1Introduction BalázsRáth ArtëmSapozhnikov Theeffectofsmallquenchednoiseonconnectivitypropertiesofrandominterlacements

El e c t ro nic J

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Electron. J. Probab.18(2013), no. 4, 1–20.

ISSN:1083-6489 DOI:10.1214/EJP.v18-2122

The effect of small quenched noise on connectivity properties of random interlacements

Balázs Ráth

Artëm Sapozhnikov

Abstract

Random interlacements (at level u) is a one parameter family of random subsets ofZ^dintroduced by Sznitman in [22]. The vacant set at leveluis the complement of the random interlacement at level u. While the random interlacement induces a connected subgraph ofZ^dfor all levelsu, the vacant set has a non-trivial phase transition inu, as shown in [22] and [19].

In this paper, we study the effect of small quenched noise on connectivity proper- ties of the random interlacement and the vacant set. For a positiveε, we allow each vertex of the random interlacement (referred to as occupied) to become vacant, and each vertex of the vacant set to become occupied with probabilityε, independently of the randomness of the interlacement, and independently for different vertices. We prove that for anyd≥3andu >0, almost surely, the perturbed random interlace- ment percolates for small enough noise parameterε. In fact, we prove the stronger statement that Bernoulli percolation on the random interlacement graph has a non- trivial phase transition in wide enough slabs. As a byproduct, we show that any electric network with i.i.d. positive resistances on the interlacement graph is tran- sient, which strengthens our result in [17]. As for the vacant set, we show that for anyd≥3, there is still a non-trivial phase transition inuwhen the noise parameter εis small enough, and we give explicit upper and lower bounds on the value of the critical threshold, whenε→0.

Keywords: Random interlacements; Bernoulli percolation; slab; vacant set; quenched noise;

long-range correlations; transience.

AMS MSC 2010:60K35; 82B43.

Submitted to EJP on June 29, 2012, final version accepted on January 4, 2013.

SupersedesarXiv:1109.5086.

1 Introduction

The model of random interlacements was recently introduced by Sznitman in [22] in order to describe the local picture left by the trajectory of a random walk on the discrete

∗The research of both authors has been supported by the grant ERC-2009-AdG 245728-RWPERCRI.

†The University of British Columbia, Department of Mathematics, Vancouver, Canada.

E-mail:rathb@math.ubc.ca

‡Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany.

E-mail:artem.sapozhnikov@mis.mpg.de

torus(Z/NZ)^d,d≥3when it runs up to times of orderN^d, or on the discrete cylinder (Z/NZ)^d×Z^,d≥2, when it runs up to times of orderN^2d, see [20], [29]. Informally, the random interlacement Poisson point process consists of a countable collection of doubly infinite trajectories on Z^d, and the trace left by these trajectories on a finite subset ofZ^d“looks like” the trace of the above mentioned random walks.

The set of vertices visited by at least one of these trajectories is the random inter- lacement at leveluof Sznitman [22], and the complement of this set is the vacant set at levelu. These are one parameter families of translation invariant, ergodic, long-range correlated random subsets ofZ^d, see [22]. We call the vertices of the random inter- lacement occupied, and the vertices of the vacant set vacant. While the set of occupied vertices induces a connected subgraph ofZ^d for all levelsu, the graph induced by the set of vacant vertices has a non-trivial phase transition inu, as shown in [22] and [19].

The effect of introducing a small amount of quenched disorder into a system with long-range correlations on the phase transition has got a lot of attention (see, e.g., [11], [28], [2], [3]). In this paper we consider how small quenched disorder affects the connectivity properties of the random interlacement and the vacant set. For ε > 0, given a realization of the random interlacement, we allow each vertex independently to switch from occupied to vacant and from vacant to occupied with probabilityε, and we study the effect it has on the existence of an infinite connected component in the graphs of occupied or vacant vertices.

We prove that for anyd ≥3 and u >0, almost surely, the set of occupied vertices percolates for small enough noise parameterε. In fact, we prove the stronger statement that Bernoulli percolation on the random interlacement graph has a non-trivial phase transition in wide enough slabs. The two main ingredients of our proof are a strong connectivity lemma for the interlacement graph proved in [17] and Sznitman’s decou- pling inequalities from [23]. As a byproduct, we show that any electric network with i.i.d. positive resistances on the interlacement graph is transient, which strengthens our result in [17].

We also prove that for any d ≥3, the set of vacant vertices still undergoes a non- trivial phase transition inuwhen the noise parameterεis small enough, and give ex- plicit upper and lower bounds on the value of the threshold, whenε→0. The bounds that we derive suggest that the vacant set phase transition is robust with respect to noise, which we state as a conjecture.

1.1 The model

Forx∈Z^d,d≥3, letPxbe the law of a simple random walkX onZ^dwithX(0) =x. LetKbe a finite subset ofZ^d. The equilibrium measure ofKis defined by

eK(x) =Px[X(t)∈/ Kfor allt≥1], forx∈K,

andeK(x) = 0forx /∈K. The capacity ofKis the total mass of the equilibrium measure ofK:

cap(K) =X

x

eK(x).

Sinced≥3, for any finite setK⊂Z^d, the capacity ofKis positive. Therefore, we can define the normalized equilibrium measure by

eeK(x) =eK(x)/cap(K).

Let W be the space of doubly-infinite nearest-neighbor trajectories inZ^{d (}d ≥ 3) which tend to infinity at positive and negative infinite times, and letW^∗be the space of equivalence classes of trajectories inW modulo time-shift. We writeWfor the canonical

σ-algebra onW generated by the coordinate maps, andW^∗for the largestσ-algebra on W^∗for which the canonical mapπ^∗from(W,W)to(W^∗,W^∗)is measurable.

Let µ be a Poisson point measure on W^∗. For a finite subset K of Z^d, denote by µK the restriction ofµto the set of trajectories fromW^∗ that intersectK, and byNK

be the number of trajectories inSupp(µK). The point measureµK can be written as µK =PN_K

i=1δπ^∗(X_i), whereXi are doubly-infinite trajectories fromW parametrized in such a way thatX_i(0)∈KandX_i(t)∈/Kfor allt <0and for alli∈ {1, . . . , NK}.

Foru >0, we say that a Poisson point measureµonW^∗has distributionPois(u, W^∗) if the following properties hold:

(1) The random variableN_K has Poisson distribution with parameterucap(K). (2) Given NK, the points Xi(0), i ∈ {1, . . . , NK}, are independent and distributed

according to the normalized equilibrium measure onK.

(3) GivenN_Kand(X_i(0))^N_i=1^K, the corresponding forward and backward paths are con- ditionally independent, (Xi(t), t ≥ 0)^N_i=1^K are distributed as independent simple random walks, and(Xi(t), t≤0)^N_i=1^K are distributed as independent random walks conditioned on not hittingK.

Properties (1)-(3) uniquely define Pois(u, W^∗), as proved in Theorem 1.1 in [22]. In fact, Theorem 1.1 in [22] gives a coupling of the Poisson point measures µ(u) with distributionPois(u, W^∗)for allu >0. We refer the reader to [22] for more details.

LetE^dbe the set of edges ofZ^{d, i.e.,}E^d={{x, y} : x, y∈Z^d,|x−y|₁= 1}. We will use the following convention throughout the paper. For a subsetJ ofE^d, the subgraph of the lattice(Z^d,E^d)with the vertex setZ^dand the edge setJ will be also denoted by J.

For a Poisson point measureµwith distributionPois(u, W^∗), therandom interlace- ment I^u =I^u(µ)(at levelu) is defined in [22] as the set of vertices ofZ^dvisited by at least one of the trajectories fromSupp(µ). This is a translation invariant and ergodic random subset ofZ^d, as shown in [22, Theorem 2.1]. The law ofI^uis characterized by the identity (see (0.10) and Remark 2.2 (2) in [22]):

P[I^u∩K=∅] =e^−ucap(K), for all finiteK⊆Z^d.

We denote byIe^u=Ie^u(µ)the set of edges ofE^dtraversed by at least one of the trajecto- ries fromSupp(µ). The corresponding random subgraphIe^uof(Z^d,E^d)(with the vertex set Z^d and the edge set Ie^u) is called the random interlacement graph(at level u). It follows from Theorem 2.1 and Remark 2.2(4) of [22] thatIe^u is a translation invariant ergodic random subgraph of(Z^d,E^d). LetV^u=Z^d\ I^ube thevacant setat levelu.

Given a parameter ε ∈ (0,1), we consider the family θx, x ∈ Z^d, of independent Bernoulli random variables (an independent noise) with parameter ε, and define ε- disordered analogues of the random interlacementI^u,ε and the vacant setV^u,ε as fol- lows. We say thatx∈ I^u,ε ifx∈ I^u andθ_x = 0orx∈ V^u andθ_x = 1. In other words, the vertices of the random interlacement get anε-chance to become vacant, and the vertices of the vacant set get anε-chance to become occupied. LetV^u,ε=Z^d\ I^u,ε. We are interested in percolative properties ofI^u,ε andV^u,ε. It follows from Remark 1.6(4) in [22] that for anyd≥3andu >0,

cov_u[1(x∈ V^u),1(y∈ V^u)](1 +|x−y|∞)^2−d, forx, y∈Z^d,

where cov_u denotes the covariance under Pois(u, W^∗). This displays the presence of long-range correlations inV^u. Non-rigorous study of the effect of small quenched noise on the critical behavior of a system with long-range correlations was initiated in [11, 28].

It was shown, among other results, in [22] that the random interlacement graphIe^u consists of a unique infinite connected component and isolated vertices. (Refinements of this result were obtained in [12, 15, 16].) In [17], we showed that the random in- terlacement graph is almost surely transient for any u > 0 in dimensions d ≥ 3. In Theorem 2.1 of the present paper, we prove that for anyu >0and small enoughε >0, the setI^u,εstill contains an infinite connected component. In fact, Theorem 2.1 implies thatI^u andIe^u still have an infinite connected component in wide enough slabs, even after a small positive density of vertices ofI^u, respectively edges of Ie^u, is removed.

One might interpret all these results as an evidence of the heuristic statement that the geometry of the interlacement graph is similar to that of the underlying latticeZ^{d. Re-} cently, this question has been settled in [4] by a clever refinement of the techniques in [16, 17]. It was proved in [4] (and later in [8] with a different, model independent proof) that the graph distance in I^u is comparable to the graph distance in Z^{d, and} a shape theorem holds for balls with respect to graph distance on I^u. First results about heat-kernel bounds for the random walk onI^u have been recently obtained in [14, Theorem 2.3].

An important role in understanding the local picture left by the trajectory of a ran- dom walk on the discrete torus(Z/NZ)^d,d≥3or the discrete cylinder(Z/NZ)^d×Z^, d≥2is played by

u_∗= inf{u≥0 : P[0↔ ∞inV^u] = 0}

(see, e.g., [21, 26]). It follows from [22, (1.53) and (1.55)] that for u < u⁰, the set V^u0 is stochastically dominated byV^u. Therefore, for all u > u_∗,P[0 ↔ ∞inV^u] = 0. Moreover, by [19, 22],u_∗∈(0,∞), i.e., there is a non-trivial phase transition forV^uinu atu_∗. In Theorem 2.3 of this paper, we prove that for small enoughε, theε-disordered vacant setV^u,ε still undergoes a non-trivial phase transition in u. In Theorem 7.4 we give explicit upper and lower bounds on the phase transition threshold for V^u,ε, as ε→0. These bounds suggest that the phase transition is actually robust with respect to noise. We state it as a conjecture in Remark 7.5.

2 Main results

For p ∈ (0,1), we define the random subset Be^p of E^d by deleting each edge with probability(1−p)and retaining it with probabilityp, independently for all edges, and, similarly, the random subsetB^p ofZ^d by deleting every vertex of Z^d with probability (1−p) and retaining it with probabilityp, independently for all vertices. We look at the random subgraphs of(Z^d,E^d)with vertex setZ^dand edge setIe^u∩Be^p, and the one induced by the set of verticesI^u∩ B^p⊂Z^d.

Our first theorem states that the graphsI^uandeI^uhave infinite connected subgraphs in a wide enough slab, moreover, Bernoulli bond percolation onIe^u and Bernoulli site percolation onI^urestricted to this slab have a non-trivial phase transition.

Theorem 2.1. Let d ≥ 3 and u > 0. There exist p < 1 and R ≥ 1 such that, almost surely, the random graphsI^u∩ B^p andIe^u∩Be^p contain infinite connected components in the slabZ²×[0, R)^d−2.

As a byproduct of the proof of Theorem 2.1, we obtain the following generalization of the main result in [17].

Theorem 2.2. Let d ≥ 3 and u > 0. Let R

ee, ee∈ E^d be independent identically dis- tributed positive random variables. The electric network{ee : ee∈Ie^u}with resistances Reeis almost surely transient, i.e., the effective resistance between any vertex inIe^uand infinity is finite.

Theorem 2.2 is a generalization of the main result of [17], since the transience of the unique infinite connected component of the random interlacement graphIe^ufollows from the case whenR

eeare almost surely equal to1 (see, e.g., [6]). The result of The- orem 2.2 is equivalent (see the main result of [13]) to the following statement: for any u >0, there existsp <1 such that the graphIe^u∩Be^p contains a transient component, i.e., the simple random walk on it is transient. The proof of this fact will come as a byproduct of the proof of Theorem 2.1.

The main idea of the proofs of Theorems 2.1 and 2.2 is renormalization. We parti- tion the graphZ^d into disjoint blocks of equal size. A block is called good if the graph Ie^u contains a unique large connected component in this block and all the edges of the block are inBe^p, otherwise it is called bad. A more precise definition will be given in Sec- tion 5. It will be shown that paths of good blocks contain paths ofIe^u∩Be^p. In particular, percolation of good blocks implies percolation ofIe^u∩Be^p. Using the strong connectivity result of [17], stated as Lemma 3.1 below, we show that a block is good with probability tending to1, as the size of the block increases. We then use the decoupling inequalities of [23], stated as Theorem 3.2 below, to show in Lemma 5.2 that∗-connected compo- nents of bad blocks are small. With the result of Lemma 5.2, the existence statement of Theorem 2.1 follows using a standard duality argument, and the proof of Theorem 2.2 is reminiscent of the proof of Theorem 1 in Section 3 of [17].

In our next theorem, we show that for small enoughε >0, theε-disordered vacant setV^u,εundergoes a non-trivial phase transition inu. Let

u_∗(ε) = inf{u≥0 : P[0↔ ∞inV^u,ε] = 0}.

Theorem 2.3. Letd≥3. For anyε∈(0,1/2)andu > u_∗(ε), P[0↔ ∞inV^u,ε] = 0.

In other words, for ε ∈ (0,1/2), the ε-disordered vacant set V^u,ε undergoes a phase transition inuatu_∗(ε). Moreover, there existsε0>0such that for allε≤ε0,

0< u_∗(ε)<∞.

The first statement of Theorem 2.3 is proved in Lemma 7.1. It follows from a stan- dard coupling argument and the fact that the setV^u0 is stochastically dominated byV^u foru < u⁰ (see [22, (1.53) and (1.55)]). The second statement of Theorem 2.3 follows from the more general statement of Theorem 7.4, in which we give explicit upper and lower bounds onu_∗(ε), asε→0. The proof of Theorem 7.4 uses renormalization, and is very similar in spirit to the proof of Theorem 2.1.

The bounds onu∗(ε)that we obtain in Theorem 7.4 are in terms of certain thresholds describing local behavior ofV^u in sub- and supercritical regimes (see (7.4) and Defini- tion 7.3, respectively). In particular, they are purely in terms ofV^uand notV^u,ε. As we discuss in Remark 7.5, these thresholds are conjectured to coincide withu_∗, therefore it is reasonable to believe that the phase transition ofV^uis stable with respect to small random noise. In other words, the following conjecture holds:

ε→0limu_∗(ε) =u_∗.

Finally, note that it is essential foru∗(ε)<∞that the parameterεis small. For example, since V^u,1/2 has the same law as the Bernoulli site percolation with parameter 1/2, which is supercritical in dimensionsd≥3(see [1]), we obtain thatu_∗(1/2) =∞.

We now describe the structure of the remaining sections of the paper. We recall the strong connectivity lemma of [17] and the decoupling inequalities of [23] in Section 3.

In Section 4 we construct and study seed events which are used in Section 5 to define good blocks. Lemma 5.2, the main ingredient of the proofs of Theorems 2.1 and 2.2, is proved in Section 5. The proofs of Theorems 2.1 and 2.2 are given in Section 6, and the proof of Theorem 2.3 is given in Section 7, where we also give explicit bounds onu_∗(ε), asε→0.

3 Notation and known results

In this section we introduce basic notation and collect some properties of the random interlacements, which are recurrently used in our proofs.

3.1 Notation

For a ∈ R^{, we write} |a| for the absolute value of a, and bac for the integer part of a. For (x1, . . . , xd) = x ∈ Z^{d, we write} |x|∞ for the l^∞-norm of x, i.e.,

|x|∞= max (|x1|, . . . ,|xd|), and|x|1for thel¹-norm ofx, i.e.,|x|1=Pd

i=1|xi|. ForR >0 andx∈Z^{d, let}B(x, R) ={y∈Z^d : |x−y|_∞≤R}be thel^∞-ball of radiusRcentered atx, andB(R) =B(0, R).

For x ∈ Z^d and integers m < n, we write x+ [m, n)^d for the set of vertices y = (y1, . . . , yd) ∈ Z^{d with} m ≤ yi −xi < n for all i ∈ {1, . . . , d}. For ee ∈ E^{d, we write} ee ∈ x+ [m, n)^d if both of its endvertices are inx+ [m, n)^d. If Je⊆ E^d, we denote by Je∩(x+ [m, n)^d)the set of edges ofJewith both endvertices inx+ [m, n)^d. Forx, y∈Z^d, we writex↔yinJe, ifxandyare in the same connected component of the graphJe.

Let(Ω1,F1,P^u), withΩ1={0,1}^Edand the canonicalσ-algebraF1, be the probabil- ity space on whicheI^uis defined. Forω ∈Ω1, we say thatee∈E^{dis in}Ie^u whenω

ee= 1. Let(Ω₂,F2,P_p), withΩ₂ ={0,1}^Ed and the canonicalσ-algebraF2, be the probability space on which Be^p is defined. Forω ∈ Ω2, we say thatee ∈ E^{d is in} Be^p when ω

ee = 1. Finally, let(Ω,F,P) = (Ω1×Ω2,F1× F2,P^u⊗Pp)denote the probability space on which the random interlacement graph Ie^u and Bernoulli bond percolation configuration Be^p are jointly defined.

Throughout the paper, we use the following notational agreement. For eventsA1∈ F1 andA2 ∈ F2, we denote the corresponding events A1×Ω2 and Ω1×A2 inF also byA1 andA2, respectively. We denote by1(A)the indicator of eventAand by A^c the complement ofA. Fori∈ {1,2}, given a random subsetJe(ω)ofE^{d, with}ω∈Ω_i, and an eventA∈ F_i, we define

A(Je) ={ω∈Ωi : χ

Je(ω)∈A}, (3.1)

where foree ∈ E^d, χ

Je(ω)(ee)equals 1 if ee∈ Je(ω), and 0 otherwise. Conversely, for an elementω∈ {0,1}^Ed, let

Gω={ee : ω

ee= 1}. (3.2)

(By our convention, we also denote byGω the graph with the vertex set Z^{d and the} edge set {ee : ω

ee= 1}.) An event A ∈ F1 is called increasing, if for any ω ∈ A, all the elements ω⁰ with G_ω⁰ ⊇ G_ω are in A. The event A is called decreasing, if A^c is increasing. Throughout the text, we write c and C for small positive and large finite constants, respectively, that may depend on d and u. Their values may change from place to place.

3.2 Strong connectivity property

The following strong connectivity lemma follows from Proposition 1 in [17].

Lemma 3.1. Letd≥3,u >0, andε >0. There exist constantsc =c(d, u, ε)>0 and C=C(d, u, ε)<∞such that for allR≥1,

P





\

x,y∈I^u∩[0,R)^d

n

x↔y in Ie^u∩[−εR,(1 +ε)R)^do



≥1−Cexp(−cR^1/6).

Lemma 3.1 may seem more general than Proposition 1 in [17], but, in fact, the two results are equivalent. In order to see this, the reader may check how Proposition 1 is derived from Lemma 13 in [17].

3.3 Decoupling inequalities Let

l(d) = 30·4^d. (3.3)

(The choice ofl(d)will be justified in the proof of Lemma 5.2.) LetL0 andl0 ≥l(d)be positive integers. We introduce the geometrically increasing sequence of length scales

Ln=lⁿ₀L0, n≥1.

Forn≥0, we introduce the renormalized lattice graphGnby Gⁿ=LnZ^d={Lnx : x∈Z^d}.

Forx∈G^nandn≥0, let

Λx,n=Gⁿ⁻¹∩(x+ [0, Ln)^d).

Let Ψ

ee,ee∈ E^d denote the canonical coordinates on{0,1}^Ed. Forx∈ G0, letG_x = G_x,0 =G_x,0,L0 be aσ(Ψ

ee,ee∈x+ [−L₀,3L₀)^d)-measurable event. We call events of the formGx,0,L₀seed events. We denote the family of events(Gx,0,L₀ : L0≥1, x∈G⁰)byG. Examples of seed events important for this paper will be considered in Section 4. The reader should think about the eventsGx,0,L₀as “bad” events. Now we recursively define bad events on higher length scales using seed events. Forn≥1andx∈Z^d, denote by G_x,n =G_x,n,L0 the event that there existx₁, x₂ ∈ Λ_x,n with|x1−x₂|∞ > L_n/l(d)such that the eventsG_x1,n−1andG_x2,n−1occur:

G_x,n= [

x₁,x₂∈Λ_x,n;|x₁−x₂|∞>_l(d)^Ln

G_x1,n−1∩G_x2,n−1 . (3.4)

(For simplicity, we omit the dependence ofGx,n on L0 from the notation.) Note that Gx,nisσ(Ψ

ee,ee∈x+ [−Ln,3Ln)^d)-measurable. (This can be shown by induction onn.) Recall the definition (3.1). The following theorem is a special case of Theorem 3.4 in [23] (modulo some minor changes that we explain in the proof).

Theorem 3.2. For alld≥3,u >0andδ∈(0,1), there existsC =C(d, u, δ)<∞such that for alln≥0,L₀≥1, andl₀≥Ca multiple ofl(d), we have

1. ifG_xare decreasing events, then for allu⁰ ≥(1 +δ)u,

Ph

G_0,n(eI^u0)i

≤ l^2d₀ sup

x∈G0∩[0,Ln)^d

Ph

G_x(eI^u)i +1

4

!2ⁿ

, (3.5)

2. ifGxare increasing events, then for allu⁰≤(1−δ)u,

Ph

G0,n(eI^u0)i

≤ l^2d₀ sup

x∈G0∩[0,Ln)^d

Ph

Gx(eI^u)i +1

4

!2ⁿ

. (3.6)

Proof of Theorem 3.2. We refer the reader to Section 3 of [23] for the notation. Our eventsG_x,ncorrespond to the eventsG_x,Lnof [23],Λ_x,nplays the role ofΛ, thusc(G, l) = 1andλ=din Definition 3.1 of [23]. There are a number of comments we would like to make before applying results derived in Section 3 of [23]:

(1) Even though the events Gx,L_n in [23] pertain to the occupancy of vertices (i.e., they are subsets of{0,1}^Zd), Theorem 3.4 in [23] also applies in the setting when the eventsGx,L_n pertain to the occupancy of edges (i.e., they are subsets of{0,1}^Ed), see Theorem 2.1, Remark 2.5(3) and Corollary 2.1’ of [23].

(2) The constantl(d)is taken to be100 in Definition 3.1 in [23], but Theorem 3.4 in [23] works for any large enough constantl(d), withl0> l(d)also large enough.

(3) The eventsG_x,ndefined by (3.4) are not cascading in the sense of Definition 3.1 in [23], because(3.4)of [23] only holds forl=l₀rather than for alllwhich is a multiple of100. Nevertheless, the statement and the proof of Theorem 3.4 in [23] only involve eventsGx,L_n, withLn=lⁿ₀L0for some previously fixedL0≥1andl0(wherel0is large enough).

Taking the above remarks into account, we can apply Theorem 3.4 of [23] to the eventsG_x,n. In order to derive (3.5) and (3.6) from Theorem 3.4 of [23], we choose l₀ large enough, so thatu⁺_∞≤(1 +δ)u,u⁻_∞ ≥(1−δ)u, andl₀^2dε(u⁻_∞)≤1/4. (See, e.g., the calculations in (3.37) of [23].)

Remark 3.3. Currently, Theorem 3.4 in [23] (and, as a result, Theorem 3.2 of this paper) is proved only for increasing and decreasing events. It would be interesting to show that the result of Theorem 3.4 in [23] holds for a more general class of events.

Corollary 3.4. Let d ≥ 3, u > 0 and δ ∈ (0,1). Let Gx be all increasing events and u⁰ = (1−δ)u, or all decreasing events andu⁰= (1 +δ)u. If

lim inf

L0→∞ sup

x∈G0∩[0,Ln)^d

Ph

Gx(eI^u)i

= 0, (3.7)

then there existl₀, L₀≥1such that for alln≥0, Ph

G0,n(eI^u0)i

≤2⁻²ⁿ. (3.8)

Moreover, if the limit in (3.7) (as L₀ → ∞) exists and equals to 0, then there exists C=C(d, u, δ)<∞such that the inequality (3.8)holds for alll₀≥Ca multiple ofl(d), L₀≥C⁰(d, u, δ, l₀, G)(for some constantC⁰(d, u, δ, l₀, G)), andn≥0.

4 Seed events

In this section we apply Corollary 3.4 to two families of (decreasing and increasing) bad events defined in terms of Ie^u. We also recursively define a similar (but simpler) family of bad events in terms ofBe^pand derive results analogous to Corollary 3.4 for this family given thatpis close enough to1. The corresponding seed events will be used in Section 5 to define good vertices in G⁰. The good vertices will have the property that the existence of an infinite path of good vertices inG0implies the existence of an infinite path in the graphIe^u∩Be^p, as stated formally in Lemma 5.1.

We define the density of the interlacement at levelu(see, e.g., (1.58) in [22]) by m(u) =P(0∈ I^u) = 1−e^−u/g(0),

wheregis the Green function of the simple random walk onZ^{dstarted at}0. The function mis continuous.

Note thatx∈ I^u if and only if{x, y} ∈Ie^ufor somey∈Z^{d, thus}I^uis a measurable function of Ie^u. It follows from Theorem 2.1 and Remark 2.2(4) of [22] that Ie^u is a translation invariant ergodic random subset ofE^d. By an appropriate ergodic theorem (see, e.g., Theorem VIII.6.9 in [9]), we get

L→∞lim 1 L^d

X

x∈[0,L)^d

1

∃y∈[0, L)^d : {x, y} ∈Ie^u _P-a.s.

= m(u). (4.1)

4.1 Bad decreasing events

In this subsection we define and study a family of bad decreasing σ(Ψ

ee, ee ∈ x+ [0,2Ln)^d)-measurable eventsE^u_x,nwith (see (3.4))

E^u_x,n= [

x1,x2∈Λx,n;|x1−x2|∞>_l(d)^Ln

E^u_x1,n−1∩E^u_x2,n−1 ,

forn ≥1, andPh

E^u_0,n(eI^u)i

≤2⁻²ⁿ. In order to define the bad decreasing seed event E^u_x=E^u_x,0, we define its complement, the “good” increasing eventE_x^u= (E^u_x)^c.

Definition 4.1. Fixu >0. Recall the definition of the graphGωin(3.2). LetE_x^ube the measurable subset of{0,1}^Ed such thatω∈E_x^uiff

(a) for alle∈ {0,1}^d, the graphGω∩(x+eL0+ [0, L0)^d)contains a connected compo- nent with at least ³₄m(u)L^d₀vertices,

(b) all of these2^dcomponents are connected in the graphG_ω∩(x+ [0,2L₀)^d). Note thatE_x^u is an increasingσ(Ψ

ee, ee∈x+ [0,2L0)^d)-measurable event. Moreover, ifJe(ω)is a random translation invariant subset ofE^{d, then}P[E_x^u(Je)] =P[E₀^u(Je)]for all x∈Z^d.

Lemma 4.2. For anyu >0there existsδ >0such that

P[E₀^u(eI^u/(1+δ))]→1, as L0→ ∞. (4.2) Proof of Lemma 4.2. Let u > 0. By the continuity ofm(u), we can choose ε > 0 and δ >0so that

(1−4ε)^dm u

1 +δ

>3 4m(u).

With such a choice ofεandδ, forL0≥1, we obtain m

u 1 +δ

(L₀−4bεL₀c)^d> 3

4m(u)L^d₀. (4.3)

Letu⁰=u/(1 +δ). We consider the boxes

Be=eL0+ [ 2bεL0c, L0−2bεL0c)^d, e∈ {0,1}^d.

The volume of Be is |Be| = (L0 −4bεL0c)^d. Using (4.1) and (4.3), we get that with probability tending to1 asL0 → ∞, each of the boxesBe,e∈ {0,1}^d contains at least

3

4m(u)L^d₀vertices ofI^u0.

Now by Lemma 3.1, all the vertices ofI^u0∩Beare connected inIe^u0∩(eL0+[bεL0c, L0− bεL0c)^d)for alle∈ {0,1}^dwith probability tending to1asL₀→ ∞. This shows that the event in Definition 4.1 (4.1) holds with probability tending to1asL₀→ ∞.

Again by Lemma 3.1, the vertices ofI^u0∩(eL₀+ [bεL₀c, L₀− bεL₀c)^d),e∈ {0,1}^dare all connected ineI^u0∩[0,2L0)^d. This, together with the previous conclusion, implies that the event in Definition 4.1 (4.1) holds with probability tending to1asL0→ ∞. Hence we have established (4.2).

Corollary 4.3. For eachu >0, there existsC =C(d, u)<∞such that for all integers l₀≥Ca multiple ofl(d)(see (3.3)),L₀≥C⁰(d, u, l₀)(for some constantC⁰(d, u, l₀)), and n≥0,

Ph

E^u_0,n(eI^u)i

≤2⁻²ⁿ.

Proof. Indeed, it immediately follows from Corollary 3.4 and Lemma 4.2.

4.2 Bad increasing events

In this subsection we define and study a family of bad increasing σ(Ψ

ee, ee ∈ x+ [0,2L_n)^d)-measurable eventsF^u_x,nwith (see (3.4))

F^u_x,n= [

x₁,x₂∈Λx,n;|x1−x2|∞>_l(d)^Ln

F^u_x1,n−1∩F^u_x2,n−1 ,

forn≥ 1, and Ph

F^u_0,n(eI^u)i

≤ 2⁻²ⁿ. In order to define the bad increasing seed event F^u_x=F^u_x,0, we define its complement, the “good” decreasing eventF_x^u= (F^u_x)^c.

Definition 4.4. Letu >0. LetF_x^ube the measurable subset of{0,1}^Edsuch thatω∈F_x^u iff for alle∈ {0,1}^d, the graphGω∩(x+eL0+[0, L0)^d)contains at most⁵₄m(u)L^d₀vertices in connected components of size at least2, i.e.,

X

y∈x+eL₀+[0,L₀)^d

1 ∃z∈ x+eL0+ [0, L0)^d : {y, z} ∈Gω

≤ 5

4m(u)L^d₀. (4.4) Note thatF_x^uis a decreasingσ(Ψ

ee, ee∈x+ [0,2L0)^d)-measurable event. Moreover, if Je(ω)is a random translation invariant subset ofE^{d, then}P[F_x^u(Je)] =P[F₀^u(Je)]. Lemma 4.5. For anyu >0there existsδ∈(0,1)such that

P[F₀^u(eI^u/(1−δ))]→1, as L0→ ∞. (4.5) Proof of Lemma 4.5. Letu >0. By the continuity ofm(u), we can chooseδ >0so that

m u

1−δ

< 5 4m(u).

Therefore, (4.1) implies that, with probability tending to1 asL0 → ∞, the inequality (4.4) withGωreplaced byIe^u/(1−δ)is satisfied for alle∈ {0,1}^d. This implies (4.5).

Corollary 4.6. For eachu >0, there existsC =C(d, u)<∞such that for all integers l₀≥Ca multiple ofl(d)(see (3.3)),L₀≥C⁰(d, u, l₀)(for some constantC⁰(d, u, l₀)), and n≥0,

Ph

F^u_0,n(eI^u)i

≤2⁻²ⁿ.

Proof. Indeed, it immediately follows from Corollary 3.4 and Lemma 4.5.

4.3 Bad Bernoulli events

In this subsection we define and study a family of bad decreasing σ(Ψ

ee, ee ∈ x+ [0,2L_n)^d)-measurable eventsD_x,nin the spirit of the definition (3.4):

D_x,n= [

x₁,x₂∈Λx,n;|x1−x2|∞>_l(d)^Ln

D_x1,n−1∩D_x2,n−1 ,

forn≥1, andPh

D0,n(Be^p)i

≤2⁻²ⁿ whenp <1is close enough to1. We define the bad decreasing seed eventDx=Dx,0as the measurable subset of{0,1}^Edsuch thatω∈Dx

iff there is an edge in the boxx+ [0,2L0)^dwhich is not inGω(remember that an edgeee is inx+ [m, n)^dif both its endvertices are inx+ [m, n)^d), i.e.,

Dx=n

ω∈ {0,1}^Ed : (x+ [0,2L0)^d)∩E^d*Gω

o

. (4.6)

Note thatD_x is a decreasing σ(Ψ

ee, ee∈ x+ [0,2L₀)^d)-measurable event. Moreover, if Je(ω)is a random translation invariant subset ofE^{d, then}P[D_x(Je)] =P[D₀(Je)]. Lemma 4.7. For any integersL₀ ≥1andl₀ >2l(d)there existsp <1such that for all n≥0,

Ph

D0,n(Be^p)i

≤2⁻²ⁿ.

Proof of Lemma 4.7. Since the probability of D₀(Be^p) is at most 1−p^d(2L0)d, we can choosep=p(L0, l0)<1so that

l^2d₀ Ph

D₀(Be^p)i

<1/2.

Note that for x₁, x₂ ∈ Gn−1, |x₁ − x₂|_∞ ≥ L_n/l(d), the events D_x1,n−1(Be^p) and Dx2,n−1(Be^p)are independent and have the same probability. Therefore, since|Λx,n| ≤ l₀^d, we get

Ph

D_0,n(Be^p)i

≤l^2d₀ Ph

D_0,n−1(Be^p)i2

≤. . .

≤ l₀^{2d1+2+...+2n−1} Ph

D0(Be^p)i2ⁿ

≤ l^2d₀ Ph

D0(Be^p)i2ⁿ

.

The result follows from the choice ofp.

5 Connected components of bad boxes are small

Forx, y∈G0, we say thatxandy are nearest-neighbors inG0 if|x−y|1 =L₀, and

∗-neighbors inG0 if|x−y|_∞ =L₀. We say thatπ= (x(1), . . . , x(m))⊂G0 is a nearest- neighbor path inG⁰, if for all j, x(j)and x(j+ 1) are nearest-neighbors in G^{0, and a}

∗-path inG⁰, if for allj,x(j)andx(j+ 1)are∗-neighbors inG^0.

Let u >0and p∈(0,1). Recall the definitions of the bad seed eventsE^u_x = (E_x^u)^c, F^u_x = (F_x^u)^c andDx from Definition 4.1, Definition 4.4 and (4.6), respectively. We say thatx∈G0is abadvertex if the event

D_x(Be^p)∪E^u_x(eI^u)∪F^u_x(eI^u)

occurs. Otherwise, we say that xis good. The following lemma will be useful in the proofs of Theorems 2.1 and 2.2.

Lemma 5.1. Let xand y be nearest-neighbors inG⁰, and assume that they are both good.

(a) Each of the graphs(eI^u∩Be^p)∩(z+ [0, L₀)^d), withz∈ {x, y}, contains the unique connected componentC_z with at least ³₄m(u)L^d₀ vertices, and

(b)CxandCyare connected in the graph(eI^u∩Be^p)∩((x+ [0,2L₀)^d)∪(y+ [0,2L₀)^d)). In particular, this implies that if there is an infinite nearest-neighbor pathπ= (x₁, . . .)of good vertices inG⁰, then the set∪^∞_i=1(xi+[0,2L0)^d)contains an infinite nearest-neighbor path ofIe^u∩Be^p.

Proof. Letxandybe nearest-neighbors inG0, and assume that they are both good. By Definition 4.1, the graphsIe^u∩(x+ [0, L₀)^d)and eI^u∩(y+ [0, L₀)^d)contain connected components of size at least ³₄m(u)L^d₀, which are connected in the graph Ie^u ∩((x+ [0,2L0)^d)∪(y+ [0,2L0)^d)).

By Definition 4.4, each of the graphseI^u∩(x+[0, L0)^d)andIe^u∩(y+[0, L0)^d)contains at most ⁵₄m(u)L^d₀vertices in connected components of size at least2. Since2·³₄ > ⁵₄, there can be at most one connected component of size≥³₄m(u)L^d₀in each of the graphsIe^u∩ (x+ [0, L0)^d)andIe^u∩(y+ [0, L0)^d). This impies that each of the graphsIe^u∩(z+ [0, L0)^d), with z ∈ {x, y}, contains the unique connected component Cz with at least ³₄m(u)L^d₀ vertices, andCxandCyare connected in the graphIe^u∩((x+ [0,2L0)^d)∪(y+ [0,2L0)^d)). Finally, by (4.6), ((x+ [0,2L₀)^d)∪(y+ [0,2L₀)^d))⊆Be^p. Therefore, all the edges of the graphIe^u∩((x+ [0,2L₀)^d)∪(y+ [0,2L₀)^d))are present inBe^p.

Forx∈G^{0, and}M < N which are divisible byL0, letH^∗(x, M, N)be the event that B(x, M)is connected to the boundary ofB(x, N)by a∗-path of bad vertices inG^{0. Let} H^∗(x, N) =H^∗(x,0, N)be the event thatxis connected to the boundary ofB(x, N)by a∗-path of bad vertices inG0.

Lemma 5.2. For anyu >0, there existL₀ ≥1,p <1,c >0andC <∞(all depending onu) such that for allN divisible byL0, we have

P[H^∗(0, N)]≤Ce^−Nc. (5.1)

Proof of Lemma 5.2. We may assume thatN ≥2L₀. It suffices to show that forn≥0, P[H^∗(0, Ln,2Ln)]≤Ce^−Lcn. (5.2) Indeed, choosenso that2Ln≤N <2Ln+1= 2l0Ln. Then

P[H^∗(0, N)]≤P[H^∗(0, Ln,2Ln)]≤Ce^−Lcn ≤C⁰e^−Nc

0

.

Letu >0. Choosel₀(> l(d)), L₀ ≥1, andp <1 such that Corollaries 4.3 and 4.6 and Lemma 4.7 hold. Forn≥0andx∈Gⁿ, we say thatxisn-badif the event

Dx,n(Be^p)∪E^u_x,n(eI^u)∪F^u_x,n(eI^u)

occurs. Otherwise, we say thatxisn-good. (In particular,xis0-bad if and only ifxis bad.) By the definition ofD_x,n(Be^p),E^u_x,n(eI^u)andF^u_x,n(eI^u),

ifx∈G^nisn-good, then there exist at most three(n−1)-bad vertices z₁, . . . , z_s∈Gn−1∩(x+ [0, L_n)^d)(with0≤s≤3) such that

|z_i−z_j|_∞> L_n/l(d)for alli6=j.

(5.3)

In order to prove (5.2), it suffices to show that for alln≥0andx∈Gn, H^∗(x, L_n,2L_n)⊆ [

y∈Gn∩(x+[−2Ln,2L_n)^d)

{yisn-bad}. (5.4)

Indeed, since the number of vertices inGn∩[−2L_n,2L_n)^d={−2L_n,−L_n,0, L_n}^dequals 4^d, we obtain by translation invariance that

P[H^∗(0, Ln,2Ln)]≤4^d

P[D0,n(Be^p)] +P[E^u_0,n(eI^u)] +P[F^u_0,n(eI^u)]

≤4^d·3·2⁻²ⁿ≤Ce^−Lcn. We prove (5.4) by induction on n. The statement is obvious for n = 0. We assume that (5.4) holds for all integers smaller thann≥1, and will show that it also holds for

Figure 1: One way to definey_iis as the closest vertex inGn−1∩∂B(0, L_n+ 5L_n−1i)to the point of the first intersection ofπ with∂B(0, Ln+ 5L_n−1i). Concentric boxes are not drawn to scale here: the innermost box isB(0, Ln), the outermost box isB(0,2Ln), and the intermediate boxes areB(0, Ln+ 5Ln−1i), fori∈ {1, . . . , m0}. The smallest and second smallest boxes along the pathπareB(yi, Ln−1)andB(yi,2Ln−1), respectively, fori∈ {0, . . . , m0}.

n. It suffices to prove the induction step forx = 0. The proof goes by contradiction.

Assume that H^∗(0, Ln,2Ln) occurs and all the vertices in {−2Ln,−Ln,0, Ln}^d are n- good. Letπbe a∗-path of bad vertices inG0fromB(0, L_n)to the boundary ofB(0,2L_n). Letm₀=bl₀/5c −1. Note that the pathπintersects the boundary of each of the boxes B(0, Ln+5L_n−1i), fori∈ {0, . . . , m0}. Therefore, there existy0, . . . , ym₀ ∈Gn−1such that for alli∈ {0, . . . , m0}, (a)|yi|_∞=Ln+ 5L_n−1iand (b)π∩B(yi, L_n−1)6=∅(see Figure 1).

By the definition ofm0andyi’s, all the boxesB(yi,2Ln−1)are disjoint and contained in [−2Ln,2Ln)^d, and the pathπconnectsB(yi, Ln−1)to the boundary ofB(yi,2Ln−1), i.e., the eventH^∗(y_i, L_n−1,2L_n−1)occurs for alli∈ {0, . . . , m0}. We will show that

there existsjsuch that

all the4^d vertices inGn−1∩(y_j+ [−2Ln−1,2L_n−1)^d)are(n−1)-good, (5.5) which will contradict our assumption that (5.4) holds forn−1.

Since all the vertices inGⁿ∩[−2Ln,2Ln)^daren-good by assumption, it follows from (5.3) that

there existz₁, . . . , z_3·4d∈[−2Ln,2L_n)^dsuch that

all the vertices in(Gn−1∩[−2L_n,2L_n)^d)\ ∪^3·4_i=1^dB(z_i, L_n/l(d))are(n−1)-good. (5.6) Note that each of the ballsB(z,2L_n/l(d))contains at most(4(L_n/l(d))+1)/(5L_n−1)≤ l0/l(d) different yi’s. Therefore, the union of the balls ∪^3·4_i=1^dB(zi,2Ln/l(d))(with zi’s defined in (5.6)) contains at most3·4^d·l₀/l(d)differenty_i’s, which is strictly smaller thanm₀by the choice ofl(d)in (3.3). We conclude that there existsj∈ {0, . . . , m₀}such that

yj∈ ∪/ ^3·4_i=1^dB(zi,2Ln/l(d)).

We assume thatl₀is chosen large enough so thatL_n/l(d)>2L_n−1, i.e.,l₀>2l(d). With this choice ofl₀,

B(yj,2Ln−1)⊆[−2Ln,2Ln)^d\ ∪^3·4_i=1^dB(zi, Ln/l(d)). (5.7) Therefore, (5.5) follows from (5.6) and (5.7), which is in contradiction with the assump- tion that (5.4) holds forn−1. This implies that (5.4) holds for alln≥0. The proof of Lemma 5.2 is completed.

6 Proofs of Theorem 2.1 and Theorem 2.2

In this section, we derive Theorems 2.1 and 2.2 from Lemmas 5.1 and 5.2.

Proof of Theorem 2.1. The two results of Theorem 2.1 can be proved similarly (note that the results of Sections 3-5 can be trivially adapted to site percolation onI^u), therefore we only provide a proof for the case of bond percolation onIe^u.

ChooseL0andp <1such that Lemma 5.2 holds. Remember the definitions of a bad vertex and the eventH^∗(0, N)from Section 5. LetM be a positive integer. Note that the probability that there exists a∗-circuit of bad vertices inG⁰∩(Z²× {0}^d−2)around [0, L₀M)²× {0}^d−2is at most

∞

X

N=M

P[H^∗(0, L0N)]≤C

∞

X

N=M

e^−Nc ≤1/2,

for large enough M. If there is no such circuit, then, by planar duality (see, e.g., [10, Chapter 3.1]), there is a nearest-neighbor path π = (x₀, x₁, . . .) of good vertices in G⁰∩(Z²× {0}^d−2) that connects [0, L0M)²× {0}^d−2 to infinity. Namely, for all i, xi ∈ G⁰∩(Z² × {0}^d−2), |xi −xi+1|1 = L0, xi is good, x0 ∈ [0, L0M)²× {0}^d−2, and

|xn|∞ → ∞ as n → ∞. It follows from Lemma 5.1 that the graph {ee : ee ∈ x+ [0,2L0)^d for somex∈ π} ⊂Z²×[0,2L0)^d−2 contains an infinite connected component ofeI^u∩Be^p. Therefore, the probability that an infinite nearest-neighbor path inIe^u∩Be^p visits[0, L₀M+ 2L₀)²×[0,2L₀)^d−2is at least1/2. By the ergodicity ofIe^u∩Be^p, an infinite nearest-neighbor path in(eI^u∩Be^p)∩(Z²×[0,2L0)^d−2)exists with probability1.

Proof of Theorem 2.2. We will use the main result of [13] that for an infinite graph G= (V, E)and i.i.d. positive random variablesR

ee,ee∈E, the following statements are equivalent: (a) almost surely, the electric network{R

ee : ee∈E}is transient, and (b) for somep <1, independent bond percolation onGwith parameterpcontains with positive probability a cluster on which simple random walk is transient. (In the proof, we will only use the easy implication, namely, that (b) implies (a).)

Therefore, in order to prove Theorem 2.2, it suffices to show that for somep < 1, with positive probability, the graphIe^u∩Be^pcontains a transient subgraph. The proof of this fact is similar to the proof of Theorem 1 in [17], so we only give a sketch here.

Let d ≥ 3 andu > 0. Denote by S^d thed-dimensional Euclidean unit sphere. We will show that, for anyε∈(0,1), there exists an eventHof probability1such that ifH occurs, then

the graphIe^u∩Be^pcontains an infinite connected subgraph

which, for eachv∈S^d, contains an infinite path in the set∪^∞_n=1B(nv,2n^ε). (6.1) (The setS∞

n=1B(nv,2n^ε)is roughly shaped like a paraboloid with an axis parallel tov.) After that, one can proceed, as in Section 3 of [17], to show that this infinite connected subgraph ofIe^u∩Be^pis transient.