Local percolative properties of the vacant set of random interlacements with small intensity

www.imstat.org/aihp 2014, Vol. 50, No. 4, 1165–1197

DOI:10.1214/13-AIHP540

© Association des Publications de l’Institut Henri Poincaré, 2014

Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz

, Balázs Ráth

and Artëm Sapozhnikov

aDepartment of Mathematics, Columbia University, RM 614, MC 4419, 2990 Broadway, New York City, NY 10027, USA.

E-mail:[email protected]

bDepartment of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2.

E-mail:[email protected]

cMax-Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany. E-mail:[email protected] Received 27 July 2012; revised 11 January 2013; accepted 17 January 2013

Abstract. Random interlacements at leveluis a one parameter family of connected random subsets ofZ^d,d≥3 (Ann. Math.

171(2010) 2039–2087). Its complement, the vacant set at levelu, exhibits a non-trivial percolation phase transition inu(Comm.

Pure Appl. Math.62(2009) 831–858;Ann. Math.171(2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466).

In this paper we study local percolative properties of the vacant set of random interlacements at levelufor all dimensionsd≥3 and small intensity parameteru >0. We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at levelu. In particular, this implies that finite connected components of the vacant set at leveluare unlikely to be large. These results are new ford∈ {3,4}. The case ofd≥5 was treated in (Probab.

Theory Related Fields150(2011) 529–574) by a method that crucially relies on a certain “sausage decomposition” of the trace of a high-dimensional bi-infinite random walk. Our approach is independent from that of (Probab. Theory Related Fields150(2011) 529–574). It only exploits basic properties of random walks, such as Green function estimates and Markov property, and, as a result, applies also to the more challenging low-dimensional cases. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

Résumé. Un entrelac aléatoire au niveauuest une famille à un paramètre de sous-ensembles connexes aléatoires deZ^d, d≥3, introduit dans (Ann. Math.171(2010) 2039–2087). Son complémentaire, l’ensemble vacant au niveauu, possède une transition de percolation non triviale enu, comme il a été montré dans (Comm. Pure Appl. Math.62(2009) 831–858) et (Ann. Math.171(2010) 2039–2087). La composante connexe infinie, lorsqu’elle existe, est presque sûrement unique, voir (Ann. Appl. Probab.19(2009) 454–466).

Dans ce papier, nous étudions les propriétés percolatives locales de l’ensemble vacant au niveauuen toutes dimensionsd≥3 et pour un petit paramètre d’intensitéu. Nous donnons une borne exponentielle tendue sur la probabilité qu’un grand (hyper)cube contienne deux composantes macroscopiques distinctes de l’ensemble vacant au niveauu. Nos résultats impliquent qu’il est peu probable que les composantes connexes finies de l’ensemble vacant au niveau usoient grandes. Ces résultats ont été prouvés dans (Probab. Theory Related Fields150(2011) 529–574) pourd≥5. Notre approche est différente (de celle de (Probab. Theory Related Fields150(2011) 529–574)) et est valide pourd≥3.

L’un des ingrédients principaux de la preuve est une certaine propriété d’indépendence conditionelle des entrelacs aléatoires, qui est intéressante en elle-même.

MSC:60K35; 82B43

Keywords:Random interlacement; Random walk; Large finite cluster; Supercriticality; Conditional independence

1. Introduction

Random interlacementsI^uat levelu >0 onZ^d,d≥3, is a one parameter family of random connected subsets ofZ^d, introduced by Sznitman [11], which arises as the local limit asN→ ∞of the set of sites visited by a simple random walk on the discrete torus(Z/NZ)^d,d≥3 when it runs up to timeuN^d, see [17]. The law ofI^u⊆Z^dis uniquely characterized by the equations:

P

I^u∩K=∅

=e^−u·cap(K) for any finiteK⊆Z^d, (1.1)

where cap(K)denotes the discrete capacity ofK, defined in (2.6) below. It is proved among other results in [11] that for anyu >0,I^u is almost surely connected, and its law is invariant and ergodic with respect to the lattice shifts. In fact, in [11], a more constructive definition ofI^uis given, which we recall in Section2.3. Informally, it states thatI^u is the trace of a certain cloud of bi-infinite random walk trajectories inZ^d, withumeasuring the density of this cloud.

The vacant setV^uat leveluis the complement ofI^uinZ^d. We viewV^uas a random graph by drawing an edge between any two vertices of the vacant set atL₁-distance 1 from each other. The vacant set exhibits a non-trivial structural phase transition inu, i.e., there existsu_∗∈(0,∞)such that

(i) for anyu > u_∗, almost surely, all connected components ofV^uare finite, and (ii) for anyu < u_∗, almost surely,V^ucontains an infinite connected component.

In particular, the finiteness ofu_∗ for d ≥3 and the positivity of u_∗ for d ≥7 were proved in [11], and the latter result was extended to all dimensionsd≥3 in [10]. It is also known thatV^ucontains at most one infinite connected component (see [13]); in particular, for anyu < u_∗, the infinite connected component is almost surely unique.

In this paper, we are interested in the local structure of the vacant set in the regime of smallu. More specifically, we show that with high probability, the unique infinite connected component ofV^uis “visible” in large hypercubic subsets ofZ^d (as the unique macroscopic connected component in the restriction ofV^u to large hypercubes ofZ^d).

Our main result is the following theorem.

Theorem 1.1 (Local uniqueness forV^u). For anyd≥3,there existu₁>0,c=c(d) >0andC=C(d) <∞such that for all0≤u≤u₁andn≥1,we have

P

the infinite connected component ofV^u intersectsB(0, n)

≥1−Ce^−nc (1.2)

and P

any two connected subsets ofV^u∩B(0, n)with diameter≥n/10are connected inV^u∩B(0,2n)

≥1−Ce^−nc. (1.3)

Statement (1.2) has already been known (it easily follows from [12], Theorem 5.1), but we include it here for completeness. Ford≥5, statement (1.3) follows from the stronger statement of [14], Theorem 3.2. Our contribution to the result of Theorem1.1is twofold. Firstly, the result (1.3) is new ford∈ {3,4}. Secondly, our proof of (1.3) is conceptually different from that of [14], and applies to all dimensionsd≥3. Let us briefly explain the strategy in the proof of [14] and why it cannot be used in low dimensions. The proof in [14] crucially relies on the fact that if d≥5, the trace of a bi-infinite random walk contains many bilateral cut-points (see [14], (6.1), (6.26)). This gives a decomposition of the random walk trace into a chain of relatively small well-separated “sausages.” Heuristically, a chain of sausages cannot separate two macroscopic connected subsets of a box. Random interlacements at level uis the trace of a certain Poisson cloud of doubly infinite random walk trajectories in Z^d, and, therefore, can be viewed as the countable union of doubly infinite chains of “sausages” inZ^d. Thus, in order to show that random interlacements at levelucannot separate two macroscopic connected subsets of a large box, one needs to show that locally it generally looks like the trace of only bounded number of random walks. This is achieved in [14] with a renormalization argument. The sausage decomposition property fails ford ≤4 (see, e.g., [7], Theorem 2.6). In fact, in dimensiond=3, even the trace of a single random walk is a “two-dimensional” object, and, therefore, could in principle form a large separating surface in a box. This is not the case, as we discuss in Section 6. Our proof of (1.3) only exploits basic properties of random walks (Green function estimates, Markov property) and works for all dimensionsd≥3.

The results of Theorem1.1are in the spirit of the local uniqueness property of supercritical Bernoulli percolation (see, e.g., [4], (7.89)). In fact, the analogues of (1.2) and (1.3) for Bernoulli percolation hold through the whole supercritical phase. We believe that the bounds (1.2) and (1.3) also hold for allu < u_∗, but with constantsc=c(d, u) >

0 andC=C(d, u) <∞depending onu. Our current understanding of the model is not good enough to be able to rigorously justify this belief.

The main technical challenges in the proof of Theorem 1.1come from the long-range dependence of the ran- dom interlacements (see, e.g., [11], Remark 1.6(4)), the lack of the BK inequality (see, e.g., [4], (2.12), and [12], Remark 1.5(3)) and the absence of finite energy property (see, e.g., [11], Remark 2.2(3)).

As an immediate corollary of Theorem1.1we obtain that finite connected components of the vacant set at levelu are unlikely to be large whenuis small enough.

Corollary 1.2. For anyd ≥3, there existc=c(d) >0 and C =C(d) <∞ such that for allu≤u₁ (defined in Theorem1.1),we have

P

n≤diam C^u(0)

<∞

≤Ce^−nc (1.4)

and P

n≤C^u(0)<∞

≤Ce^−nc, (1.5)

wherediam(C^u(0))and|C^u(0)|denote the diameter and the cardinality of the connected component of the origin in V^u,respectively.

Again, whend≥5, the result of Corollary1.2follows from [14], Theorems 3.5 and 3.6. The analogue of Corol- lary 1.2 for supercritical Bernoulli percolation is well known, and as Theorem1.1, it is a property of the whole supercritical phase of Bernoulli percolation (see, e.g., [2,6] and [4], Chapter 8). Moreover, the analogue of (1.4) for Bernoulli percolation holds with exponential decay rate (see, [4], (8.20)), and the analogue of (1.5) holds with stretched exponential decay with the explicit exponentc=(d−1)/d(see, e.g., [4], (8.66)).

Let us now mention some applications of Theorem1.1. In [9], Theorem1.1is used to study the stability of the phase transition of the vacant set under a small quenched noise. The setup is the following. 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. In [9], Theorem 5 it is proved that for anyuwhich satisfies (1.2) and (1.3), the perturbed vacant set at levelustill has an infinite connected component if the noise is small enough. In particular, this statement together with Theorem1.1imply that the perturbed vacant set at small levelustill has an infinite connected component. The use of Theorem1.1significantly simplifies the original proof of [9], Theorems 3 and 5, given in the first version of [9].

In [3], Theorem 2.3, we use Theorem1.1as an ingredient to prove that the graph distance in the unique infinite connected component of the vacant set at small level uis comparable to the graph distance onZ^d, and establish a shape theorem for balls with respect to graph distance on the infinite connected component.

We believe that the methods of this paper can be applied in order to further explore the fragmentation of the torus (Z/NZ)^d by the trace of a simple random walk, in a similar fashion to [15], where a strong coupling between the random walk trace on the torus and random interlacements is used to transfer results of [14] to the torus. We further discuss this possibility as well as the analogue of Theorem1.1for the set of sites avoided by a simple random walk onZ^din Section6.

We will now briefly sketch the main ideas of the proof of Theorem1.1. A more detailed description of the main steps of the proof will be given at the beginning of Sections3,4, and5. Before reading those descriptions, we advise the reader to become familiar with basic definitions and results concerning random interlacements in Sections2.3 and2.4.

The proof uses coarse graining (see Section3) and a conditional independence property for random interlacements (see Section4). The need for coarse graining comes from the fact that the complement of the infinite connected component of the vacant set is almost surely connected, no matter how small the parameteruis. (This is immediate from the fact thatI^uis almost surely connected for any givenu, see [11], (2.21).) The reader familiar with Bernoulli percolation may notice that this would not be the case if the vertices were made vacant independently from each other.

In this case, the usual Peierls argument would easily give the analogue of Theorem1.1for Bernoulli percolation, when the vacant set has density close to one.

To overcome the problem arising from the connectedness ofI^u, we partitionZ^d intoL_∞-boxes(B(x, R): x∈ (2R+1)·Z^d), with someR≥0. We use a variant of Sznitman’s decoupling inequalities [12] to show that whenRis large enough, there is a unique infinite connected subset of good boxes which are “sufficiently vacant.” Moreover, the remaining (bad) boxes form only finite connected subsets ofZ^d, with stretched exponential decay of the probability that a connected component of bad boxes is large. Our definition of good boxes also assures that the infinite connected component of good boxes contains an infinite connected subset ofV^u, which intersects every good box of the above set. For concreteness, in this proof sketch, we call this infinite connected subset ofV^u the “fat” set. As a result, we obtain that with high probability, any nearest-neighbor path of Z^d with large diameter often intersects the infinite connected component of good boxes, and therefore gets within distanceRfrom the fat set.

However, the possibility of having a long nearest-neighbor path inV^u which avoids the fat set (but unavoidably, with high probability, getsR-close to it sufficiently often) still remains. We use a conditional independence property of random interlacements (see Section4) to show that, roughly speaking, conditionally on the fact that a vacant path connects to a good box of the infinite connected set of good boxes and also conditioning on the configuration outside this box, there is still a uniformly positive chance that this vacant path is connected inside the specified good box to the fat set. The difficulty in the proof of this claim comes from the fact that random interlacements do not posess the so- called finite energy property (see, e.g., [11], Remark 2.2(3)). In words, the fact thatI^uis a connected set implies that depending on the realization ofI^uoutside a box, not every configuration can be realized byI^uinside this box. (This is a big constraint, and, for example, causes some difficulties in the proof of the uniqueness of an infinite connected component ofV^u, see [13].) Our definition of good boxes is chosen specifically to overcome this problem. Coming back to the proof sketch, since each long path must visit many good boxes in the infinite connected component, we conclude that with high probability each long path inV^umust be connected to the fat set. This gives us (1.3).

The paper is organized as follows. In Section2, we define the notation used in the paper, state some basic results about the simple random walk onZ^d, define random interlacements and recall some of its properties, the most im- portant of which is Lemma2.2. It is based on [12], Corollary 3.5, but formulated more generally (using so-called interlacement local times defined in Section2.4). Therefore, we give its proof sketch in theAppendix.

In Section3, we define coarse graining, and prove the existence of a “fat” infinite connected subset ofV^u, whenu is small enough (see Corollary3.7).

In Section4, we prove a conditional independence property of random interlacements (see Lemma4.4).

In Section5, we prove Theorem1.1using the results of Sections3and4.

Finally, in Section6, we briefly mention applications of the ideas developed in this paper to the vacant set of a simple random walk onZ^dand(Z/NZ)^d.

2. Notation, model, preliminaries

2.1. Basic notation

We denote byN= {0,1, . . .}the set of natural numbers, by Zthe set of integers. We denote by Rthe set of real numbers and byR₊the set of non-negative reals. Fora∈R, we write|a|for the absolute value ofa, andafor the integer part ofa.

For anyd≥1, we denote byx=(x1, . . . , xd)a generic element ofZ^d, also referred to asvertexofZ^d. We denote by|x| =max1≤i≤d|x_i|the sup-norm ofx∈Z^dand by|x|1= ^di=1|x_i|theL₁-norm ofx. ForK⊂Z^d, we denote by

|K|the cardinality ofK. We writeK⊂⊂Z^dwhenK⊂Z^dand|K|<∞.

We say thatx, x∈Z^dare nearest neighbors (respectively,∗-neighbors) if|x−x|1=1 (respectively,|x−x| =1).

We also denote|x−x|1=1 byx∼x. We say thatπ=(z1, . . . , zn)is a nearest neighbor path (respectively,∗-path) ifzi andzi+1are nearest neighbors (respectively,∗-neighbors) for all 1≤i≤n−1, and we use the notation|π| =n (not to be confused with the cardinality of the set{z₁, . . . , z_n}). We say thatV ⊆Z^d is connected (respectively,∗- connected) if any pairx₁, x₂∈V can be connected by a nearest neighbor path (respectively,∗-path) with vertices inV.

Forx∈Z^d andR∈Nwe denote by B(x, R)= {y∈Z^d: |x−y| ≤R}the closed ball of radiusR aroundx with respect to the sup-norm. For any setV ⊆Z^d, we denote byV^c=Z^d\V.

The interior boundary ofK⊆Z^d,∂_intKis the set of vertices ofKthat have some neighbor inK^c. The exterior boundary ofK⊆Z^d,∂extKis the set of vertices ofK^cthat have some neighbor inK.

Given a probability space(Ω,F,P)andA∈F, we denote by1Athe indicator of the eventA. IfXis an integrable random variable on(Ω,F,P), we denoteE[X;A] =E[X·1A].

For−∞ ≤a < b≤ +∞, we denote byB([a, b])the Borelσ-algebra on[a, b].

Our agreement about the constants used in the paper is the following. We denote small positive constants bycand large finite constants byC. When needed, we emphasize the dependence of a constant on parameters. If the constant only depends ond, then we sometimes do not mention it at all. The value of a constant may change within the same formula.

2.2. Simple random walk and potential theory

The spaceW₊stands for the set of infinite nearest-neighbor trajectories, defined for non-negative times and tending to infinity:

W₊=

w: N→Z^d, w(n)∼w(n+1), n∈N, lim

n→∞w(n)= ∞

. (2.1)

We endowW₊with theσ-algebraW+generated by the canonical coordinate mapsX_n,n∈N. For eachk∈N, we define the shift mapθ_k:W₊→W₊byθ_k(w)(·)=w(· +k). Forx∈Z^d, letP_xdenote the law of simple random walk onZ^dwith starting pointx. Simple random walk onZ^d,d≥3, is transient and the setW₊has full measure under any P_x. From now on we will viewP_xas a measure on(W₊,W+), and we write(X(t): t∈N)for a random element of W₊with distributionP_x.

ForU⊆Z^dandw∈W₊, we define H_U(w)=inf

n≥0:X_n(w)∈U

, the entrance time inU, (2.2)

H_U(w)=inf

n≥1:X_n(w)∈U

, the hitting time ofU, (2.3)

TU(w)=inf

n≥0: Xn(w) /∈U

, the exit time fromU. (2.4)

Ford≥3, the Green functiong:Z^d×Z^d→ [0,∞)of the simple random walkXis defined as g(x, y)=^∞

t=0

P_x

X(t )=y

, x, y∈Z^d.

Translation invariance yieldsg(x, y)=g(0, y−x). It follows from [8], Theorem 1.5.4, that for anyd≥3, there exist cg=cg(d) >0 andCg=Cg(d) <∞such that

cg·

|x−y| +12−d

≤g(x, y)≤Cg·

|x−y| +12−d

forx, y∈Z^d. (2.5)

The equilibrium measure ofK⊂⊂Z^dis defined by e_K(x)=

Px[HK= ∞], x∈K,

0, x /∈K.

The capacity ofKis the total mass of the equilibrium measure ofK:

cap(K)=

x

e_K(x). (2.6)

SinceZ^d is transient (d≥3), for any∅=K⊂⊂Z^d, the capacity ofKis positive. Therefore, we can define for such Kthe normalized equilibrium measure by

e_K(x)=e_K(x)/cap(K). (2.7)

The following relations forPx[HK<∞]will be useful: for anyK⊂⊂Z^dandx∈Z^d,

(i) (see, e.g. [11], (1.8)) P_x[H_K<∞] =

y∈K

g(x, y)e_K(y), (2.8)

(ii) (see [11], (1.9))

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). (2.9)

2.3. Definition of random interlacements

Now we recall the definition of the interlacement point process from [11], Section 1. We consider the space of doubly infinite nearest-neighbor trajectoriesW:

W=

w: Z→Z^d, w(n)∼w(n+1), n∈Z, lim

n→±∞w(n)= ∞

. (2.10)

We endowW with theσ-algebraWgenerated by the coordinate mapsXn,n∈Z. Consider the spaceW^∗of trajectories inWmodulo time shift

W^∗=W/∼, wherew∼w ⇐⇒ w(·)=w(· +k)for somek∈Z

and denote byπ^∗ the canonical projection fromW toW^∗ which assigns to each w∈W the∼-equivalence class π^∗(w)ofw. The mapπ^∗induces aσ-algebra onW^∗given byW^∗= {A⊂W^∗: (π^∗)⁻¹(A)∈W}.

ForK⊂⊂Z^d, we denote byW_Kthe set of trajectories inW that enter the setK, and denote byW_K^∗ the image of W_Kunderπ^∗. Note thatW_K∈WandW_K^∗ ∈W^∗.

For anyw^∗∈W^∗andu∈R+we call the pair(w^∗, u)a labeled trajectory. The space of point measures on which one canonically defines random interlacements is given by

Ω=

ω=

i≥1

δ_(w∗

i,ui): w_i^∗∈W^∗, u_i∈R₊and∀K⊂⊂Z^d, u≥0:ω

W_K^∗× [0, u]

<∞

. (2.11)

The space Ω is endowed with the σ-algebraFΩ generated by the evaluation maps of form ω→ω(D)for D∈ W^∗⊗B(R₊). We recall the definition of the measureQ_K on(W,W)from [11], (1.24): for anyA, B ∈W₊ and x∈Z^dlet

Q_K

(X_−n)_n≥0∈A, X₀=x, (X_n)_n≥0∈B

=P_x[A|H_K= ∞] ·e_K(x)·P_x[B]. (2.12) According to [11], Theorem 1.1, there exists a uniqueσ-finite measureνon(W^∗,W^∗)which satisfies the identity

ν(E)=QK

π^∗₋1

(E)

for allK⊂⊂Z^dandE∈W^∗withE⊆W_K^∗. (2.13)

The interlacement point processis the Poisson point process on W^∗×R₊ with intensity measure ν(dw^∗)du, defined on the probability space(Ω,FΩ,P). Givenω= i≥1δ_(w∗

i,ui)∈Ω andu≥0, therandom interlacement at leveluis the random subset ofZ^ddefined by

I^u(ω)=

i≥1,u_i<u

range w^∗_i

, (2.14)

where range(w^∗)= {w(n): n∈Z}for anyw∈π⁻¹(w^∗). Thevacant set at leveluis defined as V^u(ω)=Z^d\I^u(ω) forω∈Ω, u≥0.

For the sake of consistency, we mention that the law ofI^uis uniquely characterized by (1.1), see [11], Proposition 1.5 and Remark 2.2(2).

2.4. Discrete interlacement local times

In this section we define the interlacement local time fieldL^u(ω)at levelu, which counts the accumulated number of visits of the interlacement trajectories with label smaller thanuto each vertexx∈Z^d, see (2.15). We introduce this notion so that we can control the number of excursions of the interlacement trajectories inside a box in Section5.

We denote bya generic element of the product spaceN^Zd. For anyx∈Z^d, denote byΨ_x:N^Zd→Nthe canonical coordinate function defined byΨx()=(x). We consider the measurable space(N^Zd,F)whereFis theσ-algebra generated by the functionsΨ_x,x∈Z^d. For, ∈N^Zd, we say that≤if(x)≤(x)for allx∈Z^d. We say that an eventA∈Fisincreasingif for any, ∈N^Zd the conditions∈Aand≤imply∈A.

Givenω= _i≥1δ_(w∗

i,u_i)∈Ωandu≥0, we define the discrete interlacement local time profile at levelu,L^u(ω)= (L^u_x(ω): x∈Z^d)as

L^ux(ω)=

i≥1,u_i<u

n∈Z

1_{w_i(n)=x}, x∈Z^d, (2.15)

wherew_i is any particular element ofπ⁻¹(w^∗_i). Note that the functionL^u:(Ω,FΩ)→(N^Zd,F)is measurable and thatx∈I^u(ω)if and only ifL^u_x(ω)≥1.

Given a measurable functionL:(Ω,FΩ)→(N^Zd,F)and an eventA∈F, we define A(L)=

ω∈Ω: L(ω)∈A

and A^u=A L^u

foru≥0. (2.16)

It follows from (2.15) that for any 0≤u≤u, P[L^u≤L^u] =1. Therefore, for any increasing event A∈F and u≤u, we have

P A^u

≤P A^u

. (2.17)

Finally, we record that forx∈Z^dandu≥0, E

L^ux

=u. (2.18)

Indeed, by (2.12) and (2.13),E[L^u_x] =E[ω(W_{^∗_x}× [0, u))] ·g(x, x)=cap({x})·u·g(0,0)=u.

2.5. Cascading events

In this section we adapt some results of [12] to our setting which involves increasing events ofN^Zd. The result of Lemma2.2below is new, but very similar to [12], Corollary 3.5, which is stated for increasing events in{0,1}^G×Z, whereGis an infinite, connected, bounded degree weighted graph, satisfying certain regularity conditions (for exam- ple,G=Z^d−1, withd≥3). We will use Lemma2.2in the proof of Lemma3.6.

We begin with the definition of uniformly cascading events. We adapt [12], Definition 3.1, to our setting which involves local times.

Definition 2.1. Letλ >0.We say that a familyG=(G_x,L,R)_x∈Zd,L≥1,R≥0of events on(N^Zd,F)cascades uniformly (inR)with complexity at mostλ >0if there existsC(λ) <∞such that

G_x,L,Risσ (Ψ_y, y∈B(x,10L))-measurable for eachx∈Z^d,R≥0,andL≥1, and for eachlmultiple of100,x∈Z^d,R≥0,L≥1,there existsΛ⊆Z^dsuch that

Λ⊆B(x,9lL), (2.19)

|Λ| ≤C(λ)·l^λ, (2.20)

G_x,lL,R⊆

x₁,x₂∈Λ:|x₁−x₂|≥(l/100)L

G_x1,L,R∩G_x2,L,R. (2.21)

Lemma 2.2. LetG=(Gx,L,R)_x∈Zd,L≥1,R≥0be a family of increasing events on(N^Zd,F)cascading uniformly(in R)with complexity at mostλ >0.

LetL₀≥1,l₀large enough multiple of100,andL_n=lⁿ₀L₀. (2.22) Letu_L0=L₀^2−d,and recall the notation of(2.16).If

R≥0,Linf0≥1sup

x∈Z^dP G^u_x,L^L0

0,R

=0, (2.23)

then there existl0>1,R≥0,L0≥1andu >0such that sup

x∈Z^dP G^u_x,L

n,R

≤2⁻²ⁿ for alln≥0. (2.24)

The proof of Lemma2.2is essentially the same as the proof of [12], Corollary 3.5. For completeness, we include its sketch in theAppendix.

3. Coarse graining ofZ^d

In this section we show that when u is small enough, the infinite connected component of V^u contains a ubiqui- tous infinite connected subset, which has a well-prescribed structure and useful properties. We do so by partitioning Z^d into large boxes. We then define a notion of good boxes in Definition3.3. These boxes are defined to be “suf- ficiently vacant.” In Lemma3.6, we show that large∗-connected components of bad boxes are unlikely, where we use Lemma2.2to deal with the long-range correlations present in the model. We then combine it with the result of [5], Lemma 2.23, on the connectedness of the exterior∗-boundary of a ∗-connected finite subset ofZ^d to obtain in Corollary3.7that there is a unique infinite connected subset of good boxes (denoted byG^∞in Corollary3.7(2)), and all the remaining bad components are very small. It then follows from the definition of good boxes that the infinite connected component of good boxes contains the desired infinite connected subset ofV^u (see Corollary3.7(3)). An important consequence of Corollary3.7, which we will use in the proof of Theorem1.1(see (5.2) and (5.5)), is that with high probability, any long nearest-neighbor path inZ^dwill get within distanceRfrom the above defined infinite connected subset ofV^umany times.

3.1. Setup and auxiliary results

We consider the hypercubic latticeZ^dwithd≥3. For an integerR≥0, let

Z=(2R+1)·Z^d. (3.1)

We say thatx, y∈Zare (1) nearest-neighbors inZ, if|x−y|1=2R+1, and (2)∗-neighbors inZ, if|x−y| = 2R+1. We denote byB(x, N )=B(x, (2R+1)N )∩Zthe closed ball of radiusN inZ. The interior boundary of K⊆Z, denoted by∂intK, is the set of vertices ofKthat have some nearest neighbor inZ\K. Note that forR=0, the set∂_intKis different from∂_intK, defined in Section2.1.

With each vertexx∈Z, we associate the hypercube Q

x

=B x, R

⊂⊂Z^d. (3.2)

This gives us a partition ofZ^dinto disjoint hypercubes.

Definition 3.1. Letbe the subset of vertices inQ(0)such that at least two of their coordinates have values in the set{−R,−R+1,−R+2, R−2, R−1, R},and let(x)=x+,for allx∈Z.We call(x)the frame ofQ(x).

Fig. 1. The frame of Q(x)inZ³.

Note that the setis connected inZ^d, and for any x₁, x₂ ∈Znearest-neighbors inZ, the set(x₁)∪(x₂)is connected inZ^d.

In the cased=3, the set Q(x)is the usual cube, and the set(x)is just the 2-neighborhood of its edges in the sup-norm, restricted to the vertices inside Q(x).

Lemma 3.2. There existsC=C(d) <∞such that for allR≥2,

cap()≤CR^d−2/logR. (3.3)

Proof. The proof easily follows from (2.5), (2.6), (2.8), and (2.9). LetR≥2. Takex∈Z^dwith|x| =2R. Note that for anyy∈,R≤ |x−y| ≤3R. We have

cap()^(2.6)=

y∈

e(y)(2.5),(2.8)

≤ CR^d−2·P_x[H<∞]^(2.9)≤ CR^d−2·

y∈

g(x, y)

inf

z∈

y∈

g(z, y).

By (2.5), we get

y∈

g(x, y)≤CR^2−d· || ≤CR^2−d· d

2

·6²·(2R+1)^d−2≤C.

It remains to show that infz∈ y∈g(z, y)≥c·logR. By the definition of , for any z∈ and any integer 1≤k≤R, we have

y∈: |y−z| =k≥k^d−3. Therefore, uniformly inz∈, we obtain

y∈

g(z, y)≥ R k=1

y∈:|y−z|=k

g(z, y)^(2.5)≥ R k=1

c·k^2−d·k^d−3≥c·logR.

Putting all the bounds together we get (3.3).

3.2. Good vertices

Definition 3.3. Let∈N^Zd.We say thatx∈ZisR-good forif (1) (x)=0for allx∈(x),

(2) _{x∈∂intQ(x})(x)≤R^d−1.

Ifxis notR-good,then we call itR-bad for.

Remark 3.4. The choice ofR^d−1on the right-hand side of(2)is quite arbitrary.Any functionf =f (R)which grows faster than linearly would serve our purposes(see the proof of Lemma3.5).Condition(2)of Definition3.3will be important in Section5,where we use it to give an upper bound on the number of excursions of the interlacement trajectories inside∂_intQ(x).

Note that for anyR≥0 andx∈Z,

the event{xisR-good}is decreasing andσ (Ψ_y, y∈B(x, R))-measurable. (3.4) Lemma 3.5. ForR≥1,letuR=R^2−d.Then

P

0isR-good forL^uR

→1, asR→ ∞.

Proof. By the definition ofR-good vertices, it suffices to prove that P

⊆V^uR

→1 and P

x∈∂intQ(0)

L^ux^R≤R^d−1

→1, asR→ ∞.

The first statement follows from Lemma3.2. Indeed, P

⊆V^uR

=e^{−uR·cap()(3.3)}≥ e^−c/logR→1.

As for the second statement, by the Markov inequality, P

x∈∂intQ(0)

L^ux^R> R^d−1

≤R^1−d·

x∈∂intQ(0)

E L^ux^R

=R^1−d·∂intQ(0)·E

L^u₀^R(2.18)

≤ C·uR→0.

This completes the proof of Lemma3.5.

ForV₁, V₂⊆Z^d and∈N^Zd, we write “V1↔V₂by a∗-path inZofR-bad vertices for”, if there is a sequence π=(x₁, . . . , x_n)inZofR-bad vertices forsuch that

x₁ ∈V1, x_n ∈V2, ∀1≤i≤n−1:x_i+1−x_i=2R+1. (3.5) The next lemma proves that∗-connected components ofR-bad vertices forL^uinZare small for large enoughRand small enoughu. Then a standard relation between nearest-neighbor and ∗-connectivities implies the existence of a unique infinite connected component ofR-good vertices (see Corollary3.7).

Lemma 3.6. There existR≥0,u1>0,c=c(d) >0andC=c(d) <∞such that for allu≤u1andN≥1,we have P

0↔∂_intB(0, N )by a∗-path inZofR-bad vertices forL^u

≤Ce^−Nc. (3.6)

Proof. First of all, note that theF-measurable event

: 0↔∂_intB(0, N )by a∗-path inZofR-bad vertices for

is increasing. Therefore, it suffices to prove that there existR≥0,u >0,c >0 andC <∞such that for allN≥1, (3.6) holds. (Then, by (2.17), the result will hold for allusmaller thanu.)

Forx∈Z^dand integersR≥0,L≥1, consider the events

G_x,L,R=

⎧⎨

⎩

∈N^Zd: B(x, L)↔B(x,2L)^c

by a∗-path inZofR-bad vertices for

, ifL≥R,

N^Zd, ifL < R.

(3.7)

In order to prove (3.6), it suffices to show that there existL₀≥1,l₀>1,R≥0 andu >0 such that P

G_0,Ln,R L^u

≤2⁻²ⁿ for alln≥0, (3.8)

whereL_n are defined in (2.22) (see also the notation in (2.16)). This will immediately follow from Lemma2.2, as soon as we show that

(G_x,L,R)_x∈Zd,L≥1,R≥0is a family of increasing events cascading uniformly with complexity at mostd, (3.9) and that the family of events(Gx,L,R)_x∈Zd,L≥1,R≥0satisfies (2.23).

We begin with the proof of (3.9). The eventsG_x,L,R are clearly increasing. ForL≥R, we have∈G_x,L,Rif and only if there exists a∗-pathπ=(y₁, . . . , y_n)inZofR-bad vertices forsatisfying

y₁−x≤L, 2L <y_n −x, ∀1≤i≤n: y_i−x≤2L+2R+1. (3.10) Treating the casesL≥R andL < Rseparately and using (3.4) and (3.10), one can show that the eventG_x,L,R is σ (Ψ_y, y∈B(x,10L))-measurable. Letlbe a multiple of 100,x∈Z^d,R≥0,L≥1. Let

Λ=L·Z^d∩B(x,3lL).

The setΛimmediately satisfies (2.19) and (2.20) (withλ=d), so we only need to check thatΛsatisfies (2.21). By (3.7), it is enough to consider the non-trivial caseL≥R.

If∈Gx,lL,R, then there exists a∗-pathπ=(y₁, . . . , y_n)inZofR-bad vertices forsatisfying|y₁ −x| ≤lL and 2lL <|y_n −x| ≤2lL+2R+1≤3lL, so that we can findx1, x2∈Λsuch that |y₁ −x1| ≤L,|y_n −x2| ≤L.

Note that |x1−x2| ≥lL−2L > ₁₀₀^l L. Moreover, the pathπ connects B(xi, L)to B(xi,2L)^c fori∈ {1,2}. Thus ∈G_x1,L,R∩G_x2,L,R, which implies (2.21) and hence (3.9).

It remains to prove that(G_x,L,R)_x∈Zd,L≥1,R≥0satisfies (2.23). Let us chooseL₀=R. By (3.10) and (3.5) we have

G_x,R,R⊆

x∈B(x,R)∩Z

∈N^Zd: xisR-bad for .

Since|B(x, R)∩Z| =1, the condition (2.23) follows from Lemma3.5. Thus we can apply Lemma2.2to infer (3.8),

which completes the proof of Lemma3.6.

The following result states that there exists a ubiquitous infinite component of good vertices inZ. It is a conse- quence of Lemma 3.6and [5], Lemma 2.23, about the connectedness of the exterior∗-boundary of a∗-connected subset ofZ^d.

Corollary 3.7. FixR,u₁,c=c(d) >0,andC=C(d) <∞as in Lemma3.6.For allu≤u₁,we have (1) for alln, N≥1,

P

⎡

⎢⎢

⎣

B(0, N+n)\B(0, N )contains a setS⊂Zsuch that Sis connected inZ,eachx∈SisR-good forL^u,and every∗-path inZfromB(0, N+1)to∂_intB(0, N+n)

intersectsS

⎤

⎥⎥

⎦≥1−C·B(0, N+1)·e^−nc, (3.11)

(2) there exists a unique infinite connected component ofR-good vertices forL^uinZ,which we denote byG^∞,and for alln≥1,

P

G^∞contains a vertex inB(0, n)

≥1−C·

N≥n

e^−Nc, (3.12)

(3) the set

x∈G^∞(x)is an infinite connected subset ofV^u. Proof. (1) Taken, N≥1. Let

S=B(0, N )

∪

x∈B(0, N+n): x is connected toB(0, N+1)by a∗-path inB(0, N+n)ofR-bad vertices forL^u ,

and consider the exterior∗-boundary ofSinB(0, N+n):

S=

y∈B(0, N+n)\S: yis a∗-neighbor inZof somex∈S .

Note that every vertex in S isR-good. If S∩∂intB(0, N +n)=∅, then every∗-path in Zfrom B(0, N+1) to

∂intB(0, N +n)intersectsS. Non-trivially, it was proved in [5], Lemma 2.23 (see also a short proof in [16], Theo- rem 4), that ifS∩∂_intB(0, N+n)=∅, thenScontains aconnectedcomponentSinZsuch that every∗-path inZ fromB(0, N+1)to∂_intB(0, N+n)intersectsS. By translation invariance ofL^u and (3.6), withc=c(d) >0 and C=C(d) <∞as in Lemma3.6, and for alln, N≥1, we have

P

B(0, N+1)is connected to∂_intB(0, N+n) by a∗-path inZofR-bad vertices forL^u

≤B(0, N+1)·C·e^−nc. Together with the above observations, this implies the first statement of Corollary3.7.

(2) The existence ofG^∞as well as (3.12) follow from (3.6) and planar duality (see, e.g., the proof of [9], Theo- rem 2.1). The uniqueness ofG^∞follows from (3.11) and the Borel–Cantelli lemma.

(3) The fact that

x∈G^∞(x)is an infinite connected subset ofV^u follows from (2), Definition3.1of, and

Definition3.3ofR-good vertices.

4. Conditional independence for random interlacements

In this section we prove (in Lemma4.4) that the behavior of the interlacement trajectories with labels at mostuinside a finite setKis independent of their behavior outside ofK, given the information about entrance and exit points of all the excursions intoKof all the interlacement trajectories with labels at mostu. As part of the proof, we will also identify the conditional law of the excursions inside and outsideK(see (4.11) and (4.12), respectively).

We begin by introducing notation and recalling some properties of the interlacement point measures, which we will use to identify the above mentioned laws of excursions. We then properly define the excursions (in Section4.2) and theσ-algebras of events generated by excursions inside, outside, and on the boundary ofK(in Section4.3). Finally, (in Section4.4) we state and prove the conditional independence of theσ-algebras.

4.1. More preliminaries about interlacements

Recall the notation and the definition of the interlacement point process from Section2.3. Letω= _i≥0δ_(w∗ i,ui)be an interlacement point process onW^∗×R₊. ForK⊂⊂Z^d andu >0, let

ω_K,u=

i≥0

δ_(w∗ i,ui)1_{w∗

i∈W_K^∗,ui≤u} and ω−ω_K,u=

i≥0

δ_(w∗ i,ui)1_{w∗

i∈/W_K^∗}∪{ui>u} (4.1) be the restrictions ofωto the set of pairs(w_i^∗, ui)with, respectively,w_i^∗intersectingK andui≤u, and eitherw_i^∗ not intersectingKorui> u. By the definition ofω, the point measuresωK,uandω−ωK,uare independent Poisson