Connectivity bounds for the vacant set of random interlacements

www.imstat.org/aihp 2010, Vol. 46, No. 4, 976–990

DOI:10.1214/09-AIHP335

© Association des Publications de l’Institut Henri Poincaré, 2010

Connectivity bounds for the vacant set of random interlacements

Vladas Sidoravicius

and Alain-Sol Sznitman

aCWI, Kruislaan 413, NL-1098 SJ, Amsterdam, and IMPA, Estrada Dona Castorina 110, Jardim Botanico, CEP 22460-320, Rio de Janeiro, RJ, Brasil

bDepartement Mathematik, ETH Zürich, CH-8092 Zürich, Switzerland Received 5 February 2009; revised 7 July 2009; accepted 1 August 2009

Abstract. The model of random interlacements onZ^d,d≥3, was recently introduced in [Vacant set of random interlacements and percolation. Available athttp://www.math.ethz.ch/u/sznitman/preprints]. A non-negative parameteruparametrizes the density of random interlacements onZ^d. In the present note we investigate connectivity properties of the vacant set left by random inter- lacements at levelu, in the non-percolative regimeu > u_∗, withu_∗the non-degenerate critical parameter for the percolation of the vacant set, see [Vacant set of random interlacements and percolation. Available athttp://www.math.ethz.ch/u/sznitman/preprints], [Comm. Pure Appl. Math.62(2009) 831–858]. We prove a stretched exponential decay of the connectivity function for the vacant set at levelu, whenu > u_∗∗, whereu_∗∗is another critical parameter introduced in [Ann. Probab.37(2009) 1715–1746]. It is presently an open problem whetheru_∗∗actually coincides withu_∗.

Résumé. Le modèle des entrelacs aléatoires surZ^d,d≥3, a été récemment introduit dans [Vacant set of random interlacements and percolation. Available athttp://www.math.ethz.ch/u/sznitman/preprints]. Un nombre positif ou nulucontrôle la densité des entrelacs aléatoires surZ^d. Dans la note présente, nous étudions les propriétés de connectivité du complémentaire de l’entrelac au niveauu, dans le régime non percolatifu > u_∗, avecu_∗ le nombre positif qui est le paramètre critique de la percolation du complémentaire des entrelacs, voir [Vacant set of random interlacements and percolation. Available athttp://www.math.ethz.

ch/u/sznitman/preprints], [Comm. Pure Appl. Math.62 (2009) 831–858]. Nous montrons une propriété de décroissance sous- exponentielle de la fonction de connectivité au niveauu, lorsqueu > u_∗∗, oùu_∗∗est un autre paramètre critique introduit dans [Ann. Probab.37(2009) 1715–1746]. La question de savoir siu_∗etu_∗∗sont égaux est pour le moment ouverte.

MSC:60K35; 60G50; 82C41

Keywords:Connectivity function; Random interlacements; Percolation

0. Introduction

In this note we derive stretched exponential bounds on the connectivity function for the vacant set of random inter- lacements onZ^d,d≥3. The model of random interlacements has been introduced in [4]. It heuristically describes the microscopic structure left in the bulk by random walk on a cylinder with base a large(d−1)-dimensional discrete torus, or by random walk on a larged-dimensional torus, when the walk respectively runs for times proportional to the square of the number of sites in the base, or to the number of sites in the torus, cf. [5,10]. Further extensions to more general graphs can be found in [8,11].

The bounds presented here pertain to a very specific region of the non-percolative regime of the vacant set of random interlacements. Knowing whether this region actually coincides with the whole non-percolative regime outside the critical point is an important question with direct implications for the asymptotic behavior of the disconnection time of discrete cylinders with bases which become large, cf. [6]. The results in the present note do not answer this

question, but they indicate that in the case where the two regions differ, there is a marked transition in the decay properties of the connectivity function (roughly from a polynomial to a stretched exponential decay), as one moves across another (and then distinct) critical point, which has been introduced in [6].

We will now describe the model and state our main result. We refer to Section1for precise definitions. Random interlacements consists of a cloud of paths which constitute a Poisson point process on the space of doubly infinite Z^d-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. A non-negative parameter u plays the role of a multiplicative factor of the intensity measure of this Poisson point process. In a standard fashion one constructs on the same probability space(Ω,A,P), see below (1.6), the whole familyI^u,u≥0, of random interlacements at levelu≥0, cf. (1.10). They come as traces onZ^d of the cloud of trajectories modulo time-shift with labels at mostu. The subsetsI^u increase withuand foru >0 are random connected subsets ofZ^d, ergodic under space translations, cf. Theorem 2.1, Corollary 2,3 of [4]. The complement V^u ofI^u inZ^d is the so- called vacant set at levelu. It is known from Theorem 3.5 of [4] and Theorem 3.4 of [3] that there is a non-degenerate critical valueu_∗∈(0,∞)such that

(i) foru > u_∗,P-a.s. all connected components ofV^uare finite,

(0.1) (ii) foru < u_∗,P-a.s. there exists an infinite component inV^u.

It is also known from [7], that whenuis such thatV^u percolates (i.e. there isP-a.s. an infinite connected component inV^u), the infinite cluster is almost surely unique. It is presently unknown whetherV^u∗ percolates or not. Another critical pointu_∗∗∈ [u_∗,∞)has been introduced in [6]:

u_∗∗=inf

u≥0;α(u) >0

, where

(0.2) α(u)=sup

α≥0; lim

L→∞L^αP

B(0, L) ^V

←→u S(0,2L)

=0

foru≥0,

here the event under the probability refers to the existence of a nearest neighbor path inV^ujoiningB(0, L), the closed ball or radiusLand center 0 for the^∞-distance, withS(0,2L), the sphere with radius 2Land center 0 for the same distance. The supremum in the second line of (0.2) is by convention equal to zero when the set is empty (this is for instance the case whenu < u_∗). The critical parameteru_∗∗explicitly enters the upper bound on the disconnection time of discrete cylinders derived in Corollary 4.6 of [6]. It is an important question whether in factu_∗andu_∗∗coincide.

The present work does not address this question but shows that the probability, which appears in the second line of (0.2), has a stretched exponential decay inLforu > u_∗∗. Our main result is

Theorem 0.1. Foru > u_∗∗,the connectivity function in the vacant set at level uhas stretched exponential decay.

Namely there exist positive constants α₁, α₂,and 0< ρ <1,solely depending ond and u,such that with similar notation as in(0.2):

P 0 ^V

←→u x

≤α1exp

−α2|x|^ρ

for allx∈Z^d. (0.3)

In case u_∗ andu_∗∗ differ, the above theorem forces a sharp transition in the decay properties of the connectiv- ity function of the vacant set as u crosses the valueu_∗∗, cf. Remark3.1(1). We also discuss in Remark3.1(2) a variant of (0.3), cf. (3.12), whereI^u is replaced by itsR-neighborhood, and instead of assumingu > u_∗∗, we pick R large enough. Let us mention that when d=3, the left-hand side of (0.3) does not decay exponentially in|x|, cf. Remark3.1(3).

We now give some comments on the proof of Theorem0.1. The main difficulty stems from the long range de- pendence of random interlacements. We use an adaptation of the renormalization and sprinkling technique, which appears in Section 3 of [4]. The main difference resides in the fact that here we separate the combinatorial complexity bounds from the probabilistic estimates on crossings we derive, see (2.10), Lemma2.1and Proposition2.2. A similar separation is for instance also used in [9] for the very percolative regime of smallu. In our context this separation leads to finer probabilistic estimates in the renormalization scheme, which instead of producing polynomial decay in

Lfor quantities such asP[B(0, L) ^V

←→u S(0,2L)], whenuis sufficiently large, cf. Section 3 of [4], yield a stretched exponential decay of such quantities, whenu > u_∗∗.

We will now describe the organization of this note.

In Section1 we introduce further notation and recall some facts about random interlacements. In Section2 we develop the renormalization scheme. The main result is Proposition2.2. The main consequences of this proposition for the next section appears in Proposition2.5. In Section3we complete the proof of Theorem0.1. We provide further comments and discuss some extensions in Remark3.1.

Finally let us explain the convention we use for constants. Throughout the textcorc denote positive constants that solely depend ond, with values changing from place to place. Numbered constantsc₀, c₁, . . .are fixed and refer to the value pertaining to their first appearance in the text. Dependence of constants on additional parameters appears in the notation, so that for instancec(u)denotes a constant depending ond andu.

1. Notation and a brief review of random interlacements

In this section we introduce some notation and recall some facts concerning random interlacements. A more detailed review of random interlacements can also be found in Section 1 of [3].

We let| · | and| · |_∞ respectively denote the Euclidean and the^∞-distance on Z^d. Throughout we implicitly assume d≥3. By finite path we mean a sequence x₀, x₁, . . . , x_N inZ^d such that x_i andx_i+1 are neighbors, i.e.

|x_i+1−x_i| =1, for 0≤i < N. We sometimes simply write path when this causes no confusion. The notationB(x, r) andS(x, r)withx ∈Z^d andr≥0, for | · |∞-balls and spheres is explained below (0.2). For A, B subsets ofZ^d, we writeA+B for the set ofx+y withx inAandy inB, andd(A, B)=inf{|x−y|∞; x∈A, y∈B}for the mutual^∞-distance betweenAandB. WhenAis a singleton{x}, we simply writed(x, A). The notationK⊂⊂Z^d means thatK is a finite subset ofZ^d. GivenU⊆Z^d, we denote with|U| the cardinality of U, with∂U= {x ∈ U^c; ∃y∈U,|x−y| =1}the boundary ofUand with∂_intU= {x∈U; ∃y∈U^c,|x−y| =1}, the interior boundary ofU.

We denote withW₊the space of nearest neighborZ^d-valued trajectories defined for non-negative times and tending to infinity. We letW+,(Xn)n≥0,(θn)n≥0, stand for the canonicalσ-algebra, the canonical process, and the canonical shift onW₊. Since we assumed≥3, simple random walk is transient onZ^d, and we denote withPxthe restriction of the canonical law of simple random walk starting atx∈Z^d, to the set W₊, which has full measure. We write Ex for the corresponding expectation. We also definePρ =

x∈Z^dρ(x)Px, whenρ is a measure onZ^d and write Eρ for the corresponding expectation. GivenU⊆Z^d, we letHU =inf{n≥0;Xn∈U},HU =inf{n≥1;Xn∈U}, andTU=inf{n≥0;Xn∈/U}, respectively stand for the entrance time inU, the hitting time ofU, and the exit time ofU.

We denote withg(·,·)the Green function of the walk:

g(x, y)=

n≥0

Px[Xn=y], x, y∈Z^d. (1.1)

It is a symmetric function and due to translation invariance g(x, y)=g(y −x), where g(y)=g(0, y). Given K⊂⊂Z^d, we writeeKfor the equilibrium measure ofKand cap(K)for its total mass, the so-called capacity ofK:

eK(x)=Px[HK= ∞]1K(x), x∈Z^d, cap(K)=

x∈K

Px[HK= ∞]. (1.2)

The capacity is subadditive (this fact easily follows from (1.2)):

cap

K∪K

≤cap(K)+cap K

forK, K ⊂⊂Z^d. (1.3)

Further the probability to enterKcan be expressed as:

Px[HK<∞] =

y∈K

g(x, y)eK(y) forx∈Z^d, (1.4)

and one has the bounds, cf. (1.9) of [4]

y∈K

g(x, y)

sup

z∈K

y∈K

g(z, y)

≤P_x[H_K<∞] ≤

y∈K

g(x, y)

zinf∈K

y∈K

g(z, y)

forx∈Z^d. (1.5)

With classical bounds on the Green function, cf. [2], p. 31, it then follows that cL^d−2≤cap

B(0, L)

≤cL^d−2 forL≥1. (1.6)

We now turn to random interlacements. They are defined on a probability space(Ω,A,P), where a certain canoni- cal Poisson point process can be constructed, and we refer to (1.16), (1.42) of [4] or (1.9)–(1.12) of [3] for the precise definition of this probability space. In the present note we will only use the fact one can define on(Ω,A,P)families of finite Poisson point processes on(W₊,W₊),μK,u(dw),u≥0,K⊂⊂Z^d, andμ_K,u,u(dw), 0≤u < u,K⊂⊂Z^d, so that

μ_K,u,uandμ_K,u are independent with respective intensity measures

(u−u)Pe_K anduPe_K, for any 0≤u < uandK⊂⊂Z^d, (1.7)

μ_K,u=μ_K,u +μ_K,u,ufor any 0≤u < uandK⊂⊂Z^d. (1.8)

Moreover the following compatibility relations hold forK⊂K ⊂⊂Z^d: μ_K,u=

m

i=0

δ_θHK(wi)1

H_K(w_i) <∞

ifμ_K,u= m

i=0

δ_wi, (1.9)

together with similar compatibility relations with μ_K,u,u and μ_K,u,u in place of μ_K,u and μ_K,u. We refer for instance to (1.13)–(1.15) of [3], or to (1.18)–(1.21) and Proposition 1.3 of [4], for more details.

Givenω∈Ω, the interlacement at levelu≥0 is the random subset ofZ^ddefined forω∈Ω via:

I^u(ω)=

K⊂⊂Z^d

w∈Suppμ_K,u(ω)

w(N), (1.10)

where the notation Suppμ_K,u(ω)refers to the support of the finite point measureμ_K,u(ω)(dw), andN= {0,1, . . .}.

The vacant set at leveluis then defined as

V^u(ω)=Z^d\I^u(ω) forω∈Ω, u≥0. (1.11)

One finds that, cf. (1.54) of [4]:

I^u(ω)∩K=

w∈Suppμ_{K ,u}(ω)

w(N)∩K forK⊂K ⊂⊂Z^d, u≥0, ω∈Ω. (1.12)

It also follows from (1.7) that P[V^u⊇K] =exp

−ucap(K)

for allK⊂⊂Z^d, u≥0. (1.13)

This concludes this short review of random interlacements, which will suffice for the purpose of the present note.

2. The renormalization scheme

In this section we develop the renormalization scheme, which will be the main tool in the derivation of Theorem0.1.

It comes as a variation on the method developed in Section 3 of [4], and in particular uses the sprinkling technique of [4] to control the long range interactions present in the model. The main step comes in Proposition2.2and its consequences for the proof of Theorem0.1in the next section appear in Proposition2.5.

We introduce a sequence of length scales

L_n=L₀ⁿ₀ forn≥0, whereL₀≥1 and₀≥100 is a multiple of 10. (2.1) We organizeZ^din a hierarchical fashion withL0corresponding to the finest scale andL1< L2<· · ·to coarser and coarser scales. To this effect we introduce the set of labels at leveln:

I_n= {n} ×Z^d, n≥0, (2.2)

and to eachm=(n, i)∈I_n, withn≥0, we attach the boxes Cm=

iLn+ [0, Ln)^d

∩Z^d, Cm=

m∈In,d(C_m ,Cm)≤1

C_m. (2.3)

We writeS_m=∂_intC_mandS_m=∂_intC_m, form∈I_n, n≥0. Givenm∈I_n, n≥1, we considerH1(m),H2(m)⊆I_n−1 defined by

H1(m)= {m∈In−1;Cm⊆CmandCm∩Sm=∅},

(2.4) H2(m)=

m∈I_n−1;C_m∩

z∈Z^d;d(z, C_m)=L_n 2

=∅

.

Note that for alln≥1,m∈I_n:

m1∈H1(m)andm2∈H2(m)implies thatCm₁∩Cm₂=∅and

(2.5) Cm₁∪Cm₂⊆Cm.

Givenm∈I_n,n≥0, we considerΛ_mthe collection of subsetsT of

0≤k≤nI_k, such that settingT^k=T ∩I_k one has

Tⁿ= {m}, (2.6)

anym ∈T^k, 1≤k≤n, has two “descendants”m₁ m

∈H1

m and m₂

m

∈H2

m

,such thatT^k−1=

m∈T^k

m₁ m

, m₂ m

. (2.7)

In other words anyT ∈Λ_mhas a natural structure of binary tree of depthnwith rootm(∈I_n)and for 0≤k≤n,

|T^k| =2^n−k, moreover theC_m, form ∈T^k, are pairwise disjoint.

Givenn≥0,m∈I_n, andT ∈Λ_m, one can attach to eachm ∈T a binary treeTm ∈Λ_m, which, roughly speaking, consists of the descendants ofm inT:

Tm =

m ∈T;C_m ⊆C_m

, (2.8)

and for any 1≤k≤n,m ∈T^k, one has the identity Tm =

m

∪Tm1(m)∪Tm2(m) (disjoint union), (2.9)

wheremi(m)∈Hi(m),i=1,2, are as in (2.7), see also (2.5) and Fig.1.

Fig. 1. An illustration of the boxesC_miandC_mi,i=1,2.

Form∈I_n one has the following rough bound on the cardinality ofΛ_m, the collection of binary trees attached tom:

|Λ_m| ≤

c^d₀⁻¹2 c^d₀⁻¹4

· · ·

c^d₀⁻¹2ⁿ

=

c^d₀⁻¹2(2ⁿ−1)

=

c₀^2(d₀ ⁻¹⁾2ⁿ−1

. (2.10)

We then introduce the events A^u_m=

Cm←→V^u Sm

foru≥0, m∈In, n≥0, (2.11)

where the notation is similar as in (0.2). The role of Λmas a way to separate combinatorial complexity and proba- bilistic estimates comes in the next simple lemma.

Lemma 2.1(n≥0, m∈In, u≥0).

P A^u_m

≤ |Λ_m|p_n(u), wherep_n(u)= sup

T∈Λ_m

P[A^u_T]andA^u_T =

m∈T⁰

A^u_m (recallT⁰=T ∩I0). (2.12)

Proof. Observe that whenn≥1,m∈I_m, any path inV^uoriginating inC_mand ending inS_mmust go through some C_m1, m₁∈H1(m), reachS_m1and then go through someC_m2, m₂∈H2(m)and reachS_m2. Hence one has the inclusion

A^u_m⊆

mi∈Hⁱ(m),i=1,2

A^u_m

1∩A^u_m

2.

Note that whenm∈I₀,A^u_m=A^u_T, withT = {m}the unique element ofΛ_m. Therefore with a straightforward induc- tion onn, using the above inclusion, one finds thatA^u_m⊆

T∈ΛmA^u_T, for allm∈I_n,n≥0, u≥0. The claim (2.12)

readily follows.

Unlike what was done in Section 3 of [4] we will not estimateP[A^u_m],m∈I_n, by induction onn(this quantity does not depend on whichm∈I_nwe consider, due to translation invariance). Instead we will estimatep_n(u)by induction on n. This will lead to a finer specification of the possible long range dependence effects. As in [4], we will use sprinkling, i.e. we will controlp_n+1(u_n+1)in terms ofp_n(u_n), along an increasing sequenceu_n. The additional paths entering the random interlacement corresponding to the increase ofu_nintou_n+1will enable to dominate long range dependence. The main step comes in the next proposition (compare with Proposition 3.1 of [4]).

Proposition 2.2(d≥3). There exist positive constants c₁, c₂, csuch that for₀≥c,for increasing sequencesu_n, n≥0,in(0,∞)and non-decreasing sequencesrn, n≥0,of positive integers,such that

u_n+1≥u_n

1+c₁2ⁿ⁻₀^(n+1)(d−2)rn+1

forn≥0, (2.13)

one has for alln≥0,

p_n+1(u_n+1)≤p_n(u_n+1)

p_n(u_n)+u_nL^d₀⁻²

cⁿ₂⁺¹⁻₀^(n+1)(d−2)rn

. (2.14)

(Note thatp_n(·)is a non-increasing function andp_n(u_n+1)≤p_n(u_n).)

Proof. We consider somen≥0,m∈I_n+1,T ∈Λ_m, and writem1, m2for the unique elements ofH1(m),H2(m)in Tⁿ(=T ∩I_n). We also consider 0< u < u, respectively playing the role ofu_nandu_n+1, as well as an integerr≥1, playing the role ofr_n. We writeT1andT2in place ofTm1 andTm2. We also define

V =C₁∪C₂, whereC_i=

m∈Ti∩I₀

C_m ⊆C_mi fori=1,2. (2.15)

We then introduce the decomposition, see (1.7) for the notation,

μ_{V ,u}=μ_1,1+μ_1,2+μ_2,1+μ_2,2, (2.16)

where fori=j in{1,2}we have set μ_i,j=1{X₀∈C_i, H_C

j <∞}μ_{V ,u} and μ_i,i=1{X₀∈C_i, H_C

j = ∞}μ_{V ,u}.

This is similar to (3.14) of [4], except thatCi,i=1,2, now plays the role ofC_mi,i=1,2 in (3.14) of [4]. Similarly withμ_{V ,u}, andμ_{V ,u,u}, see (1.7) for the notation, we define with analogous formulas

μ_{V ,u} =μ_1,1+μ_1,2+μ_2,1+μ_2,2,

μ_{V ,u,u}=μ^∗_1,1+μ^∗_1,2+μ^∗_2,1+μ^∗_2,2 so that

μ_i,j,μ^∗_i,j, 1≤i, j≤2, are independent Poisson point processes onW₊. (2.17) Whenμ is a random point process on W₊ defined on Ω (i.e. a measurable map fromΩ into the space of point measures onW₊), it will be convenient to define forT ∈Λ_m, withm∈I_n

A_T(μ)=

m∈T∩I₀

ω∈Ω;there is a path inC_m

w∈Suppμ(ω)

w(N)

joiningC_m withS_m

. (2.18)

In particular form∈In,T ∈Λm, one finds that, cf. (1.12), A^u

T =A_T(μ_K,u) for any finiteK⊇

m∈T∩I0

C_m.

With (2.9) it follows thatA^u_T =A^u

T1∩A^u

T2, and taking into account thatw∈Suppμ_2,2implies thatw(N)∩C₁=∅, we have

A^u_T =A^u

T1∩A^u

T2=A^u

T1

(μ_1,1+μ_1,2+μ_2,1)∩A^u

T2

(μ_{V ,u})

⊆A^u

T1(μ1,1+μ1,2+μ2,1)∩A^u

T2(μ2,2).

Using the independence of theμ_i,j, 1≤i, j≤2, we find that P

A^u_T

≤P A_T

1(μ_1,1+μ_1,2+μ_2,1) P

A_T

2(μ_2,2)

=P A^u

T1

P A_T

2(μ_2,2)

≤p_n(u)P A_T

2(μ_2,2)

. (2.19)

We will now boundP[A_T

2(μ_2,2)]^(1.8)= P[A_T

2(μ_2,2+μ^∗_2,2)]in terms ofp_n(u)whenu−u is substantial enough. For this purpose cf. (2.34) below, we will dominate the influence onC₂ofμ_2,1+μ_2,1byμ^∗_2,2. This has the same spirit as what appears in Proposition 3.1 of [4], howeverC₂now replacesC_m2. The sprinkling technique will come into action during this step.

We introduce the ^∞-neighborhood of size ^L₁₀ⁿ⁺¹ of Cm₂ (^L₁₀ⁿ⁺¹ is an integer since0≥100 is a multiple of 10, cf. (2.1)):

U=

z∈Z^d;d(z,C_m2)≤L_n+1 10

. (2.20)

We then consider the times of successive returns toC₂and departures fromU:

R₁=H_C

2, D₁=T_U◦θ_R1+R₁ and by induction

(2.21) R_k+1=R₁◦θ_Dk+D_k, D_k+1=D₁◦θ_Dk+D_k fork≥1,

so that 0≤R1≤D1≤ · · · ≤Rk≤Dk≤ · · · ≤ ∞.

We then introduce the decompositions:

μ_2,1=

1≤≤r

ρ_2,1 +ρ_2,1, μ_1,2=

1≤≤r

ρ_1,2 +ρ_1,2,

(2.22) μ^∗_2,2=

1≤≤r

ρ_2,2 +ρ_2,2,

where fori=j in{1,2}and≥1, we have set

ρ_i,j =1{R< D< R₊₁= ∞}μ_i,j, ρ_i,j=1{R_r+1<∞}μ_i,j and ρ_2,2 =1{R< D< R₊₁= ∞}μ^∗_2,2, ρ_2,2=1{R_r+1<∞}μ^∗_2,2. As a result of (2.17) and the above formulas we see that:

μ_2,2, ρ_i,j,1≤≤r, ρ_i,j,1≤i, j≤2, withiorj=1, are independent

Poisson point processes onW₊. (2.23)

We denote withξ_2,1andξ_1,2the respective intensity measures ofρ_2,1andρ_1,2. We have ρ_2,1(W₊)=uP_eV[X₀∈C₂, H_C

1<∞, R_r+1<∞]

≤ucap(C₂)sup

x∈C₂

P_x[R_r+1<∞]

strong Markov

≤ ucap(C₂)

sup

x∈U^c

P_x[H_C

2<∞]r

. (2.24)

Note that with standard estimates on the Green function, cf. [2], p. 31, as well as (1.3), (1.4), (1.6), we find that sup

x∈U^c

P_x[H_C

2<∞] ≤c2ⁿL^d₀⁻² L^d_n+⁻²₁

(2.1)

= c2ⁿ⁻₀^(n+1)(d−2). (2.25)

Therefore with (2.24), and (1.3), (1.6), we find that ξ_2,1(W₊) ≤ cu2ⁿL^d₀⁻²

c2ⁿ⁻₀^(n+1)(d−2)r r≥1

≤ uL^d₀⁻²

c4ⁿ⁻₀^(n+1)(d−2)r

. (2.26)

In a similar fashion, we also have ξ_1,2(W₊)=uPe_V[X0∈C1, H_C

2<∞, Rr+1<∞]

≤uL^d₀⁻²

c4ⁿ⁻₀^(n+1)(d−2)r

. (2.27)

We will now prove that the trace onC₂of paths in the support of

1≤≤rρ_2,1 and

1≤≤rρ_1,2 is dominated by the corresponding trace of paths in the support ofμ^∗_2,2ifu−u is not too small.

We considerW_f the collection of finite paths in Z^d and for≥1, the measurable mapφ from{D< R₊₁=

∞} ⊆W₊intoW_f^×such that φ(w)=

w(R_k+ ·)_{0≤·≤Dk−Rk}

1≤k≤∈W_f^× forw∈ {D< R₊₁= ∞}. (2.28) In other wordsφ(w)keeps track of the parts of the trajectory w going from the successive returns to C2 up to departure fromU. We view theρ_i,j ,iorj=1, with≥1 fixed, as point processes on{D< R₊₁= ∞} ⊆W₊, and denote withρ_i,j the respective images underφ, which are Poisson point processes onW_f^×. We writeξ_i,j for the corresponding intensity measures. With (2.23) it follows that

μ_2,2,ρ_i,j,1≤≤r, ρ_i,j,1≤i, j≤2,

iorj=1, are independent point processes. (2.29)

Our next step is the following lemma, which is an adaption of Lemma 3.2 of [4].

Lemma 2.3. For₀≥c,for alln≥0,m∈I_n+1,T ∈Λ_m,x∈∂U,y∈∂_intC₂,one has Px[H_C

1< R1<∞, XR₁=y] ≤c2ⁿ⁻₀^(n+1)(d−2)Px[H_C

1> R1, XR₁=y], (2.30) Px[H_C

1<∞, R1= ∞] ≤c2ⁿ⁻₀^(n+1)(d−2)Px[R1= ∞ =H_C

1]. (2.31)

Proof. We begin with the proof of (2.30). Forz∈∂U,y∈∂_intC₂, one finds with the strong Markov property that P_z[H_C

1 < R₁<∞, X_R1=y]

=Ez

H_C

1 < R1, PX_H

C1[R1<∞, XR₁=y]

=P_z H_C

1< R₁, E_XH

C1

H_∂U <∞, P_XH∂U[R₁<∞, X_R1=y] ,

where we have used in the last step that forz ∈C1,P_z-a.s.,R1=H∂U+R1◦θH_∂U. As a result we obtain sup

z∈∂U

P_z[H_C

1< R₁<∞, X_R1 =y]

≤ sup

z∈∂U

P_z[H_C

1<∞] sup

z∈∂U

P_z[R₁<∞, X_R1=y]

≤c2ⁿ⁻₀^(n+1)(d−2) sup

z∈∂U

P_z[R₁<∞, X_R1=y], (2.32)

with a similar bound as in (2.25) in the last step.

Now observe thatP_z[R₁<∞, X_R1=y] =P_z[H_C

2 <∞, X_H

C2 =y],z∈C₂^cis a positive harmonic function and with Harnack inequality, cf. Theorem 1.7.2, p. 42 of [2], together with a standard covering argument we have

sup

z∈∂U

P_z[R₁<∞, X_R1=y] ≤c inf

z∈∂UP_z[R₁<∞, X_R1=y]. Thus coming back to (2.32) we find that

sup

z∈∂U

P_z[H_C

1< R₁<∞, X_R1=y]

≤c2ⁿ⁻₀^(n+1)(d−2) inf

z∈∂UPz[R1<∞, XR₁=y]

=c2ⁿ⁻₀^(n+1)(d−2) inf

z∈∂U

P_z[H_C

1< R₁<∞, X_R1=y] +P_z[H_C

1> R₁, X_R1=y] .

For large0, we find thatc2ⁿ⁻₀^(n+1)(d−2)≤¹₂, for alln≥0, withc as above and hence forx∈∂U:

Px[H_C

1< R1<∞, XR₁ =y] ≤2c2ⁿ⁻₀^(n+1)(d−2)Px[H_C

1> R1, XR₁=y], and this proves (2.30).

We then turn to the proof of (2.31) which is more straightforward. We observe that

xinf∈∂UP_x[R₁= ∞, H_C

1= ∞] = inf

x∈∂UP_x[H_V = ∞] ≥c,

using the invariance principle to let the walk move at a distance ofC_m1∪Cm₂, which is a multiple ofLn+1, as well as (1.5), (1.6) and standard bounds on the Green function. On the other hand the left-hand side of (2.31) with a similar bound as in (2.25) is smaller thanc2ⁿ⁻₀^(n+1)(d−2)and the claim follows.

The main control on the intensity measureξ_1,2 +ξ_2,1 ofρ_1,2 +ρ_2,1 in terms of the intensity measureξ_2,2 ofρ_2,2 comes from

Lemma 2.4(₀≥c).

ξ_1,2 +ξ_2,1 ≤ u u−u

1+c₁2ⁿ⁻₀^(n+1)(d−2)+1

−1 ξ_2,2 for≥1. (2.33)

Proof. This is a repetition of the proof of Lemma 3.3 of [4], withC_i,i=1,2, replacingC_mi,i=1,2 in [4], and c2ⁿ⁻₀^(n+1)(d−2)replacingc⁻_n^(d−2)in the notation of [4], thanks to (2.30), (2.31) of Lemma2.3above.

We now suppose that₀≥c, so that the Lemmas2.3and2.4apply, and that u≥

1+c12ⁿ⁻₀^(n+1)(d−2)r+1

u

(2.34)

hence u u−u

1+c₁2ⁿ⁻₀^(n+1)(d−2)r+1

−1

≤1

.

In our present notation, (2.34) coincides with (2.13). We now boundP[A_T

2(μ_2,2)]in terms ofp_n(u)≥P[A^u

T2]as follows. We first express the trace ofI^u onC₂as the union

I^u ∩C2=I ∪I∪I, (2.35)