# Loop percolation on discrete half-plane

Titus Lupu

► To cite this version:

Titus Lupu. Loop percolation on discrete half-plane. Electronic Communications in Probability, 2016, 21 (30), pp.1-9. 10.1214/16-ECP4571. hal-02738812

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Distributed under a Creative Commons Attribution 4.0 International License
# Loop percolation on discrete half-plane

Titus Lupu*

## Abstract

We consider the random walk loop soup on the discrete half-plane $Z \times \mathbb{N}^*$ and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to $\frac{1}{2}$. The absence of percolation at intensity $\frac{1}{2}$ was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.

**Keywords:** random walk loop soup; loop percolation.

**AMS MSC 2010:** 60K35.

Submitted to ECP on September 22, 2015, final version accepted on January 22, 2016.

Supersedes arXiv:1408.1045.

## 1 Introduction

We will consider discrete (rooted) loops on $Z^2$, that is to say finite paths to the nearest neighbours on $Z^2$ that return to the origin and visit at least two vertices. The rooted random walk loop measure $\mu_{Z^2}$ gives to each rooted loop of lengths $2n$ the mass $(2n)^{-1}4^{-2n}$. It was introduced in [5]. In [3] are considered loops parametrised by continuous time rather than discrete time. $\mu_{Z^2}$ has a continuous analogue, the measure $\mu_C$ on the Brownian loops on $C$. Let $P_{z,z'}^t(\cdot)$ be the standard Brownian bridge probability measure from $z$ to $z'$ of length $t$. $\mu_C$ is a measure on continuous time-parametrised loops on $C$ defined as

$$ \mu_C(\cdot) := \int_C \int_{t>0} P_{z,z}^t(\cdot) \frac{dt}{2\pi t^2} \frac{d\bar{z} \wedge dz}{2i}, $$

where $\frac{d\bar{z} \wedge dz}{2i}$ is the standard volume form on $C$. The measure $\mu_C$ was introduced in [6].

Given $\alpha > 0$ we will denote by $\mathcal{L}_{\alpha}^{Z^2}$ respectively $\mathcal{L}_{\alpha}^{C}$ the Poisson ensemble of intensity $\alpha\mu_{Z^2}$ respectively $\alpha\mu_C$, called random walk respectively Brownian loop soup. In [5] it was shown that one can approximate $\mathcal{L}_{\alpha}^{C}$ by a rescaled version of $\mathcal{L}_{\alpha}^{Z^2}$. If $A$ is a subset of $Z^2$ we will denote by $\mathcal{L}_{\alpha}^{A}$ the subset of $\mathcal{L}_{\alpha}^{Z^2}$ made of loops contained in $A$. If $U$ is an open subset of $C$ we will denote by $\mathcal{L}_{\alpha}^{U}$ the subset of $\mathcal{L}_{\alpha}^{C}$ made of loops contained in $U$. For $\delta > 0$ we will denote by $\mathcal{L}_{\alpha}^{A, \ge \delta}$ respectively $\mathcal{L}_{\alpha}^{U, \ge \delta}$ the subset of random walk loops $\mathcal{L}_{\alpha}^{A}$ respectively Brownian loops $\mathcal{L}_{\alpha}^{U}$ made of loops of diameter greater or equal to $\delta$. Similarly we will use the notation $\mathcal{L}_{\alpha}^{A, \le \delta}$ for the loops of diameter smaller or equal to $\delta$.

We will consider clusters of loops. Two loops $\gamma$ and $\gamma'$ in a Poisson ensemble of discrete or continuous loops belong to the same cluster if there is a chain of loops

<footnote>*ETH Zurich, Switzerland. E-mail: titus.lupu@eth-its.ethz.ch</footnote>
$\gamma_0, \gamma_1, \dots, \gamma_n$ in this Poisson ensemble such that $\gamma_0 = \gamma$, $\gamma_n = \gamma'$ and $\gamma_i$ and $\gamma_{i-1}$ visit a common point. For all $\alpha > 0$, loops in $\mathcal{L}_{\alpha}^{\mathbb{Z}^2}$ as well as in $\mathcal{L}_{\alpha}^{\mathbb{C}}$ form a single cluster. Thus we will consider loops on discrete half-plane $H = \mathbb{Z} \times \mathbb{N}^*$ and on continuous half-plane $H = \{z \in \mathbb{C} | \Im(z) > 0\}$, mainly from the angle of existence of an unbounded cluster.

The percolation problem for Brownian loops was studied in [10]. It was shown that there is a critical intensity $\alpha_*^H \in (0, +\infty)$ such that for $\alpha \in (0, \alpha_*^H]$, $\mathcal{L}_{\alpha}^H$ has only bounded clusters, and for $\alpha > \alpha_*^H$ the loops in $\mathcal{L}_{\alpha}^H$ form one single cluster. The critical intensity was identified to be equal to 1. But actually $\alpha_*^H = \frac{1}{2}$. In [10] the outer boundaries of outermost clusters in a sub-critical Brownian loop soup were identified to be a Conformal Loop Ensemble $CLE_{\kappa}$ with the following relation between $\alpha$ and $\kappa$.

$$ \alpha = \frac{(3\kappa - 8)(6 - \kappa)}{2\kappa}. $$

The critical value of $\kappa$ corresponds to $CLE_4$. Actually the right relation between $\alpha$ and $\kappa$ is

$$ \alpha = \frac{1}{2} \frac{(3\kappa - 8)(6 - \kappa)}{2\kappa}. $$

So the value of $\alpha$ that corresponds to $\kappa = 4$ is $\frac{1}{2}$ and not 1. The missing factor $\frac{1}{2}$ appears in the Lawler's work [7] (Proposition 2.1). The error in [10] comes from an error in the article [6] by Lawler and Werner. There the authors consider a Brownian loop soup in the half-plane and a continuous path cutting the half-plane, parametrised by the half-plane capacity. For such a path the half-plane capacity at time $t$ equals $2t$. It discovers progressively new Brownian loops and the authors map these loops conformally to the origin. In the Theorem 1 they identify the processes of these conformally mapped Brownian loops to be a Poisson point process with intensity proportional to the Brownian bubble measure. In the identification of intensity there is a factor 2 missing. Actually in the article [6], the Theorem 1 is inconsistent with the Proposition 11.

The problem of percolation by random walk loops was studied in [4], [2], [9] and [1] in more general setting than dimension 2. We will focus on the percolation by loops in $\mathcal{L}_{\alpha}^H$. The probability of existence of an infinite cluster of loops follows a $0-1$ law and there can be at most one infinite cluster ([9]). Moreover for $\alpha = \frac{1}{2}$ loops in $\mathcal{L}_{\alpha}^H$ do not percolate ([9]). This result was obtained through a coupling with the massless Gaussian free field. By considering just the loops that go back and forth between two neighbouring vertices we get a lower bound on clusters of loops by clusters of an i.i.d. Bernoulli percolation. In particular this implies that for $\alpha$ large enough loops in $\mathcal{L}_{\alpha}^H$ percolate. Hence as the parameter $\alpha$ increases there is a phase transition and a critical value $\alpha_*^H \in [\frac{1}{2}, +\infty)$ of the parameter. Using the results on the clusters of Brownian loops from [10] and the approximation result from [5] we will show in section 2 the following:

**Theorem 1.1.** For all $\alpha > \frac{1}{2}$ there is an infinite cluster of loops in $\mathcal{L}_{\alpha}^H$. In particular $\alpha_*^H = \frac{1}{2}$. Moreover, given $\alpha > \frac{1}{2}$, there is $n \in \mathbb{N}^*$ large enough, such that $\mathcal{L}_{\alpha}^{H, \le n}$ percolates too.

That is to say the critical intensity parameter for the two-dimensional Brownian loop soups and random walk loop soups is the same.

We will consider 1-dependent edge percolations on $H$, $(\omega(e))_{e \text{ edge}}$. By 1-dependent percolation we mean that if two disjoint subsets of edges $E_1$ and $E_2$ are at graph distance at least 1 then $(\omega(e))_{e \in E_1}$ and $(\omega(e))_{e \in E_2}$ are independent. According the results on locally dependent percolation by Liggett, Schonmann and Stacey in [8], for all 1-dependent edge percolations on $H$ with $p$ the probability of an edge to be open, there is an universal $\tilde{p}(p) \in [0, 1)$ such that the 1-dependent edge percolation contains an i.i.d. Bernoulli percolation with probability $\tilde{p}(p)$ of an edge to be open. Moreover the following
constraint holds:

$$ \lim_{p \to 1^{-}} \tilde{p}(p) = 1. $$

## 2 Critical intensity parameter

Let $\alpha, \delta > 0$. Given $U$ an open subset of $\mathbb{H}$, we will denote by $\mathcal{L}_{\alpha}^{U, \ge \delta}$ respectively $\mathcal{L}_{\alpha}^{U \cap H, \ge \delta}$ the subset of $\mathcal{L}_{\alpha}^{H}$ respectively $\mathcal{L}_{\alpha}^{H}$ made of loops contained in $U$ and with diameter greater or equal to $\delta$. We will use the notations $\mathcal{L}_{\alpha}^{U}$ and $\mathcal{L}_{\alpha}^{U \cap H}$ when there is a condition on the range but not on the diameter.

Let $Q_{ext}$ and $Q_{int}$ be the following rectangles:

$$ Q_{ext} := (0, 6) \times (0, 3), \quad Q_{int} := (1, 5) \times (1, 2). $$

We consider the subset of Brownian loops $\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}$, which is a.s. finite. We introduce the events $C_1(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$, $C_2(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ and $C_3(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ depending on the loops in $\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}$. The event $C_1(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ will be satisfied if there is a cluster $K_1$ of loops in $\mathcal{L}_{\alpha}^{Q_{int}, \ge \delta}$ such that in $\mathcal{L}_{\alpha}^{(0,6) \times (1,2), \ge \delta}$ there is a loop that intersects $K_1$ and $\{1\} \times (1, 2)$ and a loop that intersects $K_1$ and $\{5\} \times (1, 2)$. The two loops may be the same. $C_2(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ will be satisfied if there is a cluster $K_2$ in $\mathcal{L}_{\alpha}^{(1,2)^2, \ge \delta}$ such that in $\mathcal{L}_{\alpha}^{(1,2) \times (0,3), \ge \delta}$ there is a loop that intersects $K_2$ and $(1, 2) \times \{1\}$ and a loop that intersects $K_2$ and $(1, 2) \times \{2\}$. The event $C_3(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ is similar to the event $C_2(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ where the square $(1, 2)^2$ is replaced by the square $(4, 5) \times (1, 2)$ and the rectangle $(1, 2) \times (0, 3)$ by the rectangle $(4, 5) \times (0, 3)$. Next figure illustrates the event $\bigcap_{i=1}^{3} C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of the event $\bigcap_{i=1}^{3} C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$. One should imagine that the smooth loops are actually Brownian. Only a set of loops that is sufficient for the event is represented. Full line loops stay inside $Q_{int}$. Dashed loops cross the boundary of $Q_{int}$.</figcaption></figure>

We will call the event $\bigcap_{i=1}^{3} C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ special crossing event with exterior rectangle $Q_{ext}$ and interior rectangle $Q_{int}$. We will also consider translations, rotations and rescaling of $Q_{ext}$ and $Q_{int}$ and deal with special crossing events corresponding to the new rectangles. We are interested in the event $\bigcap_{i=1}^{3} C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ because then the loops in $\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}$ achieve the three crossings drawn on the figure 2:

Next we show that if $\alpha > \frac{1}{2}$ and $\delta$ is small enough then the probability of the event $\bigcap_{i=1}^{3} C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ is close to 1.
<figure><img src="image_3.png" /><figcaption>Figure 2: The three crossings we are interested in.</figcaption></figure>

**Lemma 2.1.** Let $Q$ be a rectangle of form $Q = (-a, a) \times (0, b)$. Let $\alpha > 0$. Let $(B_t)_{t \ge 0}$ be the standard Brownian motion on $\mathbb{C}$ started from 0 and let $\mathcal{L}_{\alpha}^Q$ be a Poisson ensemble of loops independent from $B$. Then for all $\varepsilon > 0$ there is $t \in (0, \varepsilon)$ such that $B$ at time $t$ intersects a loop in $\mathcal{L}_{\alpha}^Q$.

*Proof.* First we consider a loops soup in $\mathbb{H}$, $\mathcal{L}_{\alpha}^{\mathbb{H}}$, independent of $B$. Let

$$T := \inf\{t > 0 | B_t \text{ is in the range of a loop in } \mathcal{L}_{\alpha}^{\mathbb{H}}\}.$$

$T$ is a.s. finite. Indeed a loop in $\mathcal{L}_{\alpha}^{\mathbb{H}}$ delimits a domain with non-empty interior. Since the Brownian motion on $\mathbb{C}$ is recurrent, $B$ will visit this domain and thus intersect the loop. Let $\lambda > 0$. The Poisson ensemble of loops $\mathcal{L}_{\alpha}^{\mathbb{H}}$ is invariant in law under the Brownian scaling

$$(\gamma(t))_{0 \le t \le t_{\gamma}} \mapsto \lambda^{-\frac{1}{2}}(\gamma(\lambda t))_{0 \le t \le \lambda^{-1} t_{\gamma}}.$$

So does the Brownian motion $B$. Thus $\lambda T$ has the same law as $T$. It follows that $T = 0$ a.s.

The set of loops $\mathcal{L}_{\alpha}^{\mathbb{H}} \setminus \mathcal{L}_{\alpha}^{Q}$ is at positive distance from 0 thus $B$ cannot intersect it immediately. It follows that $B$ intersects immediately $\mathcal{L}_{\alpha}^{Q}$. $\square$

**Lemma 2.2.** Let $a, \alpha > 0$. There is a.s. a loop in $\mathcal{L}_{\alpha}^{(-a,a)^2}$ that intersects the real line $\mathbb{R}$.

*Proof.* Let $\mathcal{L}_{\alpha}^{(n)}$ be the subset of $\mathcal{L}_{\alpha}^{(-a,a)^2}$ made of loops $\gamma$ of duration $t_{\gamma}$ comprised between $2^{-n-1}$ and $2^{-n}$. The family $(\mathcal{L}_{\alpha}^{(n)})_{n \ge 0}$ is independent. By Brownian scaling, the probability that a loop in $\mathcal{L}_{\alpha}^{(n)}$ intersects $\mathbb{R}$ is the same as a loop in $\mathcal{L}_{\alpha}^{(-a2^{n/2}, a2^{n/2})^2}$ of duration comprised between $\frac{1}{2}$ and 1 intersects $\mathbb{R}$. This is at least as big as the similar probability for $\mathcal{L}_{\alpha}^{(0)}$. Since the latter probability is non-zero, the intersection events occurs a.s. for infinitely many of $\mathcal{L}_{\alpha}^{(n)}$. $\square$

**Lemma 2.3.** Let $a, \alpha > 0$. There is a.s. a loop in $\mathcal{L}_{\alpha}^{(-a,a)^2}$ that intersects the real line $\mathbb{R}$ and a loop in $\mathcal{L}_{\alpha}^{(-a,a) \times (0,a)}$.

*Proof.* Consider the subset of $\mathcal{L}_{\alpha}^{(-a,a)^2}$ made of loops intersecting $\mathbb{R}$. It is non empty according the lemma 2.2. Moreover it is independent of $\mathcal{L}_{\alpha}^{(-a,a) \times (0,a)}$. The law of a
Brownian loop that intersects $\mathbb{R}$ is locally, near the point of intersection, absolutely continuous with respect to the law of a Brownian motion started from there. Applying lemma 2.1, we get that it intersects a.s. a loop in $\mathcal{L}_{\alpha}^{(-a,a)\times(0,a)}$. $\square$

**Lemma 2.4.** Let $\alpha > \frac{1}{2}$. Then

$$ \lim_{\delta \to 0^+} \mathbb{P}\left(\bigcap_{i=1}^{3} C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})\right) = 1. $$

*Proof.* It is enough to show that the probability of each of the $C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ converges to 1 as $\delta$ tends to 0. Since the three cases are very similar, we will do the proof only for $C_1(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$. According to lemma 2.3 there is a loop $\gamma$ in $\mathcal{L}_{\alpha}^{(0,6)\times(1,2)}$ that intersects $\{1\} \times (1,2)$ and a loop $\gamma'$ in $\mathcal{L}_{\alpha}^{Q_{int}}$. Similarly there is a loop $\tilde{\gamma}$ in $\mathcal{L}_{\alpha}^{(0,6)\times(1,2)}$ that intersects $\{5\} \times (1,2)$ and a loop $\tilde{\gamma}'$ in $\mathcal{L}_{\alpha}^{Q_{int}}$. Since $\alpha > \frac{1}{2}$, $\gamma'$ and $\tilde{\gamma}'$ belong to the same cluster in $\mathcal{L}_{\alpha}^{Q_{int}}$ ([10]). Thus there is a chain of loops $(\gamma_0, \dots, \gamma_n)$ in $\mathcal{L}_{\alpha}^{Q_{int}}$, with $\gamma_0 = \gamma'$ and $\gamma_n = \tilde{\gamma}'$, joining $\gamma'$ and $\tilde{\gamma}'$. If $\delta$ is the minimum of diameters of $(\gamma_0, \dots, \gamma_n)$ and $\gamma$ and $\tilde{\gamma}$ then $C_1(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ is satisfied. Let $\delta$ be maximal value of $\delta$ such that $C_1(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})$ is satisfied. $\delta$ is a well defined random variable with values in $(0, +\infty)$. Then

$$ \lim_{\delta \to 0^+} \mathbb{P}(C_1(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})) = \lim_{\delta \to 0^+} \mathbb{P}(\delta \le \bar{\delta}) = 1. $$

$\square$

Next we recall the result on approximation of Brownian loops by random walk loops from [5]. Let $N \in \mathbb{N}^*$. We consider the discrete loops $\gamma$ on $\mathbb{Z} \times \mathbb{N}^*$. We define on these loops a map $\Phi_N$ to continuous loops on $\mathbb{H}$. Given $\gamma$ a discrete loop and $(z_0, \dots, z_{n-1}, z_0)$ the sequence of the vertices it visits, the continuous loop $\Phi_N\gamma$ satisfies:

* the duration of $\Phi_N\gamma$ is $\frac{n}{2N^2}$;

* for $j \in \{0, \dots, n-1\}$, $\Phi_N\gamma(\frac{j}{2N^2}) = \frac{z_j}{N}$;

* $\Phi_N\gamma(\frac{n}{2N^2}) = \Phi_N\gamma(0) = \frac{z_0}{N}$;

* between the times $\frac{j}{2N^2}$, $j \in \{0, \dots, n\}$, $\Phi_N\gamma$ interpolates linearly.

The number of jumps $n$ of a discrete loop $\gamma$ will be denoted $s_\gamma$. The life-time of a continuous loop $\tilde{\gamma}$ will be denoted by $t_{\tilde{\gamma}}$. Let $\theta \in (\frac{2}{3}, 2)$ and $r \ge 1$. There is a coupling between $\mathcal{L}_{\alpha}^{\mathbb{H}}$ and $\mathcal{L}_{\alpha}^{\mathbb{H}}$ such that except on an event of probability at most $cste \cdot (\alpha + 1)r^2N^{2-3\theta}$ there is a one to one correspondence between the two sets

* $\{\gamma \in \mathcal{L}_{\alpha}^{\mathbb{H}} | s_\gamma > 2N^\theta, |\gamma(0)| < Nr\}$,

* $\{\tilde{\gamma} \in \mathcal{L}_{\alpha}^{\mathbb{H}} | t_{\tilde{\gamma}} > N^{\theta-2}, |\tilde{\gamma}(0)| < r\}$,

such that given a discrete loop $\gamma$ and the continuous loop $\tilde{\gamma}$ corresponding to it,

$$ \left| \frac{s_\gamma}{2N^2} - t_{\tilde{\gamma}} \right| \le \frac{5}{8} N^{-2}, \quad \sup_{0 \le u \le 1} \left| \Phi_N \gamma \left( u \frac{s_\gamma}{2N^2} \right) - \tilde{\gamma}(u t_{\tilde{\gamma}}) \right| \le cste \cdot N^{-1} \log(N). $$

Next we state without proof a lemma that follows immediately from this approximation.

**Lemma 2.5.** Let $\alpha > 0$ and $\delta > 0$. As $N$ tends to $+\infty$ the random set of interpolating continuous loops

$$ \{\Phi_N\gamma | \gamma \in \mathcal{L}_{\alpha}^{NQ_{ext} \cap \mathbb{H}, \ge N\delta}\} $$

converges in law to the set of Brownian loops $\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}$.
We need to show that the above convergence for the uniform norm also implies a
convergence of the intersection relations, that is to say that

$$
\{(\gamma, \gamma')|\gamma, \gamma' \in \mathcal{L}_{\alpha}^{NQ_{ext} \cap H, \ge N\delta}, \gamma \text{ intersects } \gamma'\}
$$

converges in law to

$$
\{(\tilde{\gamma}, \tilde{\gamma}') | \tilde{\gamma}, \tilde{\gamma}' \in \mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}, \tilde{\gamma} \text{ intersects } \tilde{\gamma}'\}.
$$

Let $j \in \mathbb{N}$. Let $\gamma$ be a continuous path on $\mathbb{C}$ (not necessarily a loop) of lifetime $t_{\gamma}$. For $r > 0$ let

$$
T_r(\gamma) := \inf\{s > 0 \mid |\gamma(s)| \ge r\} \in (0, +\infty].
$$

If $T_r(\gamma) < +\infty$ let

$$
e^{i\omega_r} := \frac{\gamma(T_r(\gamma))}{r}.
$$

Let $I_j$ be the real interval

$$
I_j := \left( \frac{7}{12} 2^{-j}, \frac{9}{12} 2^{-j} \right).
$$

For $0 < r_1 < r_2$ let $\mathcal{A}(r_1, r_2)$ be the annulus

$$
\mathcal{A}(r_1, r_2) := \{z \in \mathbb{C} | r_1 < |z| < r_2\}.
$$

For $r > 0$ let $HD(r)$ be the half-disc

$$
HD(r) := B(0, r) \cap \{z \in \mathbb{C} | \Re(z) > 0\}.
$$

We will say that the path $\gamma$ satisfies the condition $\mathcal{C}_j$ if

$$
\cdot T_{\frac{11}{12}2-j}(\gamma) < +\infty,
$$

* after time $T_{\frac{11}{12}2-j}(\gamma) < +\infty$, $\gamma$ hits $e^{i(\omega_{2-j-1}+\frac{\pi}{2})}I_j$ at a time $\tilde{t}_j$ before hitting the circle $S(0, 2^{-j})$,

* on the time interval $(T_{2-j-1}(\gamma), \tilde{t}_j)$ $\gamma$ stays in the half-disc $e^{i\omega_{2-j-1}}HD(2^{-j})$,

* from time $\tilde{t}_j$ the path $\gamma$ stays in the annulus $\mathcal{A}(\frac{7}{12}2^{-j}, \frac{9}{12}2^{-j})$ until surrounding the disc $B(0, \frac{7}{12}2^{-j})$ once clockwise and hitting $e^{i(\omega_{2-j-1}+\pi)}I_j$.

Figure 3 illustrates a path satisfying the condition $\mathcal{C}_j$. If this condition is satisfied
than $\gamma$ disconnects the disc $B(0, \frac{7}{12}2^{-j})$ from infinity. Moreover if one perturbs $\gamma$ by
any continuous function $f: [0, t_\gamma] \to \mathbb{C}$ such that $\|f\|_\infty \le \frac{1}{12}2^{-j}$ then the path $(\gamma(s) +
f(s))_{0\le s \le t_\gamma}$ disconnects the disc $B(0, 2^{-j-1})$ from infinity. The disconnection is made
inside the annulus $\mathcal{A}(2^{-j-1}, 2^{-j})$.

**Lemma 2.6.** Let $(B_t)_{0 \le t \le T}$ be a standard Brownian path on $\mathbb{C}$ starting from 0. Then almost surely it satisfies the condition $\mathcal{C}_j$ for infinitely many values of $j \in \mathbb{N}$.

Proof. Let $\tilde{B}$ be the Brownian path $B$ continued for $t \in (0, +\infty)$. The events "$\tilde{B}$ satisfies the condition $\mathcal{C}_j$" are i.i.d. Indeed such an event is rotation invariant and depends only on $\tilde{B}$ on the time interval $(T_{2-j-1}(\tilde{B}), T_{2-j}(\tilde{B}))$. Moreover the probability of such an event is non-zero. Thus $\tilde{B}$ satisfies the condition $\mathcal{C}_j$ for infinitely many values of $j \in \mathbb{N}$. Since

$$
\lim_{j \to +\infty} T_{2-j}(\tilde{B}) = 0,
$$

so does $B$.

<figure><img src="image_4.png" /></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3: Representation of a path $\gamma$ satisfying the condition $C_j$.</figcaption></figure>

**Lemma 2.7.** Let $z_1, z_2 \in \mathbb{C}$ and $t_1, t_2 > 0$. Let $(b_s^{(1)})_{0\le s \le t_1}$ and $(b_s^{(2)})_{0\le s \le t_2}$ be two independent standard Brownian bridges from $z_1$ to $z_1$ and $z_2$ to $z_2$ respectively. On the event that $b^{(1)}$ intersects $b^{(2)}$ there is a.s. $\varepsilon > 0$ such that for all continuous functions $f_1 : [0, t_1] \to \mathbb{C}$ and $f_2 : [0, t_2] \to \mathbb{C}$ of infinity norm $\|f_i\|_{\infty} \le \varepsilon$, $(b_s^{(1)} + f_1(s))_{0\le s \le t_1}$ intersects $(b_s^{(2)} + f_2(s))_{0\le s \le t_2}$.

*Proof.* Let $T_2^{(1)}$ be the first time $b^{(1)}$ hits the range of $b^{(2)}$. If the two path do not intersect each other $T_2^{(1)} = +\infty$. On the event $T_2^{(1)} < +\infty$ the conditional law of $(b_{T_2^{(1)}+s}^{(1)} - b_{T_2^{(1)}}^{(1)})_{0\le s \le t_1 - T_2^{(1)} - \varepsilon}$ ($\varepsilon > 0$ a small constant) given the value $T_2^{(1)}$ is absolutely continuous with respect the law of a Brownian path starting from 0. From lemma 2.6 follows that the path $(b_{T_2^{(1)}+s}^{(1)} - b_{T_2^{(1)}}^{(1)})_{0\le s \le t_1 - T_2^{(1)}}$ satisfies the condition $C_j$ for infinitely many values of $j \in \mathbb{N}$. Let

$$ \tilde{j} := \max \left\{ j \in \mathbb{N} \middle| (b_{T_2^{(1)}+s}^{(1)} - b_{T_2^{(1)}}^{(1)})_{0\le s \le t_1 - T_2^{(1)}} \text{ satisfies the condition } C_j \text{ and } \exists s \in [0, t_2], |b_s^{(2)} - b_{T_2^{(1)}}^{(2)}| \ge \frac{13}{12} 2^{-j} \right\}. $$

$\tilde{j}$ is a r.v. defined on the event where $b^{(1)}$ and $b^{(2)}$ intersect. If $f_1$ and $f_2$ are such that $\|f_i\| \le \frac{1}{12} 2^{-\tilde{j}}$ then the path $b^{(1)} + f_1$ disconnects the disc $B(b_{T_2^{(1)}}^{(1)}, 2^{-\tilde{j}-1})$ from infinity inside the annulus $b_{T_2^{(1)}}^{(1)} + A(2^{-\tilde{j}-1}, 2^{-\tilde{j}})$ and the path $b^{(2)} + f_2$ crosses from the circle $S(b_{T_2^{(1)}}^{(1)}, 2^{-\tilde{j}-1})$ to the circle $S(b_{T_2^{(1)}}^{(1)}, 2^{-\tilde{j}})$, so the two must intersect. $\square$

Observe that two discrete loops $\gamma$ and $\gamma'$ intersect each other if and only if the continuous loops $\Phi_N\gamma$ and $\Phi_N\gamma'$ do. From lemmas 2.5 and 2.7 follows:
**Corollary 2.8.** Let $\alpha > 0$ and $\delta > 0$. As $N$ tends to $+\infty$ the random set of interpolating continuous loops

$$ \{\Phi_N \gamma | \gamma \in \mathcal{L}_{\alpha}^{NQ_{ext} \cap H, \ge N\delta}\} $$

jointly with the intersection relations

$$ \{(\gamma, \gamma') | \gamma, \gamma' \in \mathcal{L}_{\alpha}^{NQ_{ext} \cap H, \ge N\delta}, \gamma \text{ intersects } \gamma'\} $$

converges in law to the set of Brownian loops $\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}$ jointly with the intersection relations

$$ \{(\tilde{\gamma}, \tilde{\gamma}') | \tilde{\gamma}, \tilde{\gamma}' \in \mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}, \tilde{\gamma} \text{ intersects } \tilde{\gamma}'\}. $$

We consider the scaled up rectangle $NQ_{ext}$ and $NQ_{int}$. The next lemma deals with the probability that the discrete loops $\mathcal{L}_{\alpha}^{NQ_{ext} \cap H}$ realise the special crossing event with exterior rectangle $NQ_{ext}$ and interior rectangle $NQ_{int}$. See figures 1 and 2 and consider that $Q_{ext}$ is replaced by $NQ_{ext}$, $Q_{int}$ by $NQ_{int}$ and $\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta}$ by $\mathcal{L}_{\alpha}^{NQ_{ext} \cap H}$.

**Lemma 2.9.** Let $\alpha > \frac{1}{2}$. As $N$ tends to $+\infty$, the probability that the loops $\mathcal{L}_{\alpha}^{NQ_{ext} \cap H}$ realise a special crossing event with exterior rectangle $NQ_{ext}$ and interior rectangle $NQ_{int}$ converges to 1.

*Proof.* Let $\delta > 0$. The probability that the loops $\mathcal{L}_{\alpha}^{NQ_{ext} \cap H}$ realise the special crossing event with exterior rectangle $NQ_{ext}$ and interior rectangle $NQ_{int}$ is at least as large as the probability that the loops $\mathcal{L}_{\alpha}^{NQ_{ext} \cap H, \ge N\delta}$ realise the special crossing event with the same interior and exterior rectangle. From the corollary 2.8 follows that the latter probability converges as $N \to +\infty$ to

$$ \mathbb{P}\left(\bigcap_{i=1}^{3} C_i(\mathcal{L}_{\alpha}^{Q_{ext}, \ge \delta})\right). $$

We conclude by applying the lemma 2.4. □

To conclude that for $\alpha > \frac{1}{2}$, $\mathcal{L}_{\alpha}^{H}$ has an infinite cluster we will use a block percolation construction that will combine special crossing events.

**Proof of the Theorem 1.1.** From [9] we know already that $\alpha_*^H \le \frac{1}{2}$. We need to show that for $\alpha > \frac{1}{2}$, $\mathcal{L}_{\alpha}^H$ has an infinite cluster.

Let $\alpha > \frac{1}{2}$ and $N \ge 1$. We consider a dependent edge percolation $(\omega^N(e))_{e \text{ edge of } H}$ on the discrete half plane $H$. If $e$ is an edge of form $\{(j, k), (j + 1, k)\}$, $k \ge 1$, then $\omega^N(e) = 1$ (open edge) if $\mathcal{L}_{\alpha}^{(NQ_{int}+3Nj+i3Nk)\cap H}$ achieves a special crossing event with exterior rectangle $NQ_{ext} + 3Nj + i3Nk$ and interior rectangle $NQ_{int} + 3Nj + i3Nk$. If $e$ is an edge of form $\{(j, k), (j, k + 1)\}$, $k \ge 1$, then $\omega^N(e) = 1$ if $\mathcal{L}_{\alpha}^{(iNQ_{int}+3Nj+i3Nk)\cap H}$ achieves a special crossing event with exterior rectangle $iNQ_{ext} + 3Nj + i3Nk$ and interior rectangle $iNQ_{int} + 3Nj + i3Nk$, where the multiplication by $i$ means rotation by $+\frac{\pi}{2}$. $\omega^N$ is a 1-dependent edge percolation: if two disjoint subsets of edges $E_1$ and $E_2$ are such that no edge is adjacent to both $E_1$ and $E_2$, then $(\omega^N(e))_{e \in E_1}$ and $(\omega^N(e))_{e \in E_2}$ are independent. This is due to the fact that the subsets of loops involved in the definition of special crossing events for edges in $E_1$ and edges in $E_2$ are disjoint. To an open path in $\omega^N$ corresponds a cluster of $\mathcal{L}_{\alpha}^H$ whose loops form crossings of related interior rectangles. Thus if $\omega^N$ has an unbounded cluster, then so does $\mathcal{L}_{\alpha}^H$. See next picture.

The probability $\mathbb{P}(\omega^N(e) = 1)$ is uniform and we will denote it $p_N$. According to the lemma 2.9

$$ \lim_{N \to +\infty} p_N = 1. $$
Loop percolation on discrete half-plane

<figure><img src="image_6.png" /><figcaption>Figure 4: Crossings achieved by subsets of loops in $\mathcal{L}_{\alpha}^{H}$, corresponding to five open edges in $\omega^{N}$.</figcaption></figure>

Thus for $N$ large enough $\tilde{p}(p_N) > \frac{1}{2} \cdot \frac{1}{2}$ is the critical probability for the i.i.d. Bernoulli edge percolation on $H$. So for $N$ large enough $\omega^N$ contains a supercritical i.i.d. Bernoulli edge percolation and percolates itself. Thus $\mathcal{L}_{\alpha}^{H}$ percolates too. Actually, since our construction only uses loops of diameter less or equal to $6N$, we have also percolation for $\mathcal{L}_{\alpha}^{H, \le 6N}$. $\square$

References

[1] Y. Chang. Supercritical loop percolation on $Z^d$ for $d \ge 3$, 2015. arXiv:1504.07906

[2] Y. Chang and A. Sapozhnikov. Phase transition in loop percolation. *Probab. Theory Related Fields*, pages 1–47, 2015. arXiv:1403.5687

[3] Y. Le Jan. Markov paths, loops and fields. In *2008 St-Flour summer school, L.N. Math.*, volume 2026. Springer, 2011. MR-2815763

[4] Y. Le Jan and S. Lemaire. Markovian loop clusters on graphs. *Illinois J. Math.*, 57(2):525–558, 2013. MR-3263044

[5] G. F. Lawler and J. A. Trujillo Ferreras. Random walk loop soup. *Trans. Amer. Math. Soc.*, 359(2):767–787, 2007. MR-2255196

[6] G. F. Lawler and W. Werner. The Brownian loop-soup. *Probab. Theory Related Fields*, 128:565–588, 2004. MR-2045953

[7] G.F. Lawler. Partition functions, loop measure, and versions of SLE. *J. Stat. Phys.*, 134:813–837, 2009. MR-2518970

[8] T. M. Liggett, R. H. Schonmann, and A. M. Stacey. Domination by product measures. *Ann. Probab.*, 25(1):71–95, 1997. MR-1428500

[9] T. Lupu. From loop clusters and random interlacements to the free field, 2014. To appear in *Ann. Probab*. arXiv:1402.0298

[10] S. Sheffield and W. Werner. Conformal loop ensembles: the Markovian characterization and the loop-soup construction. *Ann. of Math.*, 176(3):1827–1917, 2012. MR-2979861

**Acknowledgments.** The author would like to thank Artem Sapozhnikov and Yinshan Chang for their comments on previous versions of this note.
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