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arXiv:1012.4711v1 [math.PR] 21 Dec 2010

Connectivity properties of random interlacement and

intersection of random walks

Bal´ azs R´ ath∗

Art¨ em Sapozhnikov∗

December 2010

Abstract

We consider the interlacement Poisson point process on the space of doubly-infinite Zd-valued

trajectories modulo time shift, tending to infinity at positive and negative infinite times. The set of

vertices and edges visited by at least one of these trajectories is the random interlacement at level u

of Sznitman [12]. We prove that for any u > 0, almost surely, (1) any two vertices in the random

interlacement at level u are connected via at most ⌈d/2⌉ trajectories of the point process, and (2) there

are vertices in the random interlacement at level u which can only be connected via at least ⌈d/2⌉

trajectories of the point process. In particular, this implies the already known result of [12] that the

random interlacement at level u is connected.

1Introduction

The model of random interlacements was recently introduced by Sznitman in [12] in order to describe

the local picture left by the trajectory of a random walk on the discrete torus (Z/NZ)d, d ≥ 3 when

it runs up to times of order Nd, or on the discrete cylinder (Z/NZ)d× Z , d ≥ 2, when it runs up to

times of order N2d, see [11], [14]. Informally, the random interlacement Poisson point process consists of

a countable collection of doubly infinite trajectories on Zd, and the trace left by these trajectories on a

finite subset of Zd“looks like” the trace of the above mentioned random walks.

So far, research related to random interlacements mainly focused on the description of the connectivity

properties of the vacant set (which corresponds to the set of vertices not visited by the random walker).

In this paper we investigate connectivity properties of the random interlacement, giving a detailed picture

about how the collection of doubly infinite trajectories are actually interlaced. Our methods are further

developed in [9] to study properties of percolation and random walks on the random interlacement.

1.1The model

Let W be the space of doubly-infinite nearest-neighbor trajectories in Zd(d ≥ 3) which tend to infinity

at positive and negative infinite times, and let W∗be the space of equivalence classes of trajectories in

W modulo time shift. We write W for the canonical σ-algebra on W generated by the coordinates Xn,

n ∈ Z, and W∗for the largest σ-algebra on W∗for which the canonical map π∗from (W,W) to (W∗,W∗)

is measurable. Let u be a positive number. We say that a Poisson point measure µ on W∗has distribution

∗ETH Z¨ urich, Department of Mathematics, R¨ amistrasse 101, 8092 Z¨ urich.

artem.sapozhnikov@math.ethz.ch. The research of both authors has been supported by the grant ERC-2009-AdG 245728-

RWPERCRI.

0MSC2000: Primary 60K35, 82B43.

0Keywords: Random interlacement; random walk; intersection of random walks; capacity; Wiener test.

Email: balazs.rath@math.ethz.ch and

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Pois(u,W∗) if the following properties hold: For a finite subset A of Zd, let µAbe the restriction of µ to

the set of trajectories from W∗that intersect A, and let NAbe the number of trajectories in Supp(µA).

Then µA=?NA

(1) The random variable NAhas Poisson distribution with parameter ucap(A) (see (2.2) for the defini-

tion of the cap(A)).

i=1δπ∗(Xi), where Xiare doubly-infinite trajectories from W parametrized in such a way

that Xi(0) ∈ A and Xi(t) / ∈ A for all t < 0 and for all i ∈ {1,...,NA}, and

(2) Given NA, the points Xi(0), i ∈ {1,...,NA}, are independent and distributed according to the

normalized equilibrium measure on A (see (2.7) for the definition).

(3) Given NAand (Xi(0))NA

pendent, (Xi(t),t ≥ 0)NA

are distributed as independent random walks conditioned on not hitting A.

i=1, the corresponding forward and backward paths are conditionally inde-

i=1are distributed as independent simple random walks, and (Xi(t),t ≤ 0)NA

i=1

Properties (1)-(3) uniquely define Pois(u,W∗) as proved in Theorem 1.1 in [12]. In fact, Theorem 1.1

in [12] gives a coupling of the Poisson point measures µ(u) with distribution Pois(u,W∗) for all u > 0,

but we will not need such a general statement here. We also mention a couple of properties of the

distribution Pois(u,W∗), which will be useful in the proofs. Property (4) follows from the above definition

of Pois(u,W∗), and (5) is a property of Poisson point measures.

(4) Let µ1and µ2be independent Poisson point measures on W∗with distributions Pois(u1,W∗) and

Pois(u2,W∗), respectively. Then µ1+ µ2has distribution Pois(u1+ u2,W∗).

(5) Let S1,...,Skbe disjoint elements of W∗. We denote by I(Si)µ the restriction of µ to the set of

trajectories from Si. Then I(S1)µ,...,I(Sk)µ are independent Poisson point measures on W∗.

We refer the reader to [12] for more details. For a Poisson point measure µ with distribution Pois(u,W∗),

the random interlacement I at level u is defined as

?

I = I(µ) =

w∈Supp(µ)

range(w).(1.1)

1.2The result

We consider a random point measure µ on W∗distributed as Pois(u,W∗). We denote by P the law of

µ. Our main result concerns the geometric properties of the support of µ. Remember that the support of

µ consists of a countable set of doubly-infinite random walk trajectories modulo time shift. We construct

the random graph G = (V,E) as follows. The set of vertices V is the set of trajectories from Supp(µ), and

the set of edges E is the set of pairs of different trajectories from Supp(µ) that intersect. Let diam(G)

be the diameter of G. Our main result is the following theorem.

Theorem 1. For d ≥ 3, let

sd= ⌈(d − 2)/2⌉,(1.2)

where ⌈a⌉ is the smallest integer not less than a. Then

P(diam(G) = sd) = 1,

In particular, we get an alternative proof of (2.21) in [12], which states that the random interlacement I

is a connected subgraph of Zd.

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Remark 1. In dimensions 3 and 4, the result is a trivial consequence of Theorem 2.6 in [4] (see also

remark at the bottom of page 661 in [4]) which states that two independent random walks in dimension

3 or 4 intersect infinitely often with probability 1. Therefore, it remains to prove the theorem for d ≥ 5.

The structure of the proof of Theorem 1 can be non-rigorously summarized as follows: first we pick

one of the doubly infinite trajectories from Supp(µ). Denote by A(1)the set of vertices of Zdvisited by

this trajectory. The second layer A(2)consists of the vertices visited by those trajectories of Supp(µ) that

intersect A(1), and recursively let A(s)denote the set of vertices visited by the trajectories that intersect

A(s−1). We prove that P(diam(G) = sd) = 1 by showing that, almost surely, A(sd)?= I and A(sd+1)= I.

Let us recall the following well-known fact (see, e.g., Proposition 2.3 in [4]): For d ≥ 3, the probability

that a simple random walk from 0 hits x is comparable with min(1,|x|2−d). We will use this fact and the

following elementary lemma to show that A(sd)?= I.

Lemma 1. There exists a finite constant C = C(d) such that for any positive integer n and for any

z0,zn+1∈ Zd,

?

z1,...,zn∈Zd

n

?

i=0

min

?

1,|zi− zi+1|2−d?

?

≤ C|z0− zn+1|2n+2−d

= ∞

if n < sd,

otherwise.

(See, e.g. (1.38) of Proposition 1.7 in [2] for a proof of Lemma 1.) Lemma 1 gives bounds on n-fold

convolutions of the probability that a random walk from z0ever visits zn+1. We will see that P(0,x ∈ A(s))

can be estimated as a (s−1)-fold convolution of such hitting probabilities, and, therefore, we will conclude

from Lemma 1 that P(0,x ∈ A(s)) ≤ C|x|2s−d. In particular, P(0,x ∈ A(sd)) → 0 as |x| → ∞. This

contradicts A(sd)= I, since I has positive density.

In order to show that A(sd+1)= I, we argue as follows. Heuristically, A(s)is a 2s-dimensional object

as long as 2s < d. The capacity of A(s)intersected with a ball of radius R (see (2.2) for the definition

of the capacity) is comparable to R2sas long as 2s ≤ d − 2. The set A(sd)already saturates the ball in

terms of capacity, thus it is visible for an independent random walk started somewhere inside the ball of

radius R. We apply a variant of Wiener’s test (see, e.g., Proposition 2.4 in [4]) to show that any random

walk hits A(sd)almost surely.

This is the general strategy of the proof. Instead of following it directly, we benefit from property (4) of

Pois(u,W∗) by decomposing µ into a sum of sdi.i.d point measures µ(s)with distribution Pois(u/sd,W∗)

and constructing each A(s)from the “new” measure µ(s).

The paper is organized as follows. In Section 2 we collect most of the notation and facts used in the

paper. The most important of those are the definitions and properties of the Green function and the

capacity. We prove the lower bound of Theorem 1 in Section 3, and the upper bound in Section 4. The

structure of the proof of the upper bound of Theorem 1 is given at the beginning of Section 4.

2Notation and facts about Green function and capacity

In this section we collect most of the notation, definitions and facts used in the paper. For a ∈ R, we

write |a| for the absolute value of a, ⌊a⌋ for the integer part of a, and ⌈a⌉ for the smallest integer not less

than a. For x ∈ Zd, we write |x| for max(|x1|,...,|xd|). For a set S, we write |S| for the cardinality of

S. For R > 0 and x ∈ Zd, let B(x,R) = {y ∈ Zd: |x − y| ≤ R} be the ball of radius R centered at

x. We denote by I(A) the indicator of event A, and by E[X;A] the expected value of random variable

XI(A). 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.

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For x ∈ Zd, let Pxbe the law of a simple random walk X on Zdwith X(0) = x. We write g(·,·) for

the Green function of the walk:

g(x,y) =

∞

?

t=0

Px(X(t) = y), x,y ∈ Zd.

We also write g(·) for g(0,·). The Green function is symmetric and, by translation invariance, g(x,y) =

g(y − x). It follows from [5, Theorem 1.5.4] that for any d ≥ 3 there exist a positive constant cg= cg(d)

and a finite constant Cg= Cg(d) such that for all x and y in Zd,

?

Definition 2.1. Let K be a subset of Zd. The energy of a finite Borel measure ν on K is

?

cgmin1,|x − y|2−d?

≤ g(x,y) ≤ Cgmin

?

1,|x − y|2−d?

. (2.1)

E(ν) =

K

?

K

g(x,y)dν(x)dν(y) =

?

x,y∈K

g(x,y)ν(x)ν(y).

The capacity of K is

cap(K) =

?

inf

νE(ν)

?−1, (2.2)

where the infimum is over probability measures ν on K. (We assume that ∞−1= 0, i.e. the capacity of

the empty set is 0.)

The following properties of the capacity immediately follow from (2.2):

Monotonicity: for any K1⊂ K2⊂ Zd, cap(K1) ≤ cap(K2);

for any K1,K2⊂ Zd, cap(K1∪ K2) ≤ cap(K1) + cap(K2);

for any x ∈ Zd, cap({x}) = 1/g(0).

(2.3)

Subadditivity: (2.4)

Capacity of a point:(2.5)

It will be useful to have an alternative definition of the capacity in d ≥ 3.

Definition 2.2. Let K be a finite subset of Zd. The equilibrium measure of K is defined by

eK(x) = Px(X(t) / ∈ K for all t ≥ 1)I(x ∈ K), x ∈ Zd.(2.6)

The capacity of K is then equal to the total mass of the equilibrium measure of K:

cap(K) =

?

x

eK(x),

and the unique minimizer of the variational problem (2.2) is given by the normalized equilibrium measure

? eK(x) = eK(x)/cap(K).(2.7)

(See, e.g., Lemma 2.3 in [3] for a proof of this fact.)

As a simple corollary of the above definition, we get for d ≥ 3,

Px(H(K) < ∞) =

?

y∈K

g(x,y)eK(y), for x ∈ Zd.(2.8)

Here, we write H(K) for the first entrance time in K, i.e. H(K) = inf{t ≥ 0 : X(t) ∈ K}. We will

repeatedly use the following bound on the capacity of B(0,R) in d ≥ 3 (see (2.16) on page 53 in [5]):

There exist constants cb= cb(d) > 0 and Cb= Cb(d) < ∞ such that for all positive R,

cbRd−2≤ cap(B(0,R)) ≤ CbRd−2.(2.9)

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3Proof of Theorem 1: lower bound on the diameter

Remember the definition of sdin (1.2). In this section we prove that P(diam(G) ≥ sd) = 1. Since,

almost surely, diam(G) ≥ 1, we only need to consider the case d ≥ 5. For two trajectories v and w in

V , we write ρ(v,w) for the distance between v and w in G. In order to prove that the probability of the

event {diam(G) ≥ sd} is 1, we assume by contradiction that this probability is ≤ 1−δ, for some positive

δ. In other words, the probability of event

E = {ρ(v,w) ≤ sd− 1 for all v,w ∈ V }

is bounded from below by δ.

For x,y ∈ Zd, we denote by S(x,y) the subset of doubly-infinite trajectories in W∗that intersect both

vertices x and y. Remember the definition (1.1) of the random interlacement I. The next lemma gives

an estimate on the probability that E occurs and two different vertices x and y of Zdare in I:

Lemma 2. For any x,y ∈ Zd,

P({x,y ∈ I} ∩ E) ≤

sd−1

?

n=1

?

z1,...,zn∈Zd

n

?

i=0

E[µ(S(zi,zi+1))], (3.1)

where we take z0= x and zn+1= y.

We postpone the proof of Lemma 2 until the end of this section.

E[µ(S(zi,zi+1))] in (3.1) is bounded from above by 2ug(zi,zi+1). (This follows, for example, from (1.33)

in [13] applied to K = {zi} and K′= {zi+1}.) Therefore, we obtain

Each of the expectations

P({x,y ∈ I} ∩ E) ≤

sd−1

?

n=1

(2u)n+1

?

z1,...,zn∈Zd

n

?

i=0

g(zi,zi+1),

where we again assume z0 = x and zn+1 = y. Recall from (2.1) that g(x,y) ≤ Cgmin(1,|x − y|2−d).

Therefore, by Lemma 1,

sd−1

?

n=1

?

z1,...,zn∈Zd

n

?

i=0

g(zi,zi+1) ≤ C|z0− zn+1|2sd−d≤ C|z0− zn+1|−1.

In particular, P({x,y ∈ I} ∩ E) ≤ C|x − y|−1→ 0, as |x − y| → ∞. By property (1) of Pois(u,W∗), for

any R > 0,

P(I ∩ B(0,R) ?= ∅) = P?NB(0,R)≥ 1?= 1 − e−ucap(B(0,R)).

By (2.9), we can take R big enough so that

P(I ∩ B(0,R) ?= ∅) ≥ 1 −δ

3.

With this choice of R, for any z ∈ Zd, we obtain

P({I ∩ B(0,R) ?= ∅} ∩ {I ∩ B(z,R) ?= ∅} ∩ E) ≥ P(E) − 2P(I ∩ B(0,R) = ∅) ≥ δ/3.

On the other hand, for z ∈ Zdwith |z| > 3R,

P({I ∩ B(0,R) ?= ∅} ∩ {I ∩ B(z,R) ?= ∅} ∩ E) ≤

?

x∈B(0,R)

?

y∈B(z,R)

P({x,y ∈ I} ∩ E) ≤ CR2d|z|−1,

which tends to 0 as |z| tends to infinity. This is a contradiction, and we conclude that P-a.s. the diameter

of G is at least sd.

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