Loop percolation on discrete half-plane

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Titus Lupu

To cite this version:

Titus Lupu. Loop percolation on discrete half-plane. Electronic Communications in Probability, In-

stitute of Mathematical Statistics (IMS), 2016, 21 (30), pp.1-9. �10.1214/16-ECP4571�. �hal-02738812�

Electron. Commun. Probab.21(2016), no. 30, 1–9.

DOI:10.1214/16-ECP4571 ISSN:1083-589X

ELECTRONIC COMMUNICATIONS in PROBABILITY

Loop percolation on discrete half-plane

Titus Lupu^*

Abstract

We consider the random walk loop soup on the discrete half-planeZ×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 ¹₂. The absence of percolation at intensity

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.

SupersedesarXiv:1408.1045.

1 Introduction

We will consider discrete (rooted) loops on Z², that is to say finite paths to the nearest neighbours onZ²that return to the origin and visit at least two vertices. The rooted random walk loop measure µ_Z2 gives to each rooted loop of lengths 2n the mass(2n)⁻¹4⁻²ⁿ. It was introduced in [5]. In [3] are considered loops parametrised by continuous time rather than discrete time.µ_Z2 has a continuous analogue, the measure µ_Con the Brownian loops onC^{. Let}P^tz,z⁰(·)be the standard Brownian bridge probability measure fromztoz⁰of lengtht. µ_Cis a measure on continuous time-parametrised loops onC^{defined as}

µ_C(·) :=

Z

C

Z

t>0

P^tz,z(·) dt 2πt²

d¯z∧dz 2i ,

where ^d¯z∧dz_2i is the standard volume form onC. The measureµ_Cwas introduced in [6].

Givenα >0we will denote byL^Z_α²respectivelyL^C_αthe Poisson ensemble of intensity αµ_Z2 respectivelyαµ_C, called random walk respectively Brownian loop soup. In [5] it was shown that one can approximateL^C_αby a rescaled version ofL^Z_α². IfAis a subset ofZ²we will denote byL^A_α the subset ofL^Z_α² made of loops contained inA. IfU is an open subset ofCwe will denote byL^U_α the subset ofL^C_αmade of loops contained inU. Forδ >0we will denote byL^A,≥δ_α respectivelyL^U,≥δ_α the subset of random walk loops L^A_α respectively Brownian loopsL^U_α made of loops of diameter greater or equal to δ. Similarly we will use the notationL^A,≤δ_α for the loops of diameter smaller or equal toδ.

We will consider clusters of loops. Two loops γ and γ⁰ in a Poisson ensemble of discrete or continuous loops belong to the same cluster if there is a chain of loops

*ETH Zurich, Switzerland. E-mail:[email protected]

γ₀, γ₁, . . . , γ_n in this Poisson ensemble such thatγ₀=γ,γ_n =γ⁰andγ_iandγ_i−1visit a common point. For allα >0, loops inL^Z_α² as well as inL^C_αform a single cluster. Thus we will consider loops on discrete half-planeH=Z×N^∗and on continuous half-plane H={z∈C|=(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 intensityα^H_∗ ∈(0,+∞)such that forα∈(0, α^H_∗],L^H_α has only bounded clusters, and forα > α^H_∗ the loops inL^H_αH

∗ form one single cluster. The critical intensity was identified to be equal to1. But actuallyα^H_∗ = ¹₂. In [10] the outer boundaries of outermost clusters in a sub-critical Brownian loop soup were identified to be a Conformal Loop EnsembleCLEκwith the following relation betweenαandκ.

α=(3κ−8)(6−κ)

2κ .

The critical value ofκcorresponds toCLE4. Actually the right relation betweenαandκ is

α= 1 2

(3κ−8)(6−κ)

2κ .

So the value ofαthat corresponds toκ= 4is ¹₂ and not1. The missing factor ¹₂ appears in the Lawler’s work [7] (Proposition2.1). 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 timetequals2t. It discovers progressively new Brownian loops and the authors map these loops conformally to the origin. In the Theorem1they 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 factor2missing. Actually in the article [6], the Theorem1is inconsistent with the Proposition11.

The problem of percolation by random walk loops was studied in [4], [2], [9] and [1]

in more general setting than dimension2. We will focus on the percolation by loops inL^H_α. The probability of existence of an infinite cluster of loops follows a0−1law and there can be at most one infinite cluster ([9]). Moreover forα= ¹₂ loops inL^H1

2

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 forαlarge enough loops inL^H_α percolate. Hence as the parameterαincreases there is a phase transition and a critical valueα^H_∗ ∈[¹₂,+∞)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 α > ¹₂ there is an infinite cluster of loops inL^H_α. In particular α^H_∗ = ¹₂. Moreover, givenα > ¹₂, there isn∈N^∗large enough, such thatL^H,≤n_α 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 consider1-dependent edge percolations onH,(ω(e))eedge. By1-dependent percolation we mean that if two disjoint subsets of edges E₁ and E₂ are at graph distance at least1then(ω(e))_e∈E1and(ω(e))_e∈E2are independent. According the results on locally dependent percolation by Liggett, Schonmann and Stacey in [8], for all1- dependent edge percolations onHwithpthe probability of an edge to be open, there is an universalp(p)˜ ∈[0,1)such that the1-dependent edge percolation contains an i.i.d.

Bernoulli percolation with probabilityp(p)˜ of an edge to be open. Moreover the following

Loop percolation on discrete half-plane

constraint holds:

lim

p→1⁻p(p) = 1.˜ 2 Critical intensity parameter

Let α, δ > 0. GivenU an open subset of H, we will denote byL^U,≥δ_α respectively L^U∩H,≥δ_α the subset ofL^H_α respectivelyL^H_αmade of loops contained inUand with diameter greater or equal toδ. We will use the notationsL^U_α andL^U∩H_α when there is a condition on the range but not on the diameter.

LetQextandQintbe the following rectangles:

Qext:= (0,6)×(0,3), Qint:= (1,5)×(1,2).

We consider the subset of Brownian loopsL^Q_α^ext,≥δ, which is a.s. finite. We introduce the eventsC1(L^Q_α^ext,≥δ),C2(L^Q_α^ext,≥δ)andC3(L^Q_α^ext,≥δ)depending on the loops inL^Q_α^ext,≥δ. The eventC1(L^Q_α^ext,≥δ)will be satisfied if there is a clusterK1of loops inL^Q_α^int,≥δ such that inL(0,6)×(1,2),≥δ

α there is a loop that intersectsK1 and{1} ×(1,2)and a loop that intersectsK1 and{5} ×(1,2). The two loops may be the same. C2(L^Q_α^ext,≥δ) will be satisfied if there is a clusterK2inL^(1,2)α ^2,≥δ such that inL(1,2)×(0,3),≥δ

α there is a loop that intersectsK2and(1,2)× {1}and a loop that intersectsK2and(1,2)× {2}. The event C3(L^Q_α^ext,≥δ)is similar to the eventC2(L^Q_α^ext,≥δ)where the square(1,2)²is replaced by the square(4,5)×(1,2)and the rectangle(1,2)×(0,3)by the rectangle(4,5)×(0,3). Next figure illustrates the eventT3

i=1C_i(L^Q_α^ext,≥δ).

Figure 1: Illustration of the event T3

i=1Ci(L^Q_α^ext,≥δ). 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 insideQ_int. Dashed loops cross the boundary ofQ_int.

We will call the eventT3

i=1Ci(L^Q_α^ext,≥δ)special crossing event with exterior rectangle Q_ext and interior rectangle Q_int. We will also consider translations, rotations and rescaling ofQ_extandQ_intand deal withspecial crossing events corresponding to the new rectangles. We are interested in the eventT3

i=1Ci(L^Q_α^ext,≥δ)because then the loops inL^Q_α^ext,≥δachieve the three crossings drawn on the figure2:

Next we show that ifα >¹₂ andδis small enough then the probability of the event T3

i=1Ci(L^Q_α^ext,≥δ)is close to1.

ECP21(2016), paper 30.

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Figure 2: The three crossings we are interested in.

Lemma 2.1.LetQbe a rectangle of formQ= (−a, a)×(0, b). Letα >0. Let(Bt)t≥0be the standard Brownian motion onCstarted from0and letL^Q_α be a Poisson ensemble of loops independent fromB. Then for allε >0there ist∈(0, ε)such thatBat timet intersects a loop inL^Q_α.

Proof. First we consider a loops soup inH^,L^H_α, independent ofB. Let T := inf{t >0|Btis in the range of a loop inL^H_α}.

T is a.s. finite. Indeed a loop inL^H_α delimits a domain with non-empty interior. Since the Brownian motion onCis recurrent,B will visit this domain and thus intersect the loop.

Letλ >0. The Poisson ensemble of loopsL^H_α is invariant in law under the Brownian scaling

(γ(t))_0≤t≤tγ 7−→λ⁻¹²(γ(λt))_0≤t≤λ−1tγ.

So does the Brownian motionB. ThusλT has the same law asT. It follows thatT = 0 a.s.

The set of loopsL^H_α \ L^Q_α is at positive distance from 0thus B cannot intersect it immediately. It follows thatBintersects immediatelyL^Q_α.

Lemma 2.2.Leta, α >0. There is a.s. a loop inL^(−a,a)α ² that intersects the real lineR^. Proof. Let L⁽ⁿ⁾α be the subset of L^(−a,a)α ² made of loops γ of duration tγ comprised between2⁻ⁿ⁻¹and2⁻ⁿ. The family(L⁽ⁿ⁾α )_n≥0is independent. By Brownian scaling, the probability that a loop in L⁽ⁿ⁾α intersectsRis the same as a loop inL^(−a2α ^n/2,a2n/2)2 of duration comprised between ¹₂ and1intersectsR. This is at least as big as the similar probability for L⁽⁰⁾α . Since the latter probability is non-zero, the intersection events occurs a.s. for infinitely many ofL⁽ⁿ⁾α .

Lemma 2.3.Leta, α >0. There is a.s. a loop inL^(−a,a)α ²that intersects the real lineR and a loop inL(−a,a)×(0,a)

α .

Proof. Consider the subset ofL^(−a,a)α ² made of loops intersecting R. It is non empty according the lemma 2.2. Moreover it is independent ofL(−a,a)×(0,a)

α . The law of a

Loop percolation on discrete half-plane

Brownian loop that intersectsR 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 inL(−a,a)×(0,a)

α .

Lemma 2.4.Letα > ¹₂. Then

lim

δ→0⁺P\³

i=1

C_i(L^Q_α^ext,≥δ)

= 1.

Proof. It is enough to show that the probability of each of theC_i(L^Q_α^ext,≥δ)converges to1asδtends to 0. Since the three cases are very similar, we will do the proof only forC1(L^Q_α^ext,≥δ). According to lemma 2.3 there is a loopγinL(0,6)×(1,2)

α that intersects {1} ×(1,2)and a loopγ⁰ inL^Q_α^int. Similarly there is a loopγ˜inL(0,6)×(1,2)

α that intersects {5} ×(1,2)and a loopγ˜⁰ in L^Q_α^int. Sinceα > ¹₂, γ⁰ and˜γ⁰ belong to the same cluster in L^Q_α^int ([10]). Thus there is a chain of loops (γ0, . . . , γn)in L^Q_α^int, withγ0 = γ⁰ and γ_n = ˜γ⁰, joiningγ⁰ andγ˜⁰. Ifδ is the minimum of diameters of(γ₀, . . . , γ_n)andγ and

˜

γthen C₁(L^Q_α^ext,≥δ)is satisfied. Letδ¯be maximal value of δsuch that C₁(L^Q_α^ext,≥δ)is satisfied.δ¯is a well defined random variable with values in(0,+∞). Then

lim

δ→0⁺P(C1(L^Q_α^ext,≥δ)) = lim

δ→0⁺P(δ≤δ) = 1.¯

Next we recall the result on approximation of Brownian loops by random walk loops from [5]. LetN ∈N^∗. We consider the discrete loopsγonZ×N^∗. We define on these loops a mapΦN to continuous loops onH^{. Given}γa discrete loop and(z0, . . . , z_n−1, z0) the sequence of the vertices it visits, the continuous loopΦNγsatisfies:

• the duration ofΦ_Nγis _2Nⁿ₂;

• forj∈ {0, . . . , n−1},ΦNγ(_2N^j2) =^z_N^j;

• ΦNγ(_2Nⁿ2) = ΦNγ(0) = ^z_N⁰;

• between the times _2N^j2,j∈ {0, . . . , n},ΦNγinterpolates linearly.

The number of jumps n of a discrete loop γ will be denoted sγ. The life-time of a continuous loop˜γwill be denoted bytγ˜. Letθ∈(²₃,2)andr≥1. There is a coupling between L^H_α and L^H_α such that except on an event of probability at most cste·(α+ 1)r²N^2−3θthere is a one to one correspondence between the two sets

• {γ∈ L^H_α|s_γ>2N^θ,|γ(0)|< N r},

• {˜γ∈ L^H_α|tγ˜> N^θ−2,|˜γ(0)|< r},

such that given a discrete loopγand the continuous loopγ˜corresponding to it,

sγ

2N² −t˜γ

≤5

8N⁻², sup

0≤u≤1

ΦNγ

u sγ

2N²

−˜γ(utγ˜)

≤cste·N⁻¹log(N).

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

Lemma 2.5.Letα >0andδ >0. AsN tends to+∞the random set of interpolating continuous loops

ΦNγ|γ∈ L^{N Q}_α ^{ext∩H,≥N δ} converges in law to the set of Brownian loopsL^Q_α^ext,≥δ.

ECP21(2016), paper 30.

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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

{(γ, γ⁰)|γ, γ⁰∈ L^{N Q}_α ^{ext∩H,≥N δ}, γintersectsγ⁰} converges in law to

{(˜γ,˜γ⁰)|˜γ,˜γ⁰ ∈ L^Q_α^ext,≥δ,˜γintersectsγ˜⁰}.

Letj ∈N^{. Let}γbe a continuous path onC(not necessarily a loop) of lifetimet_γ. For r >0let

T_r(γ) := inf{s >0||γ(s)| ≥r} ∈(0,+∞].

IfT_r(γ)<+∞let

e^iωr :=γ(Tr(γ))

r .

LetIj be the real interval

Ij:=7 122^−j, 9

122^−j . For0< r1< r2letA(r1, r2)be the annulus

A(r1, r2) :={z∈C|r1<|z|< r2}.

Forr >0letHD(r)be the half-disc

HD(r) :=B(0, r)∩ {z∈C|<(z)>0}.

We will say that the pathγsatisfies the conditionCj if

• T11

122^−j(γ)<+∞,

• after timeT11

122^−j(γ) < +∞, γ hitse^{i(ω2−j−1+π2)}I_j at a time ˜t_j before hitting the circleS(0,2^−j),

• on the time interval(T₂^−j−1(γ),˜tj)γstays in the half-disce^iω2−j−1HD(2^−j),

• from time˜tj the pathγstays in the annulusA(₁₂⁷2^−j,₁₂⁹2^−j)until surrounding the discB(0,₁₂⁷2^−j)once clockwise and hittinge^{i(ω2−j−1+π)}Ij.

Figure 3 illustrates a path satisfying the condition Cj. If this condition is satisfied than γ disconnects the discB(0,₁₂⁷2^−j) from infinity. Moreover if one perturbsγ by any continuous functionf : [0, t_γ] →C^{such that}kfk∞ ≤ ₁₂¹2^−j then the path(γ(s) + f(s))_0≤s≤tγ disconnects the discB(0,2^−j−1)from infinity. The disconnection is made inside the annulusA(2^−j−1,2^−j).

Lemma 2.6.Let(B_t)_0≤t≤T be a standard Brownian path onCstarting from0. Then almost surely it satisfies the conditionCj for infinitely many values ofj∈N^.

Proof. LetBebe the Brownian pathBcontinued fort∈(0,+∞). The events "Besatisfies the conditionCj" are i.i.d. Indeed such an event is rotation invariant and depends only on Beon the time interval(T₂−j−1(B), Te ₂−j(Be)). Moreover the probability of such an event is non-zero. ThusBe satisfies the conditionCj for infinitely many values ofj ∈N^{. Since}

j→+∞lim T₂^−j(B) = 0,e

so doesB.

Loop percolation on discrete half-plane

Figure 3: Representation of a pathγsatisfying the conditionCj.

Lemma 2.7.Let z₁, z₂ ∈ C ^and t₁, t₂ > 0. Let (b⁽¹⁾s )_0≤s≤t1 and (b⁽²⁾s )_0≤s≤t2 be two independent standard Brownian bridges fromz₁toz₁andz₂toz₂respectively. On the event thatb⁽¹⁾ intersectsb⁽²⁾ there is a.s. ε >0such that for all continuous functions f1: [0, t1]→C^andf2: [0, t2]→Cof infinity normkfik_∞≤ε,(b⁽¹⁾s +f1(s))_0≤s≤t1intersects (b⁽²⁾s +f₂(s))_0≤s≤t2.

Proof. Let T₂⁽¹⁾ be the first time b⁽¹⁾ hits the range of b⁽²⁾. If the two path do not intersect each other T₂⁽¹⁾ = +∞. On the event T₂⁽¹⁾ < +∞ the conditional law of (b⁽¹⁾

T₂⁽¹⁾+s−b⁽¹⁾

T₂⁽¹⁾)_0≤s≤t

1−T₂⁽¹⁾−ε(ε >0a small constant) given the valueT₂⁽¹⁾is absolutely continuous with respect the law of a Brownian path starting from0. From lemma 2.6 follows that the path(b⁽¹⁾

T₂⁽¹⁾+s−b⁽¹⁾

T₂⁽¹⁾)

0≤s≤t1−T₂⁽¹⁾ satisfies the conditionCj for infinitely many values ofj∈N^{. Let}

˜

:= maxn

j∈N|(b⁽¹⁾

T₂⁽¹⁾+s−b⁽¹⁾

T₂⁽¹⁾)_0≤s≤t

1−T₂⁽¹⁾ satisfies the conditionCj

and∃s∈[0, t₂],|b⁽²⁾_s −b⁽²⁾

T₂⁽¹⁾| ≥ 13 122^−jo

.

˜

is a r.v. defined on the event whereb⁽¹⁾ andb⁽²⁾ intersect. Iff1andf2are such that kfik ≤ ₁₂¹2^−˜ then the pathb⁽¹⁾+f₁ disconnects the discB(b⁽¹⁾

T₂⁽¹⁾,2^−˜−1)from infinity inside the annulusb⁽¹⁾

T₂⁽¹⁾ +A(2^−˜−1,2^−˜)and the pathb⁽²⁾+f2 crosses from the circle S(b⁽¹⁾

T₂⁽¹⁾,2^−˜−1)to the circleS(b⁽¹⁾

T₂⁽¹⁾,2^−˜), so the two must intersect.

Observe that two discrete loops γ and γ⁰ intersect each other if and only if the continuous loopsΦ_NγandΦ_Nγ⁰do. From lemmas 2.5 and 2.7 follows:

ECP21(2016), paper 30.

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Corollary 2.8.Letα >0andδ >0. AsN tends to+∞the random set of interpolating continuous loops

ΦNγ|γ∈ L^{N Q}_α ^{ext∩H,≥N δ} jointly with the intersection relations

{(γ, γ⁰)|γ, γ⁰∈ L^{N Q}_α ^{ext∩H,≥N δ}, γintersectsγ⁰}

converges in law to the set of Brownian loopsL^Q_α^ext,≥δ jointly with the intersection relations

{(˜γ,˜γ⁰)|˜γ,˜γ⁰ ∈ L^Q_α^ext,≥δ,˜γintersectsγ˜⁰}.

We consider the scaled up rectangleN QextandN Qint. The next lemma deals with the probability that the discrete loopsL^{N Q}_α ^ext∩Hrealise thespecial crossing event with exterior rectangleN Qextand interior rectangleN Qint. See figures1and2and consider thatQextis replaced byN Qext,QintbyN QintandL^Q_α^ext,≥δ byL^{N Q}_α ^ext∩H.

Lemma 2.9.Letα > ¹₂. As N tends to+∞, the probability that the loops L^{N Q}_α ^ext∩H realise a special crossing event with exterior rectangleN Qextand interior rectangle N Qintconverges to1.

Proof. Letδ >0. The probability that the loopsL^{N Q}_α ^ext∩Hrealise thespecial crossing event with exterior rectangleN Qextand interior rectangleN Qint is at least as large as the probability that the loopsL^{N Q}_α ^{ext∩H,≥N δ} realise thespecial crossing event with the same interior and exterior rectangle. From the corollary 2.8 follows that the latter probability converges asN→+∞to

P

3

\

i=1

C_i(L^Q_α^ext,≥δ)

! .

We conclude by applying the lemma 2.4.

To conclude that forα > ¹₂,L^H_αhas an infinite cluster we will use a block percolation construction that will combinespecial crossing events.

Proof of the Theorem 1.1. From [9] we know already thatα^H_∗ ≤¹₂. We need to show that forα > ¹₂,L^H_αhas an infinite cluster.

Letα > ¹₂ andN ≥1. We consider a dependent edge percolation (ω^N(e))eedge ofH

on the discrete half plane H. If e is an edge of form {(j, k),(j + 1, k)}, k ≥ 1, then ω^N(e) = 1(open edge) ifL^{(N Qint}+3N j+i3N k)∩H

α achieves aspecial crossing event with exterior rectangleN Q_ext+ 3N j+i3N kand interior rectangleN Q_int+ 3N j+i3N k. If eis an edge of form{(j, k),(j, k+ 1)}, k ≥ 1, then ω^N(e) = 1 if L^{(iN Qint}+3N j+i3N k)∩H

α

achieves a special crossing event with exterior rectangle iN Qext+ 3N j+i3N k and interior rectangleiN Qint+ 3N j+i3N k, where the multiplication byimeans rotation by +^π₂. ω^N is a1-dependent edge percolation: if two disjoint subsets of edgesE1andE2are such that no edge is adjacent to bothE₁andE₂, then(ω^N(e))_e∈E1 and(ω^N(e))_e∈E2are independent. This is due to the fact that the subsets of loops involved in the definition ofspecial crossing events for edges inE1and and edges inE2are disjoint. To an open path inω^N corresponds a cluster ofL^H_αwhose loops form crossings of related interior rectangles. Thus ifω^N has an unbounded cluster, then so doesL^H_α. See next picture.

The probabilityP(ω^N(e) = 1)is uniform and we will denote itpN. According to the lemma 2.9

N→+∞lim pN = 1.

Loop percolation on discrete half-plane

Figure 4: Crossings achieved by subsets of loops inL^H_α, corresponding to five open edges inω^N.

Thus forN large enoughp(p˜ N)>¹₂. ¹₂ is the critical probability for the i.i.d. Bernoulli edge percolation onH. So forN large enoughω^N contains a supercritical i.i.d. Bernoulli edge percolation and percolates itself. Thus L^H_α percolates too. Actually, since our construction only uses loops of diameter less or equal to6N, we have also percolation

forL^H,≤6N_α .

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[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.

ECP21(2016), paper 30.

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