UPPER TAILS FOR INTERSECTION LOCAL TIMES OF RANDOM
WALKS IN SUPERCRITICAL DIMENSIONS
Xia Chen and Peter Mörters
University of Tennessee and University of Bath
Abstract. We determine the precise asymptotics of the logarithmic upper tail probability of the
total intersection local time of p independent random walks in Zd under the assumption p(d − 2) > d.
Our approach allows a direct treatment of the infinite time horizon.
Mathematics Subject Classification (2000): 60F10, 60G50, 60K35.
1. Introduction and main result
1.1 Motivation
Quantities measuring the amount of self-intersection of a random walk, or of mutual intersection of
several independent random walks have been studied intensively for more than twenty years, see e.g.
[DV79, LG86, La91, MR97, HK01, Ch04]. This research is often motivated by the rôle these quantities
play in quantum field theory, see e.g. [FFS92], in our understanding of self-avoiding walks and polymer
models, see e.g. [MS93], or in the analysis of stochastic processes in random environments, see e.g.
[HKM06, GKS07, AC07, FMW08]. In the latter models dependence between a moving particle and a
random environment frequently comes from the particle’s ability to revisit sites with a (in some sense)
attractive environment, and therefore measures of self-intersection quantify the degree of dependence
between movement and environment. Typically, in high dimensions this dependence gets weaker, as
the movements become more transient and self-intersections less likely.
In their influential paper [KM94], Khanin, Mazel, Shlosman and Sinai make the surprising observation
that even in high dimensions the random variables counting the total number of intersections of
two independent simple random walks have subexponential tails at infinity. They argue that this
observation is intimately related to the fact that a random walk in random potential has subdiffusive
behaviour in all dimensions. Similar observations were made at around the same time by Sznitman in
a continuous setting, see [Sz93]. In their analysis of intersection measures of high-dimensional random
walks, Khanin et al. obtain estimates for the tails at infinity, but there are still significant gaps
between the given upper and lower bounds, which despite considerable effort have not been resolved
in the past ten years. In this paper we make a contribution to the closure of these gaps.
To be precise, let (X (1) (n) : n ∈ N), . . . , (X (p) (n) : n ∈ N) be p independent identically distributed
random walks taking values in Zd . The number of intersections of the walks can be measured in two
natural ways: The intersection local time,
∞
∞
X
X
1l{X (1) (i1 ) = · · · = X (p) (ip )},
···
I :=
i1 =1
ip =1
counts the times when the paths intersect, whereas the intersection of the ranges,
X
J :=
1l{X (1) (i1 ) = · · · = X (p) (ip ) = x for some (i1 , . . . , ip )},
x∈Zd
counts the sites where the paths intersect. Obviously, we always have J ≤ I.
2
XIA CHEN AND PETER MÖRTERS
Moreover, under suitable conditions on the random walk, it is well known that
1 if p(d − 2) > d,
P{I < ∞} = P{J < ∞} =
0 otherwise.
The lower bounds here are easy and hold for all symmetric, aperiodic random walks with finite variance,
while the upper bounds hold for simple, symmetric walks by a celebrated result of Dvoretzky and
Erdős [DE51]. We assume throughout this paper that p(d − 2) > d, i.e. we are assuming that we are
in supercritical dimensions. Then, in the case of simple, symmetric random walks, Khanin et al. show
in [KM94] that there exist constants c1 , c2 > 0 such that, for all a large enough,
1
1
exp − c1 a p ≤ P I > a ≤ exp − c2 a p .
Interestingly, the upper tails of J are substantially lighter. Khanin et al. show that, for all ε > 0 and
all sufficiently large a,
d−2
d−2
exp − a d +ε ≤ P J > a ≤ exp − a d −ε .
The challenging question lies in understanding the difference of these behaviours, providing sharp
estimates for the tails, and understanding the underlying ‘optimal strategies’.
The most important recent progress on this problem was made by van den Berg, Bolthausen and den
Hollander [BBH04] for the spatially and temporally continuous analogue of J: Let W1ε (t) and W2ε (t)
be the ε-neighbourhoods of two independent Brownian paths starting at the origin and running for t
time units, and use | · | to denote Lebesgue measure. They show that, for d ≥ 3,
lim
1
t↑∞ t(d−2)/d
log P W1ε (θt) ∩ W2ε (θt) ≥ t = −Idε (θ),
and, if d ≥ 5, there exists a critical θ ∗ such that Idε (θ) = Idε (θ ∗ ) for all θ ≥ θ ∗ . This strongly suggests
that, in the supercritical case d ≥ 5,
lim
1
t↑∞ t(d−2)/d
log P W1ε (∞) ∩ W2ε (∞) ≥ t = −Idε (θ ∗ ),
but the techniques of [BBH04] are strongly reliant on Donsker-Varadhan large deviation theory and,
mostly for this reason, do not allow the treatment of infinite times. Therefore this problem, like its
discrete counterpart, remains open for the time being.
In the present paper we focus on the intersection local time I, working directly in the discrete setting.
We provide sharp estimates for the tail behaviour of I, in other words we identify the limit
lim
a↑∞
1
a1/p
log P I > a ,
which is given in terms of a natural associated variational problem. The important progress here lies
in the fact that our method avoids the use of Donsker-Varadhan large deviation theory and instead
allows a direct treatment of infinite times. The proofs provide new insight into the optimal strategies
leading to the event {I > a}. An interpretation of the results is that the bulk of the intersections
contributing to large values of I are attained in a bounded subset of Zd . Naturally, this strategy does
not lead to large values of J, which therefore have to be obtained by a different, more expensive,
strategy.
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
3
1.2 Statement of the main results
1.2.1 Intersection local times for discrete time random walks. Let (X (1) (n) : n ∈ N), . . . , (X (p) (n) :
n ∈ N) be p independent identically distributed random walks started at time n = 0 in the origin
and taking values on the lattice Zd . Suppose that the walks are aperiodic, have symmetric increments
and finite variance, and that d is sufficiently large to satisfy p(d − 2) > d, i.e. we are in supercritical
dimensions.
Let G be Green’s function associated to the random walk, defined by
G(x) :=
∞
X
P{X (1) (n) = x}.
n=1
Note that we are following the (slightly unusual) convention of not summing over the time n = 0,
which only influences the value G(0). We first state integrability and boundedness of some operators
associated with Green’s function.
Lemma 1. Let q > 1 be the conjugate of p defined by p−1 +q −1 = 1. For every nonnegative h ∈ Lq (Zd )
a bounded, symmetric operator Ah : L2 (Zd ) → L2 (Zd ) is defined by
p
p
X
G(x − y)g(y) eh(y) − 1 .
Ah g(x) = eh(x) − 1
y∈Zd
Moreover, we have G ∈ Lp (Zd ) and for the spectral radius, or operator norm,
kAh k := sup |hg, Ah gi| : kgk2 = 1 = sup hg, Ah gi : g ≥ 0, kgk2 = 1
(1)
of the operator Ah we have the inequality
kAh k ≤ kGkp ekhkq − 1 .
(2)
Note that the alternative description of the spectral radii of Ah , for nonnegative h ∈ Lq (Zd ), on the
right hand side of (1) holds trivially because Green’s function is nonnegative. The next theorem,
which is our main result, shows that this family of spectral radii can be used to describe the upper
tail constant of the intersection local time. Its proof will be given in Section 2.
Theorem 2. The upper tail behaviour of the intersection local time I is given as
1
lim 1/p log P I > a = −p inf khkq : h ≥ 0 with kAh k ≥ 1 ,
a↑∞ a
(3)
and the right hand side is negative and finite.
Remark 1.
(a) Heuristically, the optimal strategy for the random walks to realise the event {I > a} is to spend
each about a1/p time units in all points of a finite domain A ⊂ Zd , which is not growing with a. This
leads to a typical intersection local time I ≈ a on this domain alone. As a ↑ ∞ the intersection sites
do not spread out in space when the random walks are conditioned on {I > a}, hence this strategy
makes I large without making J large. This explains the fundamental difference between these two
quantities in supercritical dimensions.
(b) Our proof provides the following evidence for this behaviour: Suppose A ⊂ Zd is a finite set and
I(A) :=
∞
X
i1 =1
···
∞
X
ip =1
1l{X (1) (i1 ) = · · · = X (p) (ip ) ∈ A},
the number of intersections in the set A. Then we show, as a by-product of the proof of Theorem 2, that
4
XIA CHEN AND PETER MÖRTERS
lim
1
a↑∞ a1/p
log P I(A) > a = −p inf khkq : h ≥ 0, supp h ⊂ A with kAh k ≥ 1 .
(4)
It is easy to see that, as A ↑ Zd , the right hand side in (4) converges to the right hand side in (3), and
thus for large A the tails of I(A) come arbitrarily close to those of I.
(c) A further interesting feature of our result is that the rate of decay of P{I(A) > a} is invariant
under spatial shifts of A, in other words, at the given scale the random walks show no preference for
locating their intersections near their starting point. This fact causes considerable difficulty in the
proof of the upper bound, as the rate of P{I(Ac ) > a} does not increase when A is getting large
so that, loosely speaking, the problem is not exponentially tight. The technique to get around this
problem is one of the main technical innovations in this paper.
Remark 2. From (2) we easily obtain an upper bound for the constant in (3), namely
−p inf khkq : h ≥ 0 with kAh k ≥ 1 ≤ −p log 1 + kGk−1
.
p
For a lower bound, we define the potential operator
X
Gf (x) :=
G(x − y)f (y),
y∈Zd
and note that, as G ∈ Lp (Zd ), this defines a bounded, symmetric operator
2p
G : L 2p−1 (Zd ) −→ L2p (Zd ) .
We thus get, using eh − 1 ≥ h for h ≥ 0, that
−p inf khkq : h ≥ 0, sup hg, Ah gi ≥ 1
kgk2 =1
√ √
≥ −p inf b : sup
hg, G hg ≥ 1/b
kgk2 =1
khkq =1
Neither of these bounds are sharp.
= −p/ sup hf 2p−1 , Gf 2p−1 i : kf k2p = 1 .
1.2.2 Intersection local times for continuous time random walks. It would be desirable to bring the
variational formula on the right hand side of (3) into a form naturally interpretable as a competition
between a probabilistic cost and benefit factor, and interpret the optimiser in terms of the optimal
strategy of the walks. However, the present formula cannot be easily transformed to such a form, an
artefact which we believe is due to the discrete time structure of the random walk.
We therefore provide a more transparent result for continuous time random walk for comparison. Let
(X (1) (t) : t ≥ 0), . . . , (X (p) (t) : t ≥ 0)
be independent, identically distributed continuous time random walks and let A be their generator,
defined by
Ex f (Xt(1) ) − f (x)
.
Af (x) = lim
t↓0
t
We assume that the random walks are aperiodic, and symmetric with finite variance. Then A is a
nonpositive definite, symmetric operator. Based on Green’s function
Z ∞
P{X(t) = x} dt ,
G(x) :=
0
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
5
we define the potential operator G as in the discrete time case, see Remark 2. Under our standing
assumption of supercriticality, d(p − 2) > d, it is easy to see that G ∈ Lp (Zd ), and again that the
potential operator is a bounded operator
2p
G : L 2p−1 (Zd ) −→ L2p (Zd ) .
We define the intersection local time as
Z ∞
Z
˜
I :=
dt1 · · ·
0
∞
0
dtp 1l{X (1) (t1 ) = · · · = X (p) (tp )},
which is finite in the supercritical regime. As before, we ask for its upper tail behaviour.
Theorem 3. The upper tail behaviour of the intersection local time I˜ is given as
1
lim 1/p log P I˜ > a = −p/ sup hf 2p−1 , Gf 2p−1 i : kf k2p = 1 ,
a↑∞ a
where the right hand side is negative and finite.
(5)
Remark 3. Under some additional conditions, the formula on the right can be reformulated in an
appealing way. Denote the supremum on the right hand side of (5) by ̺. Given ǫ > 0 choose f ≥ 0
such that kf k2p = 1 and hf 2p−1 , Gf 2p−1 i ≥ ̺ − ǫ. Then one can use that −A ◦ G =id and the fact
that, by Hölder’s inequality,
− 2p kGf 2p−1 k2p ≥ kf k2p2p−1 f 2p−1 , Gf 2p−1 ≥ ̺ − ǫ,
to obtain that
̺ ≥ hGf 2p−1 , f 2p−1 i = hGf 2p−1 , −AGf 2p−1 i = (̺ − ǫ)2
≥ (̺ − ǫ)2 inf g, −Ag : kgk2p = 1 ,
D Gf 2p−1
̺−ǫ
, −A
Gf 2p−1 E
̺−ǫ
whence, for ǫ ↓ 0, we get
1
≥ inf g, −Ag : kgk2p = 1 .
̺
For the converse inequality, we assume that the generator A is such that there exists a positive
minimiser g for the variational problem on the right hand side1. A simple perturbation calculation
then yields the Euler-Lagrange equation −ρ Ag = g2p−1 , where 1/ρ denotes the minimum. Together
with −A ◦ G =id, this implies that, for some u with Au = 0, we have ρ g = Gg2p−1 + u. Note that u
is vanishing at infinity and hence, assuming that A allows a maximum principle, we infer that u = 0.
Therefore,
1 1 ̺
= g, −Ag = 2 Gg2p−1 , g2p−1 ≤ 2 .
ρ
ρ
ρ
Hence we obtain that the right hand side in (5) equals
2
√
−p inf −Ag2 : kgk2p = 1 .
In this form, the constant has a natural interpretation: At least heuristically, the optimal strategy for
each random walk is to build up a local time field
Z ∞
(j)
1l{X (j) (t) = x} dt ≈ a1/p g2 (x)
for all j ∈ {1, . . . , p},
ℓ (x) :=
0
which implies
I˜ =
p
XY
x∈Zd j=1
ℓ(j) (x) ≈
p
X Y
a1/p g2 (x) = a .
x∈Zd j=1
1This assumption is generally nontrivial to verify, even for explicitly given A.
6
XIA CHEN AND PETER MÖRTERS
The probability of a random walk achieving such a local time is
√
2 ≈ exp − a1/p −Ag2 ,
which is reminiscent of the rate functions in Donsker-Varadhan theory.
Remark 4. The proof of Theorem 3 follows a similar strategy as in the discrete time case but,
thanks to the absence of some combinatorial difficulties arising from the discrete time structure, it is
considerably easier. Following the arguments analogous to Theorem 2 gives
1
lim 1/p log P I˜ > a = −p inf khkq : h ≥ 0 with kBh k ≥ 1 ,
a↑∞ a
where the operator Bh : L2 (Zd ) → L2 (Zd ) is defined by
X
p
p
G(x − y)g(y) h(y) .
Bh g(x) = h(x)
y∈Zd
Then one can observe that
√ √
inf khkq : h ≥ 0, sup hg, Bh gi ≥ 1 = inf b : sup
hg, G hg ≥ 1/b
kgk2 =1
khkq =1
kgk2 =1
= 1/ sup hf 2p−1 , Gf 2p−1 i : kf k2p = 1 .
2. Random walks in discrete time: Proof of Theorem 2.
In Section 2.1 we provide the proof of Lemma 1. The proof of Theorem 2 is divided into six steps,
presented in Sections 2.2 to 2.7 below. We now give a brief overview over these steps and the main
techniques of the proof.
By Lemma 2.3 in [KM02] we have, for any nonnegative random variable X,
h Xk i
1
1
= −κ ⇐⇒ lim 1/p log P{X > a} = −peκ/p .
lim log E
a↑∞ a
k↑∞ k
(k!)p
(6)
Moreover, it is elementary to show that
∞
X
θk
k=1
k!
1
E X k p = ∞ for θ > eκ/p
⇐⇒
lim sup
k↑∞
h Xk i
1
log E
≥ −κ .
k
(k!)p
Hence the lower bound in Theorem 2 is proved once we show that
Ik (i) the limit κ := − lim k1 log E (k!)
exists,
p
k↑∞
(ii) if there exists h ≥ 0 with khkq = θ such that kAh k > 1 then
P∞
θk
k=1 k!
1
E I k p = ∞.
The proof of (i) is accomplished in Section 2.2 using a subadditivity argument. As a by-product of
this proof we see that for the upper tail behaviour of I it is irrelevant whether we start summation in
the definition of I at time zero or one. The proof of (ii) is based on the precise formula (11) for the
integer moments, which we derive in Section 2.3. Here the peculiarities of the discrete time structure
can be quickly explained: It is easy to see that
E Ik =
X
x1 ,...,xk ∈Zd
X
i1 ,...,ik
E
k
Y
ℓ=1
p
1l{X(iℓ ) = xℓ } ,
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
7
where X is a random walk with the same law as X (1) , . . . , X (p) . To evaluate the expectation we need
to keep track of indices with ik = iℓ with k < ℓ as
1l{X(ik ) = xk }
if xk = xℓ ,
1l X(ik ) = xk , X(iℓ ) = xℓ =
0
if xk 6= xℓ .
This ‘diagonal effect’ cannot be neglected and we need to keep track of the number of equal indices
in (i1 , . . . , ik ). This leads to a significantly more complicated combinatorial structure of the moment
formula and, ultimately, to the difference between the tail behaviour in the discrete and continuous
time case. Having formulated a moment formula which keeps track of this effect, in Section 2.4,
we insert it in the exponential series in (ii) and verify divergence of the series using first Hölder’s
inequality and then a spectral argument. This completes the proof of the lower bound in Theorem 2.
The proof of the upper bound is more delicate. We again use (i) and the formula (11) to reduce the
problem to studying the large moment asymptotics of an explicit analytical quantity. However the
use of spectral arguments, which was so successful for the lower bound, now seems to be confined to
studying intersection local times in finite sets. Using similar ideas we show in Section 2.5 that, for
any finite set A ⊂ Zd ,
lim sup
k↑∞
1
1
log E[I(A)k ] ≤ −p log inf khkq : h ≥ 0, supp h ⊂ A with kAh k ≥ 1 .
k
k!
As indicated in Remark 1(c), the extension of the result from finite sets A to the entire lattice is
highly nontrivial, because the problem is not exponentially tight. To overcome this difficulty, we need
to project the full problem onto a finite domain by wrapping it around a torus. More precisely, in
Section 2.6 we let A = [−N, N )d for a large integer N and show that
lim sup
k↑∞
1
1
e h,N k ≥ 1 ,
log E[I k ] ≤ −p log inf khkq : h ≥ 0, supp h ⊂ A with kA
k
k!
e h,N the kernel G has been replaced by G
e N defined by
where in the definition of the operator A
e N (y) =
G
nX
Gp 2N z + y
z∈Zd
o p1
for any y ∈ Zd .
In Section 2.7 we let the period of the torus go to infinity and prove that
e h,N k ≥ 1 ≥ inf khkq : h ≥ 0 with kAh k ≥ 1 ,
lim sup inf khkq : h ≥ 0, supp h ⊂ A with kA
N →∞
which leads to the required upper bound.
2.1 Proof of Lemma 1.
We start by showing that G ∈ Lp (Zd ). Indeed, by [Uc98, (1.4)], we have (using only finite variance of
the increments and the fact that the walk is aperiodic),
G(z) ≤
X
x∈Zd
π(x)
1 + |x − z|d−2
for all z ∈ Zd ,
where (π(x) : x ∈ Zd ) is a summable family of nonnegative weights. Hence, as p(d − 2) > d,
X
z∈Zd
1
XX
p
≤
Gp (z)
x∈Zd
z∈Zd
X
X
1
1
π p (x)
1
p
p
=
< ∞.
π(x)
d−2 )p
(1 + |x − z|d−2 )p
(1
+
|z|
d
d
x∈Z
z∈Z
8
XIA CHEN AND PETER MÖRTERS
To prove that the operator Ah is bounded note that, whenever kg1 k2 = kg2 k2 = 1 and g1 , g2 ≥ 0, by
Hölder’s inequality,
p
p
X
hg1 , Ah g2 i =
g1 (x) eh(x) − 1 G(x − y) eh(y) − 1 g2 (y)
x,y∈Zd
≤
X
x,y∈Zd
×
p
p
p p1
Gp (x − y) g1 (x) eh(x) − 1 g2 (y) eh(y) − 1 2p−1
X g1 (x)
x,y∈Zd
p
eh(x) − 1 g2 (y)
p
(7)
2p 1q
eh(y) − 1 2p−1 .
Using again Hölder’s inequality, the second factor on the right hand side of (7) is bounded by
X
p
p
2p q1
2p q1 X g1 (x) eh(x) − 1 2p−1
g2 (x) eh(x) − 1 2p−1
x∈Zd
x∈Zd
≤
X
g12 (x)
x∈Zd
1
q+1
X
g22 (x)
x∈Zd
1
q+1
n X
x∈Zd
h(x)
e
q o q1
−1
2
q+1
whence we use that kg1 k2 = kg2 k2 = 1 and
∞
nX
X
q o 1q
khkkq n X h(x) kq o 1q
h(x)
e
−1
≤
≤ ekhkq − 1 .
k!
khk
q
d
d
k=1
x∈Z
x∈Z
By a shift of coordinates the first factor on the right hand side of (7) equals
X
p
p
X
p p1
,
g1 (x + y) eh(x+y) − 1 g2 (y) eh(y) − 1 2p−1
Gp (x)
x∈Zd
y∈Zd
and, using the Cauchy-Schwarz inequality, a further shift and Hölder’s inequality,
p
p
X
p
g1 (x + y) eh(x+y) − 1 g2 (y) eh(y) − 1 2p−1
y∈Zd
≤
X
p
p
2p 21 X 2p 21
g1 (x + y) eh(x+y) − 1 2p−1
g2 (y) eh(y) − 1 2p−1
≤
X
y∈Zd
y∈Zd
y∈Zd
g12 (y)
p
4p−2
X
g22 (y)
y∈Zd
p
4p−2
p−1
X
q 2p−1
q
h(y)
e
−1
≤ ekhkq − 1 q+1 .
y∈Zd
Combining all these estimates we obtain that
1
X
p p
hg1 , Ah g2 i ≤
G(y)
ekhkq − 1 ,
y∈Zd
which implies that Ah is bounded and its spectral radius satisfies (2).
2.2 Existence of the high moment limit of intersection local time
We define the ‘augmented’ intersection local time I0 by
∞
X
1l{X (1) (i1 ) = · · · = X (p) (ip )} .
I0 :=
i1 ,...,ip =0
1
E[I0k ]
log
exists.
k→∞ k
(k!)p
Lemma 4. The limit κ0 := − lim
,
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
9
Proof. By the subadditivity lemma it suffices to show that, for any k, l ≥ 1,
k+l 1/p
1/p l 1/p
k+l
E I0
≤
E I0k
E I0
.
k
(8)
By X we
a random walk with the same law as X (1) , . . . , X (p) and define its local time in x by
Pdenote
∞
ℓ(x) := j=0 1l{X(j) = x}. Then
E I0k+l =
X
p
Y
k+l
ℓ(xj )
E
X
x1 ,...,xk+l ∈Zd
=
x1 ,...,xk+l
∈Zd
j=1
∞
X
=
∞
X
k+l
Y
E
j=1
i1 ,...,ik+l =0
X
i1 ,...,ik+l =0 x1 ,...,xk+l ∈Zd
p−1
Y
k+l
ℓ(xj )
1l{X(ij ) = xj }
E
j=1
p−1
Y
Y
k+l
k+l
.
ℓ(xj )
1l{X(ij ) = xj }
E
E
j=1
j=1
For any two vectors (j1 , . . . , jk ) and (jk+1 , . . . , jk+l ) we write
(j1 , . . . , jk ) ≺ (jk+1 , . . . , jk+l )
⇐⇒
max{j1 , . . . , jk } ≤ min{jk+1 , . . . , jk+l } .
Instead of choosing the vector (i1 , . . . , ik+l ) directly, we may first pick k out of k + l coordinates, then
pick two vectors (j1 , . . . , jk ) ≺ (jk+1 , . . . , jk+l ) and fill the chosen k coordinates successively with the
entries of (j1 , . . . , jk ) and the remaining coordinates with the entries of (jk+1 , . . . , jk+l ). Using this
procedure and the permutation invariance we get
E I0k+l ≤
k+l
k
X
(i1 ,...,ik )
≺(ik+1 ,...,ik+l )
X
x1 ,...,xk+l
p−1
Y
Y
k+l
k+l
.
ℓ(xj )
1l{X(ij ) = xj }
E
E
j=1
j=1
Thus, using Hölder’s inequality in the second step,
E I0k+l ≤
≤
k+l
k
k+l
k
X
x1 ,...,xk+l
X
x1 ,...,xk+l
X
E
(i1 ,...,ik )
≺(ik+1 ,...,ik+l )
X
E
(i1 ,...,ik )
≺(ik+1 ,...,ik+l )
As the last factor equals (E[I0k+l ])
p−1
p
1p
k+l
E I0k+l
≤
k
k+l
Y
j=1
k+l
Y
j=1
p−1
Y
k+l
ℓ(xj )
1l{X(ij ) = xj }
E
j=1
p 1 p
1l{X(ij ) = xj }
X
x1 ,...,xk+l
p p−1
Y
k+l
p
ℓ(xj )
.
E
j=1
we get
X
x1 ,...,xk+l
∈Zd
X
E
(i1 ,...,ik )
≺(ik+1 ,...,ik+l )
k+l
Y
j=1
p 1
p
.
1l{X(ij ) = xj }
Writing i∗ = max{i1 , . . . , ik } we have
E
k+l
Y
j=1
Y
Y
k+l
k
1l{X(ij ) = xj } E
1l{X(ij − i∗ ) = xj − xi∗ } ,
1l{X(ij ) = xj } = E
j=1
j=k+1
(9)
10
XIA CHEN AND PETER MÖRTERS
and therefore
X
E
(i1 ,...,ik )
≺(ik+1 ,...,ik+l )
k+l
Y
j=1
=
1l{X(ij ) = xj }
Y
k
1l{X(ij ) = xj }
E
∞
X
Y
Y
k+l
k
ℓ(xj − xi∗ ) .
1l{X(ij ) = xj }
E
E
i1 ,...,ik =0
=
1l{X(ij ) = xj − xi∗ }
∞
X
i1 ,...,ik =0
j=1
j=1
∞
X
E
ik+1 ,...,ik+l =0
k+l
Y
j=k+1
j=k+1
Thus
X
x1 ,...,xk+l ∈Zd
x1 ,...,xk+l ∈Zd h=1
=
X
x1 ,...,xk ∈Zd
E
Y
p
k+l
Y
j=1
(i1 ,...,ik )
≺(ik+1 ,...,ik+l )
p Y
X
=
X
∞
X
h
ih
1 ,...,ik =0
p
1l{X(ij ) = xj }
Y
Y
k+l
k
h
1l{X(ij ) = xj }
E
ℓ(xj − xih∗ )
E
j=1
j=k+1
Y
k
h
1l{X(ij ) = xj }
E
∞
X
j=1
h
h=1 ih
1 ,...,i =0
k
X
p Y
E
xk+1 ,...,xk+l ∈Zd h=1
k+l
Y
j=k+1
ℓ(xj − xih∗ ) .
By Hölder’s inequality
X
p k+l
Y
Y
E
ℓ(xj − xih∗ )
xk+1 ,...,xk+l ∈Zd h=1
≤
j=k+1
p Y
h=1
X
xk+1 ,...,xk+l ∈Zd
Y
p 1/p
k+l
= E I0l ,
E
ℓ(xj − xih∗ )
j=k+1
and thus
X
x1 ,...,xk+l ∈Zd
X
E
j=1
(i1 ,...,ik )
≺(ik+1 ,...,ik+l )
≤ E I0l
k+l
Y
X
p
1l{X(ij ) = xj }
p
Y
∞
X
h
x1 ,...,xk ∈Zd h=1 ih
1 ,...,ik =0
Y
k
h
1l{X(ij ) = xj } = E I0l E I0k ,
E
j=1
which together with (9) proves (8) and thus completes the proof.
1
E[I k ]
log
exists and is equal to κ0 .
k→∞ k
(k!)p
Lemma 5. The limit κ := − lim
Proof. To make the passage from I0 to I first note that I ≤ I0 , and hence we have
1
1
E I k ≤ −κ0 .
lim sup log
p
(k!)
k→∞ k
Hence only the converse inequality needs to be proved. Define the local times
∞
X
ℓj (x) :=
1l{X (j) (k) = x}
for x ∈ Zd .
k=0
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
Then we obtain
I0 ≤ I +
p
p Y
X
j=1
11
ℓi (0) ,
i=1
i6=j
and thus, by the triangle inequality,
p
p Y
k i1/k
h X
k 1/k
k 1/k
ℓi (0)
.
E I0
≤E I
+E
j=1
(10)
i=1
i6=j
For the second term we get, using the multinomial theorem,
E
p
p Y
h X
j=1
i=1
i6=j
k i
ℓi (0)
=
X
k1 +···+kp =k
k1 ,...,kp ≥0
p
P
Y
k!
E ℓi (0) j6=i kj .
k1 ! · · · kp !
i=1
As X = ℓi (0) is a geometric random variable and hence E[eλX ] < ∞ for some λ > 0, we get
E ℓi (0)n ≤ n! E[eλX ] λ−n .
Hence, for a suitable constant C > 0,
E
p Y
p
h X
j=1
i=1
i6=j
k i
p
ℓi (0)
≤ E eλX λ−k(p−1)
X
k1 +···+kp =k
k1 ,...,kp ≥0
p
p
X
Y
k!
kj ! ≤ C k (k!)p−1 ,
k1 ! · · · kp !
j=1
i=1
j6=i
which implies
p
p Y
k i
h X
1
1
lim inf log
ℓ
(0)
= −∞ .
E
i
k→∞ k
(k!)p
i=1
j=1
i6=j
The result thus follows from (10).
2.3 The integer moments of intersection local time
We define, for any 1 ≤ m ≤ k, the family of all m-partitions of the set {1, . . . , k} as
S
Em := π = (π1 , . . . , πm ) : πj 6= ∅ with πi ∩ πj = ∅ for all i 6= j and m
j=1 πj = {1, . . . , k} ,
where we assume that the elements π1 , . . . , πm of an m-partition are ordered by increasing order of
their minimal elements. To any (i1 , . . . , ik ) ∈ Nk we associate
• a tuple (i∗1 , . . . , i∗m ) of distinct natural numbers such that {i1 , . . . , ik } = {i∗1 , . . . , i∗m } and elements in the tuple (i∗1 , . . . , i∗m ) appear in the order in which they appear first in (i1 , . . . , ik );
• an m-partition (π1 , . . . , πm ) ∈ Em with ij = i∗ℓ whenever j ∈ πℓ .
Conversely, given a tuple (j1 , . . . , jm ) of distinct natural numbers and an m-partition π we can find
(i1 , . . . , ik ) ∈ Nk such that the induced m-tuple is (j1 , . . . , jm ) and the induced m-partition is π.
Define the family of k-tuples of points in Zd associated to some π ∈ Em by
A(π) := (x1 , . . . , xk ) ∈ (Zd )k : xi = xj for all i, j ∈ πℓ and ℓ ∈ {1, . . . , m} .
For any (x1 , . . . , xk ) ∈ A(π) and for any 1 ≤ ℓ ≤ m, we use xπℓ for the common value of {xj : j ∈ πℓ }.
Observing that
X(iℓ ) = xℓ for all ℓ ∈ {1, . . . , k}
⇔
(x1 , . . . , xk ) ∈ A(π) and X(i∗ℓ ) = xπℓ for all ℓ ∈ {1, . . . , m} ,
12
XIA CHEN AND PETER MÖRTERS
we get that
E Ik =
X
X
X
k
X
X
x1 ,...,xk ∈Zd
=
x1 ,...,xk ∈Zd
E
i1 ,...,ik
k
Y
ℓ=1
m=1 π∈Em
1l{X(iℓ ) = xℓ }
p
1l{(x1 , . . . , xk ) ∈ A(π)}
X
E
j1 ,...,jm
distinct
m
Y
ℓ=1
p
1l{X(jℓ ) = xπℓ } ,
and therefore, using Sm to denote the symmetric group on m elements,
E Ik =
X
x1 ,...,xk ∈Zd
X
k
X
m=1 π∈Em
1l{(x1 , . . . , xk ) ∈ A(π)}
m
X Y
σ∈Sm ℓ=1
G xπσ(ℓ) − xπσ(ℓ−1)
p
,
(11)
where we use the convention that xπσ(0) := 0.
2.4 The lower bound in Theorem 2
Based on (11) we now complete the proof of the lower bound in Theorem 2. We will repeatedly make
use of the following two simple facts:
• For any π ∈ Em , the map
Φ : A(π) → (Zd )m ,
(x1 , . . . , xk ) 7→ (xπ1 , . . . , xπm )
is one-to-one and onto, i.e. a bijection.
• For any j1 , . . . , jm ≥ 1 with j1 + · · · + jm = k, we have
k!
1
.
# π = (π1 , · · · , πm ) ∈ Em : #(πℓ ) = jℓ for all ℓ ∈ {1, . . . , m} =
m! j1 ! · · · jm !
Recall from our outline that, given Lemma 5, in order to verify the lower bound in Theorem 2 it
suffices to establish the following lemma.
Lemma 6. If there exists h ≥ 0 with khkq ≤ θ such that kAh k > 1, then
∞
X
θk
k=1
k!
1
E I k p = ∞.
Proof. Let h ≥ 0 with khkq ≤ θ. Then, using Hölder’s inequality in the first step,
1
θk E I k p ≥
=
X
x1 ,...,xk ∈Zd
k
Y
j=1
k
X
X X
m
k
X
X Y
X
G(xπσ(ℓ) − xπσ(ℓ−1) )
1l{(x1 , . . . , xk ) ∈ A(π)}
h(xj )
m=1 π∈Em
X
Y
m
X
Y
m
m=1 σ∈Sm π∈Em x1 ,...,xk ∈Zd
=
k
X
X X
m=1 σ∈Sm π∈Em x1 ,...,xm ∈Zd
σ∈Sm ℓ=1
h(xπℓ )#(πℓ )
ℓ=1
ℓ=1
Y
m
ℓ=1
#(πℓ )
h(xℓ )
Y
m
ℓ=1
G(xπσ(ℓ) − xπσ(ℓ−1) )
G(xσ(ℓ) − xσ(ℓ−1) ) ,
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
13
where we use the convention xσ(0) := 0, and the last step follows from the one-to-one correspondence
between A(π) and Zm . Therefore,
k
X
k 1
p
θ E I
m!
≥
m=1
=
m
X Y
X
k
x1 ,...,xm ∈Zd π∈Em
k
X
X
h(xℓ )
ℓ=1
X
m=1 x1 ,...,xm ∈Zd
#(πℓ )
j1 +···+jm =k
j1 ,...,jm ≥1
Y
m
ℓ=1
k!
j1 ! · · · jm !
Y
m
G(xℓ − xℓ−1 )
jℓ
h(xℓ )
ℓ=1
Y
m
ℓ=1
G(xℓ − xℓ−1 ) ,
where we use the convention x0 := 0. Thus
Y
Y
m
m
∞ X
∞
∞
X
X
X X
1
θ k k 1p
jℓ
G(xℓ − xℓ−1 )
E I
h(xℓ )
≥
k!
j1 ! · · · jm !
j +···+jm =k
x ,...,x
m=1
k=m
k=1
=
=
∞
X
1
m
Y
m X
∞
m
X Y
h(xℓ )j
m=1 x1 ,...,xm
ℓ=1 j=1
m
∞
X X Y
h(xℓ )
e
m=1 x1 ,...,xm ℓ=1
j!
ℓ=1
ℓ=1
1
j1 ,...,jm ≥1
ℓ=1
G(xℓ − xℓ−1 )
(12)
− 1 G(xℓ − xℓ−1 ) .
Now assume that h additionally satisfies kAh k > 1. Fix ε > 0 such that kAh k > eε and g ≥ 0 with
kgk2 = 1 such that hg, Ah gi > eε . By monotone convergence we may find a finite set A ⊂ Zd such that
g1lA , Ah g1lA > eε and kg1lA k2 > e−ε ,
and additionally, but without loss of generality, g, h > 0 on A. We infer that there exists δ > 0 with
inf G(y) eh(x) − 1 ≥ δ g2 (x) 1lA (x)
for all x ∈ Zd .
y∈A
Hence
m
Y
X
x1 ,...,xm ∈Zd ℓ=1
eh(xℓ ) − 1 G(xℓ − xℓ−1 )
X
=
G(x1 )
x1 ,...,xm ∈Zd
≥δ
=δ
X
x1 ,...,xm ∈Zd
p
eh(x1 )
g(x1 )1lA (x1 )
(g1lA )
g1lA , Am−1
h
−1
m p
hY
ℓ=2
eh(xℓ−1 ) − 1 G(xℓ − xℓ−1 )
p
eh(xℓ ) − 1
ip
eh(xm ) − 1
m p
hY
ℓ=2
i
p
eh(xℓ−1 ) − 1 G(xℓ − xℓ−1 ) eh(xℓ ) − 1 g(xm )1lA (xm )
.
By the spectral theorem, see for example [He82, 79.1], for any g̃ ∈ L2 (Zd ) with kg̃k2 = 1 there exists
a probability measure µg̃ such that, for every k ≥ 0,
Z ∞
k
θ k µg̃ (dθ) ,
g̃, Ah g̃ =
−∞
and consequently, at least for even k,
Z ∞
Z
θ k µg̃ (dθ) ≥
g̃, Akh g̃ =
−∞
∞
−∞
k
θ µg̃ (dθ) = hg̃, Ah g̃ik .
14
XIA CHEN AND PETER MÖRTERS
We infer that
lim inf
m↑∞
m odd
1
m
log
X
m
Y
x1 ,...,xm ∈Zd ℓ=1
eh(xℓ ) − 1 G(xℓ − xℓ−1 )
≥ log g1lA , Ah (g1lA ) + log kg1lA k2 > 0 .
Combining this with (12) we see that the series
P θk
k!
1
E I k p diverges, as claimed.
2.5 The upper bound for intersections in a finite set
We now assume that A ⊂ Zd is a finite set and consider the intersection local time restricted to the
set A, which is defined as I(A) in Remark 1(b). The arguments of Section 2.3 show that
p
m
k
X Y
X
X X
k
G(xπσ(ℓ) − xπσ(ℓ−1) ) .
1l{(x1 , . . . , xk ) ∈ A(π)}
E[I(A) ] =
x1 ,...,xk ∈A
m=1 π∈Em
σ∈Sm ℓ=1
We now provide an upper bound for the asymptotics of this expression as k ↑ ∞, replacing (for later
reference) Green’s function wherever it occurs by an arbitrary symmetric kernel.
Lemma 7. Suppose G : Zd → (0, ∞) is symmetric. Then
p
m
k
X Y
X
X X
1
1
1l{(x1 , . . . , xk ) ∈ A(π)}
G(xπσ(ℓ) − xπσ(ℓ−1) )
lim sup log
k!
k↑∞ k
σ∈Sm ℓ=1
x1 ,...,xk ∈A m=1 π∈Em
≤ −p log inf khkq : h ≥ 0, supp h ⊂ A with kAh k ≥ 1 ,
where the operator Ah is associated with the kernel G.
Remark 5. With this lemma we have completed the proof of Remark 1(b). Indeed, the lower bound
for the asymptotics of the intersection local times I(A) in a finite set A follows from the arguments in
Sections 2.1 and 2.3 simply replacing Zd by A. For the upper bound use Lemma 7 with G = G. Proof. Fix a vector x = (x1 , . . . , xk ) of length k with entries from the set A and associate its empirical
measure Lxk by letting
Lxk :=
k
1 X
δxj .
k
j=1
For each τ ∈ Sk and π ∈ Em we denote by τ (π) ∈ Em the partition consisting of the sets τ (π)ℓ := τ (πℓ )
for ℓ ∈ {1 . . . , m}. Then, for any τ ∈ Sk and x = (x1 , . . . , xk ), we get
k
X
X
m=1 π∈Em
=
1l{(x1 , . . . , xk ) ∈ A(π)}
X
y1 ,...,yk ∈A
=
X
y1 ,...,yk ∈A
1l{x = y ◦ τ }
1l{x = y ◦ τ }
m
X Y
σ∈Sm ℓ=1
k
X
X
m=1 π∈Em
k
X
X
m=1 π∈Em
G(xπσ(ℓ) − xπσ(ℓ−1) )
1l{(y1 , . . . , yk ) ∈ A(τ (π))}
1l{(y1 , . . . , yk ) ∈ A(π)}
where we use the conventions yτ (πσ(0) ) := 0, yπσ(0) := 0.
m
X Y
σ∈Sm ℓ=1
m
X Y
σ∈Sm ℓ=1
G(yτ (πσ(ℓ) ) − yτ (πσ(ℓ−1) ) )
G(yπσ(ℓ) − yπσ(ℓ−1) ) ,
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
15
Observe that, abbreviating µ = Lxk and assuming Lyk = Lxk , we have
X
τ ∈Sk
1l{x = y ◦ τ } =
Y
kµ(x) ! ,
x∈A
and hence summing the previous expression over all permutations τ ∈ Sk gives
k!
k
X
X
m=1 π∈Em
=
Y
x∈A
1l{(x1 , . . . , xk ) ∈ A(π)}
m
X Y
σ∈Sm ℓ=1
G(xπσ(ℓ) − xπσ(ℓ−1) )
m
k
X Y
X
X
X
1l{(y1 , . . . , yk ) ∈ A(π)}
kµ(x) !
G(yπσ(ℓ) − yπσ(ℓ−1) ) .
1l{Lyk = µ}
y1 ,...,yk
m=1 π∈Em
σ∈Sm ℓ=1
Write φµ (x) = µ(x)1/q for all x ∈ A and note that
X
1l{Lyk
k
X
X
= µ}
y1 ,...,yk
1l{(y1 , . . . , yk ) ∈ A(π)}
m=1 π∈Em
≤
Y
−kµ(x)
φµ (x)
y1 ,...,yk
x∈A
×
X
m
X Y
σ∈Sm ℓ=1
m
X Y
σ∈Sm ℓ=1
φµ (y1 ) · · · φµ (yk )
G(yπσ(ℓ) − yπσ(ℓ−1) )
k
X
X
m=1 π∈Em
1l{(y1 , . . . , yk ) ∈ A(π)}
G(yπσ(ℓ) − yπσ(ℓ−1) ) .
Simplifying and finally using the expression for #Em , we obtain
X
y1 ,...,yk
φµ (y1 ) · · · φµ (yk )
=
k
X
X
k
X
X
m=1 π∈Em
X
m=1 π∈Em y1 ,...,yk
=
1l{(y1 , . . . , yk ) ∈ A(π)}
ℓ=1
m
X Y
m=1 π∈Em y1 ,...,ym
=
m!
m=1
=
σ∈Sm ℓ=1
G(yπσ(ℓ) − yπσ(ℓ−1) )
Y
X Y
m
m
#(πℓ )
1l{(y1 , . . . , yk ) ∈ A(π)}
φµ (yπℓ )
G(yπσ(ℓ) − yπσ(ℓ−1) )
k
X
X
k
X
m
X Y
#(πℓ )
φµ (yℓ )
k
X
X m=1 y1 ,...,ym
π∈Em ℓ=1
X
j1 +···+jm =k
j1 ,...,jm ≥1
X Y
m
σ∈Sm ℓ=1
ℓ=1
m
X X Y
y1 ,...,ym
σ∈Sm ℓ=1
#(πℓ )
φµ (yℓ )
Y
m
ℓ=1
m
G(yσ(ℓ) − yσ(ℓ−1) )
G(yℓ − yℓ−1 )
Y
k!
φµ (yℓ )jℓ
j1 ! · · · jm !
ℓ=1
Y
m
ℓ=1
G(yℓ − yℓ−1 ).
Further, using Stirling’s formula, we obtain a fixed polynomial P (·), depending only on the cardinality
of A, such that
Y
x∈A
Y
Y
k
1
−kµ(x)
µ(xj ) p .
φµ (x)
≤ P (k) k!
kµ(x) !
x∈A
j=1
16
XIA CHEN AND PETER MÖRTERS
Summarizing all what we have got so far
k
X
X
m=1 π∈Em
1l{(x1 , . . . , xk ) ∈ A(π)}
≤ P (k) k!
Y
k
µ(xj )
1
p
m
X Y
σ∈Sm ℓ=1
X
k
G(xπσ(ℓ) − xπσ(ℓ−1) )
X
m=1 y1 ,...,ym ∈A
j=1
X
j1 +···+jm =k
j1 ,...,jm ≥1
Y
m
m
Y
1
jℓ
G(yℓ − yℓ−1 ) .
φµ (yℓ )
j1 ! · · · jm !
ℓ=1
ℓ=1
Summing over all possible vectors x = (x1 , . . . , xk ) gives
p
m
k
X Y
X
X X
1l{(x1 , . . . , xk ) ∈ A(π)}
Ik :=
G(xπσ(ℓ) − xπσ(ℓ−1) )
m=1 π∈Em
x1 ,...,xk ∈A
p
p
≤ P (k) (k!)
sup
µ∈Pk (A)
σ∈Sm ℓ=1
X
k
X
m=1 y1 ,...,ym ∈A
Y
p
m
m
Y
1
jℓ
φµ (yℓ )
G(yℓ − yℓ−1 ) ,
j1 ! · · · jm !
X
ℓ=1
j1 +···+jm =k
j1 ,...,jm ≥1
ℓ=1
where Pk (A) stands for the set of probability densities ν on A such that ν(x) is of the form i/k. Recall
that φµ ≥ 0 satisfies kφµ kq = 1 and hence we may replace the supremum by one over all Lq -normalised
nonnegative functions on A. Thus, for every θ > 0,
Y
m
m
∞
n
Y
X
X
X
X 1
θk
1
jℓ
n
p
Ik ≤ P (k) sup
f (yℓ )
G(yℓ − yℓ−1 )
θ
k!
j1 ! · · · jm !
kf kq =1 n=1
j +···+jm =n
m=1
y1 ,...,ym ∈A
= P (k) sup
∞
X
X
Y
m X
∞
h(yℓ )j
j!
khkq =θ m=1 y ,...,y ∈A ℓ=1 j=1
m
1
m ∞
X
X Y
h(yℓ )
e
= P (k) sup
khkq =θ m=1 y ,...,y ∈A
m
1
Let g(x) =
ℓ=1
1
j1 ,...,jm ≥1
ℓ=1
Y
m
G(yℓ − yℓ−1 )
ℓ=1
m
Y
−1
ℓ=1
ℓ=1
G(yℓ − yℓ−1 ).
[#A]−1/2
X
y1 ,...,ym ∈A
for all x ∈ A and note that kgk2 = 1. Using that 0 ≤ h(x) ≤ θ we obtain
Y
m m
Y
h(yℓ )
e
−1
G(yℓ − yℓ−1 )
ℓ=1
≤
=
≤
ℓ=1
sup G(y) eθ − 1
y∈A
X
m p
Y
y1 ,...,ym ∈A ℓ=2
eh(yℓ−1 )
− 1 G(yℓ − yℓ−1 )
sup G(y) eθ − 1 #A g, Am−1
g
h
p
eh(yℓ )
−1
y∈A
sup G(y) eθ − 1 #A kAh km−1 .
y∈A
Suppose that θ < inf{khkq : supp h ⊂ A, kAh k ≥ 1}, then there exists ǫ > 0 such that every h with
khkq = θ satisfies kAh k < 1 − ǫ. We infer that
∞
X
1
θ
θk
p
Ik ≤ P (k) sup G(y) e − 1 #A sup
kAh km−1 .
k!
y∈A
khkq =θ
m=1
As the suprema on the right hand side are both finite, we obtain
Ik
1
≤ −p log θ ,
lim sup log
(k!)p
k→∞ k
as required to complete the proof.
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
17
2.6 Wrapping the random walks around the torus
We now look back to the full lattice Zd and discuss how to project the problem onto a finite set. We fix
a large integer N an let AN := [−N, N )d . Then every x ∈ Zd can uniquely be written as x = 2N z + y
for z ∈ Zd and y ∈ AN . Hence, using Hölder’s inequality in the second step,
k
X X
X
X
k
1l{(2N z1 + y1 , . . . , 2N zk + yk ) ∈ A(π)}
E I =
y1 ,...,yk ∈AN z1 ,...,zk ∈Zd
m
X Y
×
≤
− zπσ(ℓ−1) ) + (yπσ(ℓ) − yπσ(ℓ−1) )
G 2N (zπσ(ℓ)
σ∈Sm ℓ=1
X
y1 ,...,yk ∈AN
×
m=1 π∈Em
X
k
X X m=1 π∈Em σ∈Sm
m
Y
p
G 2N (zπσ(ℓ)
ℓ=1
X
z1 ,...,zk
∈Zd
p
1l{(2N z1 + y1 , . . . , 2N zk + yk ) ∈ A(π)}
− zπσ(ℓ−1) ) + (yπσ(ℓ) − yπσ(ℓ−1) )
1/p p
.
From the uniqueness of the decomposition x = 2N z + y we infer that
1l{(2N z1 + y1 , . . . , 2N zk + yk ) ∈ A(π)} = 1l{(y1 , . . . , yk ) ∈ A(π)} 1l{(z1 , . . . , zk ) ∈ A(π)}.
Therefore,
X
z1 ,...,zk ∈Zd
1l{(2N z1 + y1 , . . . , 2N zk + yk ) ∈ A(π)}
= 1l{(y1 , . . . , yk ) ∈ A(π)}
×
m
Y
ℓ=1
z1 ,...,zk ∈Zd
ℓ=1
Gp 2N (zπσ(ℓ) − zπσ(ℓ−1) ) + (yπσ(ℓ) − yπσ(ℓ−1) )
1l{(z1 , . . . , zk ) ∈ A(π)}
Gp 2N (zπσ(ℓ) − zπσ(ℓ−1) ) + (yπσ(ℓ) − yπσ(ℓ−1) )
X
= 1l{(y1 , . . . , yk ) ∈ A(π)}
= 1l{(y1 , . . . , yk ) ∈
where we define
X
m
Y
m
Y
Gp 2N (zσ(ℓ) − zσ(ℓ−1) ) + (yπσ(ℓ) − yπσ(ℓ−1) )
z1 ,...,zm ∈Zd ℓ=1
m
Y
e p (yπ
A(π)}
G
− yπσ(ℓ−1) ) ,
σ(ℓ)
N
ℓ=1
e N (y) =
G
nX
Gp 2N z + y
z∈Zd
o p1
for any y ∈ Zd .
Summarizing, we have shown that
p
X
m
k
X Y
X
X
k
e
1l{(x1 , . . . , xk ) ∈ A(π)}
GN (xπσ(ℓ) − xπσ(ℓ−1) ) .
E I ≤
x1 ,...,xk ∈AN
m=1 π∈Em
σ∈Sm ℓ=1
We can now apply Lemma 7 and obtain
1
1
e h,N k ≥ 1 ,
lim sup log E[I k ] ≤ −p log inf khkq : h ≥ 0, supp h ⊂ AN with kA
k!
k↑∞ k
e h,N : L2 (Zd ) → L2 (Zd ) is defined by
where the self-adjoint operator A
p
p
X
e N (x − y) eh(y) − 1 g(y) .
e h,N g(x) = eh(x) − 1
G
A
y∈AN
18
XIA CHEN AND PETER MÖRTERS
2.7 The upper bound in Theorem 2
The proof of Theorem 2 is complete once we have proved the following lemma.
Lemma 8.
e h,N k ≥ 1 ≥ inf khkq : h ≥ 0 with kAh k ≥ 1 .
lim sup inf khkq : h ≥ 0 with kA
N →∞
Proof. Fix positive integers M < N and define the annulus
EN := x ∈ AN : |x| ≥ N − M .
We decompose G = G+ + G− with G+ (x) = G(x)1l{|x| > M } and G− (x) = G(x)1l{|x| ≤ M }. In
e N we define symmetric, periodic functions G
e + and G
e − by
analogy to G
e ± (y) =
G
nX
Gp± 2N z + y
z∈Zd
o 1p
for any y ∈ Zd .
eN ≤ G
e+ + G
e − . The induced operators A
e h,± : L2 (AN ) → L2 (AN )
By the triangle inequality we have G
are defined by
p
p
X
e ± (x − y) eh(y) − 1 g(y) .
e h,± g(x) = eh(x) − 1
G
A
y∈AN
For the spectral radius of these operators we thus obtain
e h,N k ≤ kA
e h,+ k + kA
e h,− k .
kA
Suppose now that h : AN → [0, ∞) is given and let h0 : Zd → [0, ∞) its extension to the whole of Zd
defined by letting h0 (x) = 0 for x ∈ Zd \ AN . We next show that
nX
o 1 h #E i p−1 n X p o p1
p
p
N
khkq
p
e
G (x)
kAh,N k ≤ kAh0 k + e
−1
+2
G (x)
.
(13)
d
(2N )
d
|x|>M
x∈Z
Using (13) we complete the proof by the following argument: For any t > 0 let
ϕ(t) := inf khkq : h ≥ 0 with kAh k ≥ t .
Note that, using that G ∈ Lp (Zd ), by choosing first a large M and then an even larger N , we can
make the expression in the curly bracket on the right hand side of (13) as small as we wish. Hence
e h,N k ≥ 1 ≥ ϕ(1 − ε),
lim sup inf khkq : h ≥ 0 with kA
N →∞
for any ε > 0. As
1
1−ε
θ
eθ − 1) ≤ exp{ 1−ε
} − 1 we obtain, from the definition of Ah that
kA
h
1−ε
k≥
1
1−ε
kAh k ,
whence ϕ(1 − ε) ≥ (1 − ε) ϕ(1), so that the statement of Lemma 8 follows by letting ε ↓ 0.
To prove (13) first fix g : AN → [0, ∞) with kgk2 = 1. For notational convenience we extend both g
and h periodically to the whole lattice Zd . We claim that there exists x0 ∈ AN such that
o 2p
p
X n
p #EN
2p−1
≤ ekhkq − 1 2p−1
.
g(x + x0 ) eh(x+x0 ) − 1
(2N )d
x∈EN
(14)
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
19
Indeed, this follows readily from
X
x∈EN
y∈AN
p X
X
p−1
p
2p−1
p 2p−1
2p
2
h(x+y)
h(x+y)
2p−1
p−1
g (x + y)
e
−1
−1
≤
g(x + y) e
x∈EN
y∈AN
≤ #EN
X
h(y)
e
y∈AN
p
− 1 p−1
p−1
2p−1
x∈EN
y∈AN
p
≤ #EN ekhkq − 1 2p−1 .
We write h̄(x) = h(x + x0 ) and ḡ(x) = g(x + x0 ). Then
p
p
p
p
X
X
e − (x − y) eh(y) − 1g(y) =
e − (x − y) eh̄(y) − 1 ḡ(y)
g(x) eh(x) − 1 G
ḡ(x) eh̄(x) − 1 G
x,y∈AN
≤
X
p
ḡ(x)
x,y∈AN \EN
+2
X
x,y∈AN
p
e − (x − y) eh̄(y) − 1 ḡ(y)
eh̄(x) − 1 G
ḡ(x)
x∈EN
y∈AN
p
p
e − (x − y) eh̄(y) − 1 ḡ(y) .
eh̄(x) − 1 G
e − (x − y) = G− (x − y) when x, y ∈ AN \ EN , we infer that
From the fact that G
p
p
X
e − (x − y) eh̄(y) − 1 ḡ(y) ≤ kAh k .
ḡ(x) eh̄(x) − 1 G
0
x,y∈AN \EN
Moreover, from Hölder’s inequality,
p
p
X
e − (x − y) eh̄(y) − 1 ḡ(y)
ḡ(x) eh̄(x) − 1 G
x∈EN
y∈AN
≤
X
x,y∈AN
×
p 1
q
2p−1
p
p
h̄(x)
h̄(y)
e
e
−1 e
− 1 ḡ(x)ḡ(y)
G− (x − y)
X q
eh̄(x)
x∈EN
y∈AN
2p p−1
2p−1
p
h̄(y)
− 1 ḡ(x)ḡ(y)
−1 e
.
To treat the first factor, note that, by periodicity,
p
q
X p
2p−1
h̄(y)
h̄(x)
e (x − y)
− 1 ḡ(x)ḡ(y)
−1 e
e
G
−
x,y∈AN
=
X
x∈AN
e p (x)
G
−
X q
y∈AN
eh̄(x+y)
p
2p−1
h̄(y)
.
− 1 ḡ(x + y)ḡ(y)
−1 e
Moreover, arguing as before using Hölder’s inequality,
p
X q
2p−1
h̄(x+y)
h̄(y)
e
−1 e
− 1 ḡ(x + y)ḡ(y)
y∈AN
≤
X ≤
X
y∈AN
y∈AN
2p 1 X 2p 1
p
p
2p−1
2
2p−1
2
h̄(x+y)
h̄(y)
−1
ḡ(y) e
−1
ḡ(x + y) e
y∈AN
p X
p−1
2p−1
p 2p−1
p
2
h(y)
p−1
g (y)
e
−1
≤ ekhkq − 1 2p−1 .
y∈AN
20
XIA CHEN AND PETER MÖRTERS
For the second factor we obtain, using (14),
2p
X q
2p−1
h̄(x)
h̄(y)
e
−1 e
− 1 ḡ(x)ḡ(y)
x∈EN
y∈AN
=
X x∈EN
2p X 2p p
p
2p−1
2p−1
2p #EN
h̄(x)
h̄(y)
.
−1
−1
ḡ(x) e
ḡ(y) e
≤ ekhkq − 1 2p−1
(2N )d
y∈AN
Summarizing, we have that
X
e h,− k = sup
kA
kgk2 =1 x,y∈A
g(x)
N
khkq
≤ kAh0 k + 2 e
p
e − (x − y)
eh(x) − 1 G
−1
p
eh(y) − 1 g(y)
1h
p−1
p
#EN i p
.
G (x)
(2N )d
d
X
p
x∈Z
In a similar fashion as above, or indeed as in Lemma 1, we also obtain that
1
X
1
X
p
p
p
p
khkq
e
e
G+ (y)
kAh,+ k ≤
G (x)
e
ekhkq − 1 .
−1 =
y∈AN
|x|>M
This proves (13) and hence completes the proof of the lemma.
3. Random walks in continuous time: Proof of Theorem 3
3.1 Tail behaviour of the intersection local time
In this section we prove that
˜ > a = −p ,
log
P
I
(15)
a↑∞ a1/p
̺
where ̺ is as in Remark 3. As this is essentially a simpler version of the proof of Theorem 2, we only
give a sketch, further details can be filled in easily by copying the methods of our main result. The
proof is carried out in four steps.
lim
1
In the first step we reduce the problem to an analytic problem. By (6) it suffices to prove
h I˜k i
1
lim log E
= p log ̺ .
k↑∞ k
(k!)p
Using the convention that xσ(0) := 0, we get
Z ∞
p Z ∞
k
ik
ip
h Z ∞
hXY
Y
X
(j)
k
˜
1l{X (t) = x} dt =
dtk
dt1 · · ·
1l{X(tℓ ) = xℓ }
E
EI = E
x∈Zd j=1
=
x1 ,...,xk ∈Zd
X
h X Z
X
k
h X Y
x1 ,...,xk ∈Zd
=
0
x1 ,...,xk ∈Zd
σ∈Sk
∞
dt1
∞
t1
0
σ∈Sk ℓ=1
Z
dt2 · · ·
Z
∞
dtk E
tk−1
k
Y
ℓ=1
0
0
ℓ=1
ip
1l{X(tℓ ) = xσ(ℓ) }
ip
G(xσ(ℓ−1) − xσ(ℓ) ) ,
and hence it suffices to prove that
1
1
lim log
k→∞ k
(k!)p
X
x1 ,...,xk ∈Zd
X Y
k
σ∈Sk ℓ=1
p
G(xσ(ℓ−1) − xσ(ℓ) ) = p log ρ .
(16)
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
21
In the second step we provide the lower bound for (16). Let q be the conjugate of p, and use Hölder’s
inequality to obtain, for any nonnegative h ∈ Lq (Zd ) with khkq = 1,
1
k!
X
x1 ,...,xk
∈Zd
X Y
k
σ∈Sk ℓ=1
p 1/p
≥
G(xσ(ℓ−1) − xσ(ℓ) )
X
x1 ,...,xk
∈Zd
k
Y
ℓ=1
h(xℓ ) G(xℓ − xℓ−1 ).
d
2p/q
p
For any nonnegative f ∈ L2p (Z
√ ) with kf k2p = 1 we define h := f d and g := f and note that
2p−1
kgk2 = khkq = 1 and f
= hg. If we also fix a finite set A ⊂ Z , we can find a constant δ > 0
(depending on f and A) such that h(x) ≥ δg2 (x) for all x ∈ Zd , and G(x) ≥ δ for all x ∈ A. Then
k
X Y
x1 ,...,xk ℓ=1
h(xℓ ) G(xℓ − xℓ−1 )
≥δ
2
=δ
2
X
g(x1 )1lA (x1 )
x1 ,...,xk
g1lA , Bhk−1 g1lA
k p
hY
ℓ=2
i
p
h(xℓ−1 ) G(xℓ − xℓ−1 ) h(xℓ ) g(xk )1lA (xk )
.
From the spectral theorem we infer that
lim inf
k↑∞
1
log g, Bhk−1 g ≥ loghg1lA , Bh g1lA i + log kg1lA k2
k
A↑Zd
−→ loghf 2p−1 , Gf 2p−1 i ,
and the lower bound in (16) follows as f was arbitrary from the positive unit ball of L2p (Zd ).
In the third step we show that, for any finite set A ⊂ Zd ,
p
k
X X Y
1
1
G(xσ(ℓ) − xσ(ℓ−1) )
lim sup log
(k!)p
k→∞ k
x1 ,...,xk ∈A σ∈Sk ℓ=1
2p−1
≤ p log sup hf
, Gf 2p−1 i : for f : Zd → [0, ∞) with supp h ⊂ A and kf k2p = 1 ,
(17)
where (for later reference) we replace Green’s function wherever it occurs by an arbitrary symmetric
kernel G : Zd → [0, ∞). The same tilting technique as in Lemma 7 gives, for a suitable polynomial P ,
µ = Lxk and φ(x) = µ(x)1/q ,
k
X Y
σ∈Sk ℓ=1
G(xσ(ℓ) − xσ(ℓ−1) ) ≤ P (k) k!
k
nY
µ(xj )
j=1
1
p
o
X
y1 ,...,yk ∈A
φ(y1 ) · · · φ(yk )
k
Y
ℓ=1
G(yℓ − yℓ−1 ) .
Therefore we obtain
p
k
X X Y
1
G(x
−
x
)
σ(ℓ)
σ(ℓ−1)
(k!)p
x1 ,...,xk ∈A
=
X
σ∈Sk ℓ=1
x1 ,...,xk ∈A
= eo(k)
X
exp k
µ(z) log µ(z) + o(k)
sup
sup
kφkq =1 y ,...,y ∈A
1
k
z∈A
X
kφkq =1 y ,...,y ∈A
1
k
X
φ(y1 ) · · · φ(yk )
k
Y
ℓ=1
p
G(yℓ − yℓ−1 ) .
φ(y1 ) · · · φ(yk )
k
Y
ℓ=1
p
G(yℓ − yℓ−1 )
22
XIA CHEN AND PETER MÖRTERS
Let g(x) := [#A]−1/2 1lA (x) so that kgk2 = 1. Using that φ(x) ≤ 1 for all x ∈ A, we obtain
X
y1 ,...,yk ∈A
φ(y1 ) · · · φ(yk )
k
Y
ℓ=1
G(yℓ − yℓ−1 ) ≤ [#A] g, Bφk−1 g ≤ [#A] kBφ kk−1 ,
from which the result readily follows.
In the fourth step we extend the result from finite sets to the full lattice Zd . Projection on a torus as
in Section 2.6 gives
p
X Y
p
X Y
k
k
X
X
e
GN (yσ(ℓ) − yσ(ℓ−1) ) ,
G(xσ(ℓ) − xσ(ℓ−1) ) ≤
x1 ,...,xk ∈Zd
y1 ,...,yk ∈AN
σ∈Sm ℓ=1
where
Using (17) for this kernel gives
e N (x) :=
G
hX
z∈Zd
σ∈Sk ℓ=1
i1/p
Gp (2N z + x)
.
X Y
p
k
X
1
1
lim sup log
G(xσ(ℓ) − xσ(ℓ−1) )
(k!)p
k→∞ k
x1 ,...,xk ∈Zd σ∈Sm ℓ=1
≤ p log sup hf 2p−1 , GN f 2p−1 i : for f : Zd → [0, ∞) with supp f ⊂ AN and kf k2p = 1 ,
(18)
e N . Exactly as in the proof of (13) one
where the operator GN stands for convolution with the kernel G
shows that, for 0 < M < N ,
sup hf 2p−1 , GN f 2p−1 i : for f : Zd → [0, ∞) with supp f ⊂ AN and kf k2p = 1
≤ sup hf 2p−1 , Gf 2p−1 i : for f : Zd → [0, ∞) with kf k2p = 1
h X
i1/p
h #E i p−1 h X
i1/p
p
N
p
+
Gp (x)
+2
G
(x)
,
(2N )d
d
|x|>M
x∈Z
where EN := {x ∈ AN : |x| > N − M }. The first term on the last line can be made arbitrarily small by
choice of M . For any M the second term converges to zero as N ↑ ∞. This shows that, as N ↑ ∞, the
variational problem on the right hand side of (18) converges to ̺, thus completing the proof of (15).
4. Conclusion
We believe that the method of this paper opens up new avenues for the treatment of a wide range
of intersection problems in supercritical dimensions, which are by no means limited to random walk.
The major advantage of our approach over the established Donsker-Varadhan theory is its direct
applicability to problems with an infinite time horizon. However, studying the upper tail behaviour
of the intersection of the ranges requires additional ideas and is subject of ongoing research.
Acknowledgements: We thank the Mathematisches Forschungsinstitut Oberwolfach for enabling
us to complete this work during a ‘Research in Pairs’ visit, and an anonymous referee for carefully
reading our manuscript. Further thanks are due to Nadia Sidorova and Alexander Frolkin for providing
numerical evidence allowing us to reject a conjecture that came up in the course of the work. The
first author was supported in part by NSF grant DMS-0704024 and the second author by grant
EP/C500299/1 and an Advanced Research Fellowship of EPSRC.
UPPER TAILS FOR INTERSECTIONS OF RANDOM WALK
23
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[Uc98]
Xia Chen
Department of Mathematics
University of Tennessee
Knoxville, TN 37996-1300
USA
[email protected]
Peter Mörters
Department of Mathematical Sciences
University of Bath
Bath BA2 7AY
United Kingdom
[email protected]