A local CLT for convolution equations with an application to weakly self-avoiding random walks

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by  Luca Avena

A local CLT for convolution equations with an application
to weakly self-avoiding random walks
∗
Luca Avena† Erwin Bolthausen‡ Christine Ritzmann
University of Zu¨rich
3
1
0 Abstract
2
We prove error bounds in a central limit theorem for solutions of certain convo-

n lution equations. The mainmotivationfor investigatingthese equationsstems from
a
applications to lace expansions, in particular to weakly self-avoiding random walks
J
inhighdimensions. Asanapplicationwetreatsuchself-avoidingwalksincontinuous
5
space. The bounds obtained are sharper than the ones obtained by other methods.
2

]
R 1 Introduction
P
. Let φ be the standard normal density in Rd, B = B be a sequence of rotationally
h { k}k≥1
t invariant integrable functions, and λ > 0 a (small) parameter. Define recursively
a
m
C = δ , (1.1)
0 0
[
n

1 C = C φ+λ c B C , n 1,
n n−1 k k n−k
v ∗ ∗ ≥
k=1
1 X
7 where
0 def
c = C (x)dx.
6 n n
. Z
1
δ denotestheDirac“function”. Allour(signed)distributionswillhavedensities,except
0 0
3 those at index 0.
1 As written above, the sequence C = C is not quite recursively defined as the
: { n}n≥0
v right hand side contains the summand c B . The sequence c itself satisfies
n n n
i { }
X
c = 1, (1.2)
0
r
a n
c = c +λ c b c , n 1,
n n−1 k k n−k
≥
k=1
X
∗AMS2000classification: 60K35, 60F05. Keywordsandphrases: centrallimit theorem,convolution
equations, self-avoiding random walks.
†supported by the Swiss National Science Foundation under contract 138141, and by the
Forschungskredit of theUniversity of Zu¨rich.
‡supported by the Swiss National Science Foundation under contract 138141, and by the Humboldt
Foundation.
1
where b = B (x)dx. Therefore, if λ b < 1 for all n, these equations define the
k k n
| |
sequence c uniquely, and then, also C is well-defined. We will always assume that
n
{ R}
we are in this situation.
The main assumption is a decay property of the B for large n. We will also assume
n
Gaussiandecay propertiesinspacewhicharenaturalfortheapplications toself-avoiding
walks. The method we present here can probably be adapted to treat situations with
less severe decay assumptions in space, but we have not worked that out.
Our main interest is to prove a local central limit theorem for C /c under appro-
n n
priate conditions on B and λ. Of course, the parameter λ can be incorporated into B.
However, the approach we follow is purely perturbative. We will give conditions on B,
and then state that if in addition λ is small enough a CLT holds.
We use φ as the convoluting factor in (1.1) with C , so that for λ = 0, C simply
n−1 n
is the normal density with covariance matrix n identity. At the expense of a few
×
complications, we could also investigate the case where the first summand in (1.1) is
C S with a rotationally invariant density S.We however feel that this generalization
n−1
∗
wouldsomehowobscurethemainlineoftheargument. Tostepoutfromtherotationally
invariant case however leads to new, complicated, and interesting problems which will
be presented elsewhere.
The main motivation for our investigation of the equations comes from weakly self-
avoiding random walks, as first investigated by Brydges and Spencer in the seminal
paper [2]. Their results are for random walks on the d-dimensional lattice Zd, d 5. In
≥
contrast, we investigate weakly self-avoiding random walks on Rd with standard normal
increments. Themodelhastwoparametersλ,ρ > 0,ρbeingtherangeoftheinteraction,
and λ the strength. We set I (x) d=ef 1 , and if x = (x ,...,x ) Rd n, and
ρ {|x|≤ρ} 1 n
∈
0 ≤ i < j ≤ n, we set Uiρj(x) d=ef Iρ(xj −xi), where x0 = 0. Then, for 0(cid:0) ≤(cid:1)λ ≤ 1,
define the probability measure P on Rd n by its density with respect to Lebesgue
n,λ,ρ
measure:
1 (cid:0) (cid:1)
p (x) = K [0,n](x)Φ[0,n](x),
n,λ,ρ λ,ρ
Z
n,λ,ρ
where
K [a,b](x) d=ef 1 λUρ (x) , (1.3)
λ,ρ − ij
a≤Yi<j≤b(cid:16) (cid:17)
b
def
Φ[a,b](x) = φ(x x ). (1.4)
i i−1
−
i=a+1
Y
Z is the usual partition function, i.e. the norming factor which makes p into
n,λ,ρ n,β,ρ
a probability density. The main interest is to prove a central limit theorem for this
measure, in the simplest case for the last marginal measure. It is convenient to consider
firsttheunnormalized kernel CSAW(x), x Rd,which is definedto bethe last marginal
n ∈
2
density of Z p (x), i.e.
n,β n,β,ρ
n−1
CSAW(x )= K [0,n](x)Φ[0,n](x) dx . (1.5)
n n λ,ρ i
Z i=1
Y
By a lace expansion, the CSAW satisfy an equation
n
n
CSAW =CSAW φ+ Π CSAW, (1.6)
n n−1 ∗ k ∗ n−k
k=1
X
where the kernels Π describe the interactions through the weak self-avoidance. The
k
Π are complicated functions and are hard to evaluate precisely. However, one crucial
k
property is that the leading order decay is the same as that of the CSAW. It therefore
k
looks natural to write Π = λcSAWB , and one seeks for condition on the B ensuring
k k k k
a CLT for solutions of (1.1). We can then apply the central limit theorem obtained
for (1.1), Theorem 2.2, provided we can check the conditions on the sequence B. The
theorem we obtain as a corollary of Theorem 2.2 is
Theorem 1.1
For d 5, ρ (0,1], and 0 < ε 1 , there exists λ (d,ε) > 0 such that for λ (0,λ ]
≥ ∈ ≤ 100 0 ∈ 0
there exist a parameter δ(d,ρ,λ) > 0 and a constant K(d,ε,λ) > 0 such that for all
n N
∈
⌈n/2⌉
C (x)
n φ (x) K r φ (x)+n−d/2 jφ (x)
nδ n nδ(1+ε) jδ(1+ε)
c − ≤  
(cid:12) n (cid:12) j=1
(cid:12) (cid:12) X
(cid:12) (cid:12)  
with (cid:12) (cid:12)
n−1/2 for d= 5
r = n−1logn for d= 6 . (1.7)
n

n−1 for d 7

≥
Remark 1.2 
a) The bound leads to C /c φ = O(r ).
k n n − nδk1 n
b) The theorem does not give a local CLT as at x = 0 both φ (0) and the bound
nδ
are of order n−d/2. A moments reflection however reveals that there cannot be a
local CLT as the starting point keeps to have a noticeable influence on C (x)/c
n n
for points x at distance of order 1 from the origin. However, our bound proves
C (x)
lim limsup sup nd/2 n φ (x) = 0.
nδ
r→∞ n→∞ x:|x|≥r (cid:12) cn − (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
So the result comes as close as possible to(cid:12) a local CLT. (cid:12)
3
c) The summation up to n/2 is somewhat arbitrary, and can be replaced by αn
⌈ ⌉ ⌈ ⌉
for any α (0,1), adapting K. In fact, for 0 < α< 1
∈
n
n−d/2 jφ (x) K(α)r φ (x)
jδ(1+ε) n nδ(1+ε)
≤
j=⌈αn⌉
X
We have chosen α= 1/2 for convenience. The second summand however is impor-
tant and it takes care of the failure of the local CLT for x near the origin.
Self-avoiding random walks in continuous space have never been investigated, to our
knowledge. For our approach, continous space is actually more convenient than the
lattice. The method we use is the one developed originally in the thesis of Christine
Ritzmann [3], [1], but there are a number of improvements, generalizations, and simpli-
fications presented here.
There is a huge literature on the lace expansions and applications to many types
of models. For a survey of the state of art around 2006, see [4], but there were still
many developments since then. The emphasize here is to present an elementary and
completely self-contained proof of a sharp CLT for solutions of (1.1), together with the
perhapssimplestpossibleapplication. Noknowledgeofearlierversionsoflaceexpansions
or [3] are assumed.
The basic analysis of (1.1) here is independent of the application to self-avoiding
walks, and goes through in all dimensions.
2 Main results
We fix some notations: N is the set of natural numbers 1,2,..., and N d=ef N 0 .
0
{ } ∪{ }
For t > 0, φ is the centered normal density in Rd with covariance matrix t identity.
t
×
We write φ for φ . For a R we write a for the sequence identical to a.
1
∈
We typically drop for the convolution, so we just write AB instead of A B for
∗ ∗
two integrable functions A,B.
WewriteC Rd forthesetofcontinuous,integrablefunctionsf : Rd R,vanishing
∗
→
at ,whichareoftheformf(x)= f ( x )forsomecontinuousfunctionf : [0, ) R.
0 0
∞ (cid:0) (cid:1) | | ∞ →
We also write C+ Rd for the strictly positive ones. Occasionally, we simply write C
∗ ∗
and C+.
∗ (cid:0) (cid:1)
Here are the conditions we need for B:
Condition 2.1
WeassumethatthefunctionsB C Rd aredominatedinabsolutevaluebyfunctions
m ∗
∈
Γ C+ Rd which satisfy the following conditions:
m ∈ ∗ (cid:0) (cid:1)
(cid:0) (cid:1)
B1 There exist numbers χ (s)> 0, 1 s n, satisfying χ (s)= χ (n s), and for
n n n
≤ ≤ −
some constant K
1
n−1
(s (n s))χ (s) K , n, (2.1)
n 1
∧ − ≤ ∀
s=1
X
4
such that
Γ Γ χ (m)Γ , (2.2)
m n m+n n+m
≤
B2 There exists a constant K > 0 such that for t s 2t one has
2
≤ ≤
Γ K Γ (2.3)
s 2 2t
≤
B3 There exists K > 0 such that for m t, m N, t R+, k = 0,1,2, one has
3
≤ ∈ ∈
φ (x y) y 2kΓ (y)dy K γ(k)(m)φ (x), (2.4)
t m 3 t+m
− | | ≤
Z
where
γ(k)(m) d=ef y 2kΓ (y)dy.
m
| |
Z
B4
K d=ef nγ(0)(n) < , K d=ef γ(1)(n)< , K d=ef n−1γ(2)(n)< .
4 5 6
∞ ∞ ∞
n n n
X X X
(2.5)
A simple example where the conditions are satisfied is Γ = n−aφ , a > 2, but
n n/2
unfortunately, the application to self-avoiding walks needs a slightly more complicated
choice, as will be discussed later.
We occasionally use γ instead of γ(0)(n).
n
We remark that under the above condition, one has for
def
b = B (x)dx
n n
Z
the estimate
b γ
n n
| |≤
with
γ γ χ (m)γ . (2.6)
m n m+n n+m
≤
We formulate our basic result:
We fix an arbitrary positive ε > 0, and write
def
ψ = φ , (2.7)
n nδ(1+ε)
with δ defined below in (2.16).
In the rest of this section, we will use L as a positive constant, not necessarily the
same at different occurences, which may depend on d,ε,K K , but not on n,λ.
1 6
−
Let
2 n
ζ (n) d=ef 1+ m2−iγ(i)(m) (2.8)
1
i=0m=1
X X
5
∞
ζ (n)d=ef γ(1)(m)+mγ(0)(m) ,
2
mX=n(cid:16) (cid:17)
n n
ζ(n)d=ef n−2 ζ (j)+n−1 ζ (j).
1 2
j=1 j=1
X X
Because of (2.3) and (2.5) we have
lim ζ(n)= 0, n−1ζ(n)< , ζ(m) ζ(2n) for n m 2n. (2.9)
n→∞ ∞ ≤ ≤ ≤
n
X
Theorem 2.2
Assume Condition 2.1. Then, if λ is small enough (depending on d,ε and K -K ),the
1 6
following estimates holds
[n/2]
C (x)/c φ (x) Lλ s(ψ Γ )(x)+ζ(n)ψ (x) , (2.10)
n n nδ s n−s n
| − | ≤  
s=1
X
 
where δ = δ(B,λ) > 0 is defined in (2.16).
In the example Γ (x) = n−aφ (x), 2 < a < 3, one has ζ (n) = const n3−a,
n n/2 1
×
ζ (n)= const n2−a, and therefore ζ(n)= const n2−a, and therefore
2
× ×
C (x)/c φ (x) Ln2−aψ
n n nδ n
| − |≤
giving a local CLT with error estimate. For a > 3, we get
C (x)/c φ (x) Ln−1ψ .
n n nδ n
| − |≤
As remarked above, this Γ cannot work for the application to self-avoiding walks, and
n
in fact, a pure local CLT is not possible in that case.
A first question we address is about the behavior of the sequence c :
n
{ }
Proposition 2.3
Assume Condition 2.1 and let c be the sequence defined by (1.2). Then if λ is small
enough the following holds:
a) There exists a unique µ >0 such that α d=ef lim µ−nc exists in (0, ).
n→∞ n
∞
b) Writing a d=ef µ−nc , one has
n n
def ∞
a a < Lλγ = Lλ γ . (2.11)
| n+1− n| n j=n j
X
c)
∞
µ−1 = 1 λ a b . (2.12)
k k
− k=1
X
6
Remark 2.4
a) Pluggingtheexpression(2.12)into(1.2), weseethata = a satisfiesa = 1,
{ n}n∈N0 0
and
n
∞
a = a λa a b +λ a b (a a ), n 1. (2.13)
n n−1 n−1 k k k k n−k n−1
− k=n+1 − ≥
k=1
X X
b) From (2.11) we get
∞
a α Lλ kγ . (2.14)
n k
| − | ≤
k=n
X
Proof of Proposition 2.3. Let l (N) be the Banach space of absolutely summable
1
sequences x= x , and l (N) be the set of sequences with x d=ef sup γ−1 x <
{ n}n∈N γ k kγ n n | n|
. l (N), is a Banach space, too, and by (2.5), l l , and the embedding
∞ γ k·kγ γ ⊂ 1
is co(cid:16)ntinuous. T(cid:17)he linear map s : l1(N) → l∞(N0) is defined by s(x)0 = 0, and
s(x) d=ef n x , n 1. Evidently, s(x) x L x . We also define the
n j=1 j ≥ k k∞ ≤ k k1 ≤ k kγ
def
affine mapPping S : l1 l∞ by S(x) = 1+s(x). We define two mappings ψ1,ψ2 from
→
l (N) to the set of sequences with index set N. ψ (x) = ψ (x) = 0, and
1 1 1 2 1
∞
def
ψ (x) = S(x) b S(x) ,
1 n n−1 k k
k=n+1
X
n
def
ψ (x) = S(x) b s(x) s(x)
2 n k k n−k − n−1
k=2
X (cid:2) (cid:3)
def
for n 2. Finally we set ψ = λψ +λψ . Remark first that
1 2
≥
∞
ψ(0) = λψ (0) = λ b ,
n 1 n k
k=n+1
X
so that ψ(0) Lλ, by (2.6).
k kγ ≤
∞ ∞
ψ (x) ψ (y) s(x) s(y) b S(x) + S(y) b
| 1 n− 1 n| ≤ k − k∞ | k k| n−1 | k|
" #
k=n+1 k=n+1
X (cid:12) (cid:12) X
∞(cid:12) (cid:12)
L x y 2+L x +L y γ
≤ k − kγ k kγ k kγ k
h ik=Xn+1
ψ (x) ψ (y) L x y 1+ x + y .
k 2 − 2 kγ ≤ k − kγ k kγ k kγ
(cid:16) (cid:17)
7
Similarly, for n 2, by resummation
≥
n−1 n
ψ (x) ψ (y) = x (S(y) S(x) )b (2.15)
1 n − 1 n j k − k k
j=1 k=n−j+1
X X
n−1 n
+ (y x ) S(y) b .
j − j k k
j=1 k=n−j+1
X X
In the first summand, we estimate S(x) S(y) by L x y , so we get for this
| k − k| k − kγ
part an estimate
n−1 ∞ n
L x x y γ γ .
≤ k kγk − kγ t k
j=1 t=j k=n−j+1
XX X
n−1 ∞ n n−1 ∞ n
γ γ χ (t)γ
t k t+k t+k
≤
j=1 t=j k=n−j+1 j=1 t=j k=n−j+1
XX X XX X
∞ s−1
γ δ(s,t)χ (t),
s s
≤
s=n+1 t=1
X X
where we have used (2.6), and where δ(s,t) is the numberof indices j satisfying 1 j
≤ ≤
n 1, t j, n j+1 s t n, so that δ(s,t) t (s t),and using (2.1), we get for
− ≥ − ≤ − ≤ ≤ ∧ −
the first summand of (2.15) an estimate L x x y γ(n). In a similar way, we
≤ k kγk − kγ
get for the second summand an estimate L 1+ y x y γ(n) and therefore
≤ k kγ k − kγ
(cid:16) (cid:17)
ψ (x) ψ (y) L x y 1+ x + y ,
k 2 − 2 kγ ≤ k − kγ k kγ k kγ
(cid:16) (cid:17)
leading to
ψ(x) ψ(y) Lλ x y 1+ x + y .
k − kγ ≤ k − kγ k kγ k kγ
(cid:16) (cid:17)
From that and ψ(0) l , it follows that ψ maps l continuously into itself, and fur-
γ γ
∈
thermore, if λ is small enough, by the Banach fixed point theorem, the iterates ψn(0)
converge in l to an element ξ with ξ Lλ which is a fixed point of ψ.
γ k kγ ≤
If we write
def def ∞ −1
η = S(ξ), ̟ = 1 λ η b ,
k k
− k=1
then it is readily checked, using the fact(cid:16)that ξ Xis a fixed p(cid:17)oint of ψ, that the sequence
η ̟n satisfies (1.2), and therefore it is this sequence. So it follows that ̟ = µ, and
n
{ }
µ−nc satisfies the properties listed in a)-c).
n
Let f = f be a sequence of functions in C+ which satisfy lim sup f (x) = 0.
{ n} ∗ n→∞ x n
For any sequence g = g , g C define
n n ∗
{ } ∈
g (x)
def n
g = sup sup | |,
f
k k n x∈Rd fn(x)
8
def
and write Bf = g : g f < which equipped with f is a Banach space.
{ k k ∞} k·k
As B C , the “covariance” matrix satisfies
m ∗
∈
xTxB (x)dx = b I ,
m m d
Z
for some b R (possibly negative), I being the d d unit matrix. Evidently, b
m d m
∈ × ≤
γ(1)(m), and by Conditions 2.1 (2.5), the following number is well defined (for small
(cid:12) (cid:12)
enough λ) (cid:12) (cid:12)
µ−1+λ ∞ a b
δ d=ef m=1 m m . (2.16)
µ−1+λ ∞ ma b
Pm=1 m m
By choosing λ > 0 small enough, we can achieve that
P
1 δ Lλ, 1 µ Lλ. (2.17)
| − | ≤ | − |≤
and also
1/2 a 3/2, n,
n
≤ ≤ ∀
which we assume henceforward.
Let C be the solution of (1.1). We put A d=ef C µ−n where µ is given by (2.12).
n n
This sequence satisfies A = δ and
0 0
n
A = µ−1A φ+λ a B A . (2.18)
n n−1 k k n−k
k=1
X
Then a = A (x)dx, and A /a = C /c .
n n n n n n
Proof of Theorem 2.2. Evidently, the statement is the same (given Proposition
R
2.3) as to bound A (x) a φ (x) in the same way.
n n nδ
| − |
We define an operator Ψ on sequences of functions G = G , G C ,
{ n}n≥0 n ∈ ∗
def
Ψ(G) = G , and for n 1
0 0 ≥
n
def
Ψ(G) = a φ G G ∆(j,j),
n n nδ 0− n−j
j=1
X
with
j
∆(k,j) d=ef a φ µ−1a φ λ a a B φ (2.19)
j kδ j−1 (k−1)δ+1 m j−m m (k−m)δ
− −
m=1
X
for k j. A simple resummation gives
≥
n j
Ψ(G) = G a φ G µ−1φG λ a B G .
n n− n−j (n−j)δ j − j−1− m m j−m
" #
j=1 m=1
X X
9
From that it follows that if A satisfies A = δ and (2.18), then Ψ(A) = A, and
0 0
vice versa: If A = δ , and A satisfies the fixed point equation, then (2.18) follows by
0 0
induction on n.
We consider the Banach space (Bf, f) where
k·k
[n/2]
def
f = sψ Γ +ζ(n)ψ . (2.20)
n s n−s n
s=1
X
If E is the sequence a φ then Ψ(E) = E n a ∆(n,j). From Lemma
{ n nδ} n n − j=1 n−j
3.1 (3.4), we see that Ψ(E) E Bf with Ψ(E) E f Lλ. Furthermore, if G Bf,
− ∈ k − kP≤ ∈
with G = 0, then for n 1,
0
≥
n
Ψ(G) (x) G f (x)∆(j,j)(x) .
| n | ≤ k kf | n−j |
j=1
X
We apply Lemma 3.2 and Lemma 3.1, (3.5), we see that
Ψ(G) Lλ G .
f f
k k ≤ k k
We therefore conclude that for small enough λ > 0 the iterates Ψn(E) converge to
a a fixed point which we know has to be the sequencs A, which therefore satisfies
A E Lλ. So, we have proved the theorem.
f
k − k ≤
3 Technical Lemmas
We recall some properties of the semigroup φ . Of course, φ (x) = t−d/2φ x/√t .
t t
{ }
We often write φ˙ for the derivative in t, and we write ∂ φ for the partial derivatives in
t i t (cid:0) (cid:1)
x , and ∂2φ for the second partial derivatives, etc. We also write ∆φ d=ef d ∂2φ ,
i ij t t i=1 ii t
as usual. We will often use the heat equation φ˙ = 1∆φ . The partial derivatives in x of
t 2 t P
φ are all of the form pφ for a polynomial in x whose exact form is of no concern for us.
Here are some elementary properties we will use:
If t s 2t then
• ≤ ≤
φ 2d/2φ . (3.1)
t s
≤
If p is any polynomial in x, then for any ε> 0, there exists C > 0 such that
ε,p
•
p(x) φ(x) C φ (x) (3.2)
ε,p 1+ε
| | ≤
implying
p x/√t φ (x) C φ (x). (3.3)
t ε,p t(1+ε)
≤
(cid:12) (cid:16) (cid:17)(cid:12)
From this,see thatderiv(cid:12)atives inx(cid:12)of k-th orderof φ (x)areboundedby C t−k/2
(cid:12) (cid:12) t ε,k
φ (x), and derivatives in t of k-th order are bounded by C t−kφ (x).
t(1+ε) ε,k t(1+ε)
10