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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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Proof. Take a positive integer r. By Definition 4.1 (see also the notation there), for each i ∈ {1,...,sd−1},
µ(i)= µ(i)
r
and µ(i)
any i ∈ {1,...,sd− 1}, let N(i)be the number of trajectories in Supp(µ(i)
is the number of doubly-infinite trajectories modulo time-shift from Supp(µ) that intersect B(r). By
property (1) of Pois(u,W∗), N(i)has the Poisson distribution with parameter ucap(B(r)). By the defi-
nition of Pois(u,W∗), we know that (recall the notation from Section 1.1), for each i ∈ {1,...,sd− 1},
µ(i)
r
j=1δπ∗(X(i)
are parametrized in such a way that X(i)
j ∈ {1,...,N(i)}, and (b) they satisfy properties (2) and (3) of Pois(u,W∗). In particular, given N(i)and
(X(i)
walks.
Property (5) of Pois(u,W∗) gives that for each i ∈ {1,...,sd− 1}, all the random walks (X(i)
0)N(i)
r,∞for k ∈ {1,...,sd−1}. Therefore, Lemma 10 and Remark 2 imply that,
given N(i)and (X(i)
j=1for all i ∈ {1,...,sd− 1}, almost surely, for each pair of different random
walks (X(i)
l
1 ≤ m ≤ sd− 1, such that X(i)
j
l
∩ wsd−1?= ∅, and wi∩ wi+1?= ∅ for i ∈ {1,...,sd− 2}.
Since this holds for any r, the result follows.
r + µ(i)
r,∞, and the measures µ(i)
r,∞are independent by property (5) of Pois(u,W∗). For
r ).In other words, N(i)
=?N(i)
j), where X(i)
1,...,X(i)
N(i)are doubly-infinite trajectories from W such that (a) they
j(0) ∈ B(r) and X(i)
j(t) / ∈ B(r) for all t < 0 and for all
j(0))N(i)
j=1, the forward trajectories (X(i)
j(t),t ≥ 0)N(i)
j=1are distributed as independent simple random
j(t),t ≥
j=1are independent from µ(k)
j(0))N(i)
j(t),t ≥ 0) and (X(k)
(t),t ≥ 0), there exist doubly-infinite trajectories wm ∈ Supp(µ(m)
∩ w1?= ∅, X(k)
r,∞),
Proof of Theorem 1: upper bound on diameter. We complete the proof of Theorem 1 by showing that
P(diam(G) ≤ sd) = 1. By Remark 1, we may and will assume that d ≥ 5. Let µ(1),...,µ(sd−1)be
independent Poisson point measures on W∗with distribution Pois(u/(sd− 1),W∗). We construct the
graph G′= (V′,E′) as follows. The set of vertices V′is the set of trajectories from ∪sd−1
and the set of edges E′is the set of pairs of different trajectories from ∪sd−1
Lemma 12 implies that the diameter of G′is at most sd. On the other hand, by property (4) of Pois(u,W∗),
graphs G and G′have the same law. This completes the proof.
i=1Supp(µ(i)),
i=1Supp(µ(i)) that intersect.
Acknowledgments. We would like to thank A.-S. Sznitman for suggesting to look for an alternative
proof of the connectivity of the random interlacement, inspiring discussions, and valuable comments. We
also thank A. Drewitz for comments on the manuscript.
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