)}{\pi}\int_{t/s-1}^{\infty}u^{\lambda-1}(u+1)^{-1}\mathrm{d}u,$

by Brownian scaling and Lemma 3.6. In particular, any subsequential limit
$\hat{F}$ of $(\hat{F}_{T})_{T\geq 1}$ belongs to $\mathcal{G}_{\alpha}$ for
all $\alpha\in(0,1-\lambda)$. We can then conclude the proof as in the case
$\theta\in(0,1/2]$.

$\bullet~{}\theta\geq 1$: If $\theta>1$, Lemma 3.5 shows that $(s,t)\mapsto 1$
is the unique fixed-point of $S$ in $[0,1]^{I_{\leq}^{*}}$. This identifies
the limit. Alternatively, and in order to cover the critical case $\theta=1$,
we can proceed as follows. Let $\tau\geq 0$ be any threshold,
$\theta^{\prime}\in(1/2,1)$ be very close to one and
$\tau^{\prime}>\tau\vee\tau_{\theta^{\prime}}$. We have

$\displaystyle\liminf_{\mathbb{T}\to\infty}\mathbb{P}\left(\\{sT\\}\times\mathbb{S}^{1}\overset{\mathcal{L}_{\mathtt{C},\tau}^{\theta=1}}{\longleftrightarrow}\\{tT\\}\times\mathbb{S}^{1}\right)$
$\displaystyle\geq\lim_{\mathbb{T}\to\infty}\mathbb{P}\left(\\{sT\\}\times\mathbb{S}^{1}\overset{\mathcal{L}_{\mathtt{C},\tau^{\prime}}^{\theta^{\prime}}}{\longleftrightarrow}\\{tT\\}\times\mathbb{S}^{1}\right)$
$\displaystyle=\frac{\sin(\pi\theta^{\prime})}{\pi}\int_{t/s-1}^{\infty}u^{\theta^{\prime}-1}(u+1)^{-1}\mathrm{d}u.$

We then conclude by noticing that the above display converges to 1 as
$\theta^{\prime}\to 1^{-}$. ∎

##  4 Polar sets and capacity

This section is dedicated to the proof of Theorem 1.5. By Theorem 1.2, we
already know with great accuracy the probability that a large cluster gets
$\varepsilon$-close to a given point. In order to prove Theorem 1.5, we will
also need a two-point estimate that we encapsulate in the following result. We
will denote by

$Z_{\varepsilon}:=\mathbb{P}\Big{(}e^{-1}\partial\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\varepsilon\mathbb{D}\Big{)}.$
(4.1)

######  Proposition 4.1.

Let $r_{0}>0$ be a fixed macroscopic radius. For all $\eta>0$, there exists
$C>0$ such that for all $x,y\in\mathbb{D}$, $\varepsilon>0$,

$\frac{1}{Z_{\varepsilon}^{2}}\mathbb{P}\left(D(x,\varepsilon)\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}D(y,\varepsilon)\right)\leq
C|\log|x-y||^{2(1-\theta)+\eta}$ (4.2)

and

$\frac{1}{Z_{\varepsilon}^{2}}\mathbb{P}\left(D(x,\varepsilon)\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}D(y,\varepsilon),D(x,\varepsilon)\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\partial
D(x,r_{0})\right)\leq C|\log|x-y||^{1-\theta+\eta}.$ (4.3)

Note that if $x$ and $y$ are in the bulk of $\mathbb{D}$ (i.e. at a
macroscopic distance to $\partial\mathbb{D}$), then the reverse inequalities
also hold. The lower bounds are much simpler to establish by FKG-inequality.
We do not write these details since we will only need the upper bounds.

In order to prove this proposition, we will need precise estimates on the
probability of crossing two annuli with loops. These are the two-point
analogues of the estimate (2.3) and are contained in Section 4.1. In Section
4.2 we will then prove Proposition 4.1. Finally, we will prove Theorem 1.5 in
Section 4.3. The proof of Theorem 1.5 is fairly standard once Theorem 1.2 and
Proposition 4.1 are established.

###  4.1 Crossing two annuli with loops

######  Lemma 4.2.

Let $A>0$ be large. Let $x,y\in\mathbb{C}$ and $r_{x},r_{y},R>0$ such that
$A|x-y|\leq R\leq 1$ and $\max(r_{x},r_{y})\leq|x-y|^{A}$. Then

$\mu^{\rm
loop}_{D(x,1)}(\\{\wp\mathrm{~{}visiting~{}}D(x,r_{x})\mathrm{~{}and~{}}D(y,r_{y})\\})=\left(1+o(1)\right)\frac{(\log|x-y|)^{2}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}$
(4.4)

and

$\displaystyle\mu^{\rm
loop}_{D(x,1)}(\\{\wp\mathrm{~{}visiting~{}}D(x,r_{x}),D(y,r_{y})\mathrm{~{}and~{}}\partial
D(x,R)\\})=(1+o(1))\frac{\log\frac{1}{R}\log\frac{R}{|x-y|^{2}}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}$
(4.5)

where $o(1)\to 0$ as $A\to\infty$, uniformly in $x,y,r_{x},r_{y},R$ as above.

###### Proof.

By translation invariance we can assume that $x=0$. Rooting the loops at the
closest point to $x$, we have

$\mu^{\rm
loop}_{\mathbb{D}}=\frac{1}{\pi}\int_{0}^{1}r\mathrm{d}r\int_{0}^{2\pi}\mathrm{d}\theta\mu^{\rm
bub}_{\mathbb{D}\setminus r\bar{\mathbb{D}}}(re^{i\theta}).$ (4.6)

To justify this decomposition, we can for instance first root the loops in the
domain $\mathbb{C}\setminus\bar{\mathbb{D}}$ at the point whose distance to
the origin is maximal:

$\mu^{\rm
loop}_{\mathbb{C}\setminus\bar{\mathbb{D}}}=\frac{1}{\pi}\int_{1}^{\infty}R\mathrm{d}R\int_{0}^{2\pi}\mathrm{d}\theta^{\prime}\mu^{\rm
bub}_{R\mathbb{D}\setminus\bar{\mathbb{D}}}(re^{i\theta^{\prime}}).$

This decomposition follows from the whole plane version of (2.6) and then by
the restriction property of the loop and bubble measures. Let
$f:z\mapsto-1/z$. By conformal invariance of the loop measure,

$\mu^{\rm loop}_{\mathbb{D}}=f\circ\mu^{\rm
loop}_{\mathbb{C}\setminus\bar{\mathbb{D}}}=\frac{1}{\pi}\int_{1}^{\infty}R\mathrm{d}R\int_{0}^{2\pi}\mathrm{d}\theta^{\prime}f\circ\mu^{\rm
bub}_{R\mathbb{D}\setminus\mathbb{D}}(re^{i\theta^{\prime}}).$

The bubble measure is conformally covariant: $f\circ\mu^{\rm
bub}_{R\mathbb{D}\setminus\bar{\mathbb{D}}}(re^{i\theta^{\prime}})=\frac{1}{R^{4}}\mu^{\rm
bub}_{\mathbb{D}\setminus\frac{1}{R}\bar{D}}(-\frac{1}{R}e^{-i\theta^{\prime}})$.
The change of variable $r=1/R$ and $\theta=\pi-\theta^{\prime}$ finishes the
proof of (4.6).

Let $r\in(0,r_{x})$ and $\theta\in[0,2\pi]$. We compute

$\displaystyle\mu^{\rm bub}_{\mathbb{D}\setminus
r\bar{\mathbb{D}}}(re^{i\theta})(\\{\wp\text{~{}intersecting~{}}D(y,r_{y})\\})=\pi\int_{\partial
D(y,r_{y})}\mathrm{d}z_{y}H_{\mathbb{D}\setminus(r\mathbb{D}\cup
D(y,r_{y})}(re^{i\theta},z_{y})H_{\mathbb{D}\setminus
r\mathbb{D}}(z_{y},re^{i\theta}).$

For all $z_{y}\in\partial D(y,r_{y})$,

$H_{\mathbb{D}\setminus r\mathbb{D}}(z_{y},re^{i\theta})=(1+o(1))\frac{1}{2\pi
r}\frac{\log|x-y|}{\log r}.$

We can then integrate

$\displaystyle\int_{\partial
D(y,r_{y})}\mathrm{d}z_{y}H_{\mathbb{D}\setminus(r\mathbb{D}\cup
D(y,r_{y})}(re^{i\theta},z_{y})$ $\displaystyle=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\mathbb{P}_{(r+\varepsilon)e^{i\theta}}\left(\tau_{D(y,r_{y})}<\tau_{r\mathbb{D}}\wedge\tau_{\partial\mathbb{D}}\right).$
(4.7)

To compute the above hitting probability, we first stop the Brownian path at
the first exit time of $er\mathbb{D}$:

$\displaystyle\mathbb{P}_{(r+\varepsilon)e^{i\theta}}\left(\tau_{D(y,r_{y})}<\tau_{r\mathbb{D}}\wedge\tau_{\partial\mathbb{D}}\right)=\mathbb{E}_{(r+\varepsilon)e^{i\theta}}\left[\mathbf{1}_{\left\\{\tau_{er\partial\mathbb{D}}<\tau_{r\mathbb{D}}\right\\}}\mathbb{P}_{B_{\tau_{er\partial\mathbb{D}}}}\left(\tau_{D(y,r_{y})}<\tau_{r\mathbb{D}}\wedge\tau_{\partial\mathbb{D}}\right)\right].$

Starting from any point $w\in er\partial\mathbb{D}$, the probability that a
Brownian path hits the small disc $D(y,r_{y})$ before exiting the domain
$\mathbb{D}$ and before hitting $r\mathbb{D}$ can be accurately estimated:

$\mathbb{P}_{w}\left(\tau_{D(y,r_{y})}<\tau_{r\mathbb{D}}\wedge\tau_{\partial\mathbb{D}}\right)=(1+o(1))\frac{\log\frac{1}{|x-y|}}{\log\frac{1}{r_{y}}\log\frac{1}{r}-\left(\log\frac{1}{|x-y|}\right)^{2}}.$
(4.8)

See [19, Lemma 2.3]. This implies that the integral in (4.7) is equal to

$\displaystyle(1+o(1))\frac{\log\frac{1}{|x-y|}}{\log\frac{1}{r_{y}}\log\frac{1}{r}-\left(\log\frac{1}{|x-y|}\right)^{2}}\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\mathbb{P}_{(r+\varepsilon)e^{i\theta}}\left(\tau_{er\partial\mathbb{D}}<\tau_{r\mathbb{D}}\right)$
$\displaystyle=(1+o(1))\frac{1}{r}\frac{\log\frac{1}{|x-y|}}{\log\frac{1}{r_{y}}\log\frac{1}{r}-\left(\log\frac{1}{|x-y|}\right)^{2}}.$

Putting everything together, we obtain that

$\displaystyle\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\text{~{}intersecting~{}}r_{x}\mathbb{D}\text{~{}and~{}}D(y,r_{y})\\})$
$\displaystyle=(1+o(1))\Big{(}\log\frac{1}{|x-y|}\Big{)}^{2}\int_{0}^{r_{x}}\frac{\mathrm{d}r}{r}\frac{1}{\log\frac{1}{r}\big{(}\log\frac{1}{r_{y}}\log\frac{1}{r}-\big{(}\log\frac{1}{|x-y|}\big{)}^{2}\big{)}}$
$\displaystyle=-(1+o(1))\log\left(1-\frac{(\log|x-y|)^{2}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}\right)=(1+o(1))\frac{(\log|x-y|)^{2}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}.$

This concludes the proof of (4.4).

(4.5) follows from the observation that

$\displaystyle\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}visiting~{}}r_{x}\mathbb{D},D(y,r_{y})\mathrm{~{}and~{}}R\partial\mathbb{D}\\})$
$\displaystyle=\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}visiting~{}}r_{x}\mathbb{D},D(y,r_{y})\\})-\mu^{\rm
loop}_{R\mathbb{D}}(\\{\wp\mathrm{~{}visiting~{}}r_{x}\mathbb{D},D(y,r_{y})\\}).$

We have just computed the first right hand side term. By scaling, the second
term is equal to

$(1+o(1))\frac{\big{(}\log\frac{R}{|x-y|}\big{)}^{2}}{\log\frac{R}{r_{x}}\log\frac{R}{r_{y}}}=(1+o(1))\frac{\big{(}\log\frac{R}{|x-y|}\big{)}^{2}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}},$

implying that

$\displaystyle\mu^{\rm
loop}_{\mathbb{D}}(\\{\wp\mathrm{~{}visiting~{}}r_{x}\mathbb{D},D(y,r_{y})\mathrm{~{}and~{}}R\partial\mathbb{D}\\})=(1+o(1))\frac{\log\frac{1}{R}\log\frac{R}{|x-y|^{2}}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}.$

This concludes the proof of (4.5). ∎

######  Corollary 4.3.

Let $A>0$ large enough and $x,y,r_{x},r_{y},R$ be as in the statement of Lemma
4.2. There exists $C=C(A)>0$ such that the probability that there are loops
crossing $D(x,|x-y|/2)\setminus D(x,r_{x})$, $D(y,|x-y|/2)\setminus
D(y,r_{y})$ and $D(x,R)\setminus(D(x,|x-y|/2)\cup D(y,|x-y|/2))$ is bounded by

$C\frac{\log\frac{1}{R}\log\frac{R}{|x-y|^{2}}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}.$
(4.9)

We emphasise that the three crossings described above can be realised by a
single loop, or by two or three different loops.

###### Proof.

We handle each of the following scenarios separately:

$\bullet$ A single loop makes the three crossings. By (4.5), the probability
of that event is at most (4.9) for some $C>0$.

$\bullet$ One loop crosses $D(x,|x-y|/2)\setminus D(x,r_{x})$ and
$D(y,|x-y|/2)\setminus D(y,r_{y})$ and another loop crosses
$D(x,R)\setminus(D(x,|x-y|/2)\cup D(y,|x-y|/2))$. In that case, there must be
a loop visiting both $D(x,r_{x})$ and $D(y,r_{y})$ and another loop crossing
$D(x,R)\setminus D(x,3|x-y|/2)$. Using that for any (crossing) events $A,B$,

$\mathbb{P}\left(\exists\wp\neq\wp^{\prime}\in\mathcal{L}_{\mathbb{D}}^{\theta}:\wp\in
A,\wp^{\prime}\in
B\right)\leq\mathbb{E}\Big{[}\sum_{\wp\neq\wp^{\prime}}\mathbf{1}_{\left\\{\wp\in
A\right\\}}\mathbf{1}_{\left\\{\wp^{\prime}\in
B\right\\}}\Big{]}=\theta^{2}\mu^{\rm loop}_{D}(A)\mu^{\rm loop}_{D}(B),$

we can bound the probability of that event by

$\displaystyle\theta^{2}\mu^{\rm
loop}_{D}(\\{\wp\text{~{}visiting~{}}D(x,r_{x})\text{~{}and~{}}D(y,r_{y})\\})\mu^{\rm
loop}_{D}(\\{\wp\text{~{}crossing~{}}D(x,R)\setminus D(x,3|x-y|/2)\\}).$

By (2.3) and (4.4), this is at most

$C\frac{(\log|x-y|)^{2}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}\times\frac{\log\frac{1}{R}}{\log\frac{1}{|x-y|}}\leq
C\frac{\log\frac{1}{R}\log\frac{R}{|x-y|^{2}}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}.$

$\bullet$ One loop crosses $D(x,|x-y|/2)\setminus D(x,r_{x})$ and
$D(x,R)\setminus(D(x,|x-y|/2)\cup D(y,|x-y|/2))$ and another loop crosses
$D(y,|x-y|/2)\setminus D(y,r_{y})$. As above, the probability of that event
can be bounded by

$\displaystyle\theta^{2}\mu^{\rm
loop}_{D}(\\{\wp\text{~{}crossing~{}}D(x,R)\setminus D(x,r_{x})\\})\mu^{\rm
loop}_{D}(\\{\wp\text{~{}crossing~{}}D(y,|x-y|/2)\setminus D(y,r_{y})\\})$
$\displaystyle\leq
C\frac{\log\frac{1}{R}}{\log\frac{1}{r_{x}}}\times\frac{\log\frac{1}{|x-y|}}{\log\frac{1}{r_{y}}}C\frac{\log\frac{1}{R}\log\frac{R}{|x-y|^{2}}}{\log\frac{1}{r_{x}}\log\frac{1}{r_{y}}}$

using (2.3).

$\bullet$ One loop crosses $D(y,|x-y|/2)\setminus D(y,r_{x})$ and
$D(x,R)\setminus(D(x,|x-y|/2)\cup D(y,|x-y|/2))$ and another loop crosses
$D(x,|x-y|/2)\setminus D(x,r_{y})$. The probability of that event can be
bounded in a similar way as the previous one.

$\bullet$ The three crossings are made by three different loops. The
probability of that event is at most

$\displaystyle\theta^{3}\mu^{\rm
loop}_{D}(\\{\wp\text{~{}crossing~{}}D(x,|x-y|/2)\setminus
D(x,r_{x})\\})\mu^{\rm
loop}_{D}(\\{\wp\text{~{}crossing~{}}D(y,|x-y|/2)\setminus D(y,r_{y})\\})$
$\displaystyle\times\mu^{\rm
loop}_{D}(\\{\wp\text{~{}crossing~{}}D(x,R)\setminus D(x,3|x-y|/2)\\})$

which is again bounded by the same expression. ∎

###  4.2 Proof of Proposition 4.1

###### Proof of Proposition 4.1.

Let us first notice that (4.2) is a consequence of (4.3). Indeed, by FKG
inequality the left hand side of (4.3) is at least the left hand side of (4.2)
times the probability that
$D(x,|x-y|/4)\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\partial
D(x,r_{0})$ and that there is a loop included in $D(x,|x-y|/2)$ surrounding
$D(x,|x-y|/4)$. By Theorem 1.2, this shows that the left hand side of (4.3) is
at least the left hand side of (4.2) times $|\log|x-y||^{-1+\theta+o(1)}$.

We now focus on proving (4.3). Let $\delta>0$ small. First of all, if
$|x-y|\leq\exp\left(-|\log\varepsilon|^{1-\delta}\right)$, then we simply
bound the left hand side of (4.3) by

$\frac{1}{Z_{\varepsilon}^{2}}\mathbb{P}\left(D(x,\varepsilon)\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\partial
D(x,r_{0})\right)\leq\frac{C}{Z_{\varepsilon}}\leq|\log\varepsilon|^{1-\theta+o(1)}\leq|\log|x-y||^{(1-\theta)/(1-2\delta)}.$

In the remaining of the proof, we focus on the case
$|x-y|\geq\exp\left(-|\log\varepsilon|^{1-\delta}\right)$. For
$z\in\\{x,y\\}$, let $D_{z}$ be the disc $D(z,|x-y|/2)$, and
$\mathcal{L}_{x,y}$ the collection of loops in
$\mathcal{L}_{\mathbb{D}}^{\theta}$ that intersect $\partial D_{x}\cup\partial
D_{y}$. Define also

$r_{z}=\inf\\{r>0:\exists\wp\in\mathcal{L}_{x,y}\mathrm{~{}intersecting~{}}\partial
D(z,r)\\},\quad\quad z=x,y$

and

$R=\sup\\{r>0:\exists\wp\in\mathcal{L}_{x,y}\mathrm{~{}intersecting~{}}\partial
D(x,r)\\}.$

Conditioned on $r_{x},r_{y}$ and $R$, in order to have
$D(x,\varepsilon)\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}D(y,\varepsilon)$
and
$D(x,\varepsilon)\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\partial
D(x,r_{0})$, there must be a cluster of loops included in $D_{z}$ that crosses
$D(z,r_{z})\setminus D(z,\varepsilon)$, $z=x,y$ and a cluster of loops
included in $D\setminus(\bar{D}_{x}\cup\bar{D}_{y})$ that crosses
$D(x,r_{0})\setminus D(x,R)$. By independence of the different involved sets
of loops, we therefore obtain that the left hand side of (4.3) is at most

$\displaystyle\frac{1}{Z_{\varepsilon}^{2}}\mathbb{E}\left[\prod_{z=x,y}\mathbb{P}\left(D(z,\varepsilon)\overset{\mathcal{L}_{D_{z}}^{\theta}}{\longleftrightarrow}\partial
D(z,r_{z})\Big{|}r_{z}\right)\mathbb{P}\left(D(x,R)\overset{\mathcal{L}_{\mathbb{D}\setminus(\bar{D}_{x}\cup\bar{D}_{y})}^{\theta}}{\longleftrightarrow}\partial
D(x,r_{0})\Big{|}R\right)\right].$

Let $z\in\\{x,y\\}$. By scaling,

$\displaystyle\frac{1}{Z_{\varepsilon}}\mathbb{P}\left(D(z,\varepsilon)\overset{\mathcal{L}_{D_{z}}^{\theta}}{\longleftrightarrow}\partial
D(z,r_{z})\Big{|}r_{z}\right)=\frac{1}{Z_{\varepsilon}}\mathbb{P}\left(\frac{2\varepsilon}{|x-y|}\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}\frac{2r_{z}}{|x-y|}\partial\mathbb{D}\Big{|}r_{z}\right).$

Let us assume for a moment that $r_{z}\leq e^{-1}|x-y|/2$. By definition of
$Z_{\varepsilon}$, the above right hand side term is the inverse of the
probability that $\varepsilon\mathbb{D}$ is connected to
$e^{-1}\partial\mathbb{D}$, conditionally on the fact that
$2\varepsilon/|x-y|\mathbb{D}$ is connected to $2r_{z}/|x-y|$. By FKG
inequality, this is at most

$\mathbb{P}\left(\varepsilon\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}2\varepsilon/|x-y|\partial\mathbb{D}\right)^{-1}\mathbb{P}\left(\frac{2r_{z}}{|x-y|}\mathbb{D}\overset{\mathcal{L}_{\mathbb{D}}^{\theta}}{\longleftrightarrow}e^{-1}\partial\mathbb{D}\right)^{-1}$
(4.10)

times the inverse of two welding events that happen with high probability
which ensure that the connections have been made at
$2\varepsilon/|x-y|\partial\mathbb{D}$ and at $\frac{2r_{z}}{|x-y|}$