equality 2.17 is the result of the first expansion for A ~x
J
. In this expansion, we wish to treat the connections from the top of the first pivotal to
~x
J \I
as being independent from the connections from
o
to ~x
I
that do not use the first pivotal bond. In the second expansion for A ~x
J
, we wish to extract a factor
τ~x
I
− y
2
for some y
2
from the connection from o to ~x
I
that is still present in By
1
,
~x
I
. This leads to a result of the form
X
y
1
By
1
,
~x
I
τ~x
J \I
− y
1
= X
y
1
,y
2
Cy
1
, y
2
τ~x
J \I
− y
1
τ~x
I
− y
2
+ a
~x
J \I
,
~x
I
, 2.18
where a ~x
J \I
, ~x
I
is an error term, and, to first approximation, Cy
1
, y
2
represents the sausage at o together with the pivotal bonds ending at y
1
and y
2
, with the two branches removed. In particular,
Cy
1
, y
2
is independent of I. The leading contribution to Cy
1
, y
2
is p
ǫ
y
1
p
ǫ
y
2
with y
1
6= y
2
,
corresponding to the case where the sausage at o is the single point o. For details, see Section 4, where 2.18 is derived.
We will use a new expansion for the higher-point functions, which is a simplification of the expansion for oriented percolation in Z
d
× Z
+
in [20]. The difference resides mainly in the second expansion, i.e., the expansion of A
~x
J
.
2.3 The main identity and estimates
In this section, we solve the recursion 2.12 by iteration, so that on the right-hand side no r-point function appears. Instead, only s-point functions with s
r appear, which opens up the possibility for an inductive analysis in r. The argument in this section is virtually identical to the argument in
[19, Section 2.3], and we add it to make the paper self-contained. We define
νx =
∞
X
n=0
B
⋆ n
x , 2.19
where B
⋆ n
denotes the n-fold space-time convolution of B with itself, with B
⋆ 0
x = δ
o ,x
. The sum over n in 2.19 terminates after finitely many terms, since by definition Bx, t
6= 0 only if t ∈ ǫN, so that in particular Bx, 0 = 0. Therefore, B
⋆ n
x = 0 if n t
x
ǫ, where, for x = x, t ∈ Λ,
t
x
= t denotes the time coordinate of x . Then 2.12 can be solved to give τ~x
J
= ν
⋆
A ~x
J
. 2.20
The function ν can be identified as follows. We note that 2.20 for r = 2 yields that
τx = ν
⋆
Ax . 2.21
Thus, extracting the n = 0 term from 2.19, using 2.15 to write one factor of B as A
⋆
p
ǫ
cf., 2.14 for the terms with n
≥ 1, it follows from 2.21 that
νx = δ
o ,x
+ ν
⋆
Bx =
δ
o ,x
+ ν
⋆
A
⋆
p
ǫ
x =
δ
o ,x
+ τ
⋆
p
ǫ
x . 2.22
Substituting 2.22 into 2.20, the solution to 2.12 is then given by
τ~x
J
= A
~x
J
+ τ
⋆
p
ǫ ⋆
A
~x
J
, 2.23
813
which recovers 2.16 when r = 2, using 2.15. For r ≥ 3, we further substitute 2.17–2.18 into
2.23. Let
ψy
1
, y
2
= X
v
p
ǫ
v Cy
1
− v, y
2
− v, 2.24
ζ
r
~x
J
= A ~x
J
+ τ
⋆
p
ǫ ⋆
a
~x
J
, 2.25
where a
~x
J
= a ~x
J
; 1 + X
I ⊂J
1
:I 6=∅
a ~x
J \I
, ~x
I
. 2.26
Then, 2.23 becomes τ
r
~x
J
= X
v ,y
1
,y
2
τ
2
v ψy
1
− v, y
2
− v
X
I ⊂J
1
:I 6=∅
τ
r1
~x
J \I
− y
1
τ
r2
~x
I
− y
2
+ ζ
r
~x
J
, 2.27 where r
1
= |J \ I| + 1 and r
2
= |I| + 1. Since 1 ≤ |I| ≤ r − 2, we have that r
1
, r
2
≤ r − 1, which opens up the possibility for induction in r.
The first term on the right side of 2.27 is the main term. The leading contribution to ψy
1
, y
2
is ψ
2 ǫ,2ǫ
y
1
, y
2
≡ ψ y
1
, 2 ǫ, y
2
, 2 ǫ
= X
u
p
ǫ
u p
ǫ
y
1
− u p
ǫ
y
2
− u 1 − δ
y
1
, y
2
, 2.28
using the leading contribution to C described below 2.18. We will analyse 2.27 using the Fourier transform. For I
⊆ J, we write ~k
I
= k
i i
∈I
, k
I
= X
i ∈I
k
i
, ~t
I
= t
i i
∈I
, t
I
= min
i ∈I
t
i
, 2.29
and abbreviate them to ~k, k, ~t and t, respectively, when I = J . With this notation, the Fourier transform of 2.27 becomes
ˆ τ
r
~t
~k =
t −2ǫ
X
• s
=0
ˆ τ
2
s
k X
∅6=I⊂J
1
t
J \I
−s
X
• s
1
=2 ǫ
t
I
−s
X
• s
2
=2 ǫ
ˆ ψ
s
1
,s
2
k
J \I
, k
I
ˆ τ
r1
~t
J \I
−s
1
−s
~k
J \I
ˆ τ
r2
~t
I
−s
2
−s
~k
I
+ ˆ ζ
r
~t
~k, 2.30
where P
• t
≤s≤t
′
is an abbreviation for P
s ∈[t,t
′
]∩ǫZ
+
. The identity 2.30 is our main identity and will be our point of departure for analysing the r-point functions for r
≥ 3. Apart from ψ and ζ
r
, the right-hand side of 2.27 involves the s-point functions with s = 2, r
1
, r
2
. As discussed below 2.27, we can use an inductive analysis, with the r = 2 case given by the result of Theorem 1.1 proved in
[16]. The term involving ψ is the main term, whereas ζ
r
will turn out to be an error term. The analysis will be based on the following important proposition, whose proof is deferred to Sec-
tions 5–7. In its statement, we denote
∂
2
∂ k
2
by ∇
2 k
and use the notation
b
ǫ
s
1
,s
2
= ǫ
n
s1,s2
1
{s
1
≤s
2
}
1 + s
1 d−22
×
1 + s
2
− s
1 −d−22
d 2,
log1 + s
2
d = 2, 1 + s
2 2−d2
d 2,
2.31
814
where n
s
1
,s
2
= 3 − δ
s
1
,s
2
− δ
s
1
,2 ǫ
δ
s
2
,2 ǫ
. 2.32
We note that the number of powers of ǫ is precisely such that, for d 4,
∞
X
• s
1
,s
2
=2 ǫ
b
ǫ
s
1
,s
2
= O ǫ.
2.33 We also rely on the notation
β = L
−d
, 2.34
and, for d ≤ 4, we write β
T
= L
−d
T
. Then, the main bounds on the lace-expansion coefficients are as follows:
Proposition 2.2 Bounds on the lace-expansion coefficients. The lace-expansion coefficients sat- isfy the following properties:
ψ
2 ǫ,2ǫ
y
1
, y
2
= X
u
p
ǫ
u p
ǫ
y
1
− u p
ǫ
y
2
− u 1 − δ
y
1
, y
2
. 2.35
i Let d 4, κ ∈ 0, 1 ∧ ∆ ∧
d −4
2
, λ = λ
ǫ
c
and r ≥ 3. There exist C
ψ
, C
r
ζ
∞ independent of ǫ and L
= L d such that, for all L ≥ L
, q ∈ {0, 2}, k
i
∈ [−π, π]
d
i = 1, . . . , r −1, s
i
, t
j
∈ ǫZ
+
i = 1, 2, j = 1, . . . , r − 1, the following bounds hold:
|∇
q k
i
ˆ ψ
s
1
,s
2
k
1
, k
2
| ≤ C
ψ
σ
q
1 + s
i q
2
δ
s
1
,s
2
+ ββb
ǫ
s
1
,s
2
+ b
ǫ
s
2
,s
1
, 2.36
|ˆ ζ
r
~t
~k| ≤ C
r
ζ
1 + ¯t
r −2−κ
, 2.37
where ¯t denote the second-largest element of {t
1
, . . . , t
r −1
}. ii Let d
≤ 4 with α ≡ bd −
4 −d
2
0, κ ∈ 0, α and r ≥ 3. Let β
T
= β
1
T
−bd
and λ
T
= 1 + OT
−µ
with µ ∈ 0, α − δ, as in Theorem 1.1ii. There exist C
ψ
, C
r
ζ
∞ independent of ǫ and L
= L d such that, for L
1
≥ L with L
T
defined as in 1.7, q ∈ {0, 2}, k
i
∈ [−π, π]
d
i = 1, . . . , r − 1, s
i
, t
j
≤ ǫZ
+
∩ [0, log T ] i = 1, 2, j = 1, . . . , r − 1, the following bounds hold: |∇
q k
i
ˆ ψ
s
1
,s
2
k
1
, k
2
| ≤ C
ψ
σ
q
1 + s
i q
2
δ
s
1
,s
2
+ β
T
β
T
b
ǫ
s
1
,s
2
+ b
ǫ
s
2
,s
1
, 2.38
|ˆ ζ
r
~t
~k| ≤ C
r
ζ
T
r −2−κ
. 2.39
We will prove the identity 2.35 in Section 4.4, the bounds 2.36 and 2.38 in the beginning of Section 6, and the bounds 2.37 and 2.39 in the beginning of Section 7.
It follows from 2.36 and 2.33 that for d 4, the constant V
ǫ
defined by V
ǫ
= 1
ǫ
∞
X
• s
1
,s
2
=2 ǫ
ˆ ψ
s
1
,s
2
0, 0, 2.40
with λ = λ
ǫ
c
, is finite uniformly in ǫ 0. In Proposition 2.4 below, we will prove the existence of
lim
ǫ↓0
V
ǫ
. The constant V of Theorem 1.2 should then be given by that limit. By 2.28, kDk
∞
= 815
O β and λ
ǫ
c
= 1 + O β uniformly in ǫ, we have
ˆ ψ
2 ǫ,2ǫ
0, 0 = 1 − ǫ + λ
ǫ
c
ǫ
1 − ǫ + λ
ǫ
c
ǫ
2
− 1 − ǫ
2
+ λ
ǫ
c
ǫ
2
X
x
Dx
2
| {z
}
2−ǫ+Oβǫλ
ǫ c
ǫ
= 2 − ǫ + Oβ
ǫ.
Combining this with 2.36 yields V
ǫ
= 2 − ǫ + Oβ. 2.41
This establishes the claim on V of Theorem 1.2i. For d ≤ 4, on the other hand, β = β
T
converges to zero as T
↑ ∞, so that V
ǫ
is replaced by 2 − ǫ in Theorem 2.1ii.
2.4 Induction in r
