for d ≤ 4, and the same bound with β
T
and ˆ β
T
both replaced by β for d 4. Applying Lemma 5.3
to this bound, we can estimate 7.41, similarly to 7.39. However, due to the sum not the supremum over x
i
in 7.41, kτ
s
′
−s
′′
k
∞
in 7.39 is replaced by kτ
s
′
−s
′′
k
1
≤ K, where the running variable s
′′
is at most s, due to the restriction in Construction ˜ ℓ
≤s
x
i
, t
i
. Therefore, 7.41 is bounded by 7.42 multiplied by Os, which reduces the power of the denominator to d
−22, as required. This completes the proof of Lemma 7.2.
7.3 Proof of 7.7
Recall the definition 4.52 of a
N
y
1
,
~x
I
; 3
±
and denote by a
N ,N ′
y
1
,
~x
I
; 3
±
the contribution from B
N ′
δ
b
N +1
, y
1
; C
N
cf., Figure 7. We note that a
N ,N ′
y
1
, ~x
I
; 3
±
≥ 0 for every N, N
′
≥ 0. Similarly to the argument around 6.89, we have
a
N ,N ′
y
1
, ~x
I
; 3
±
≤ X
b
N +1
X
c ,v
Diagrammatic bound on ˜ M
N +1
b
N +1
1
H
t y1
b
N
, ~x
I
; C
±
∩ {b
N
−→c}
× p
b
N +1
P
N ′
b
N +1
, v ; c p
v ,y
1
, 7.43
where we recall C
+
= {b
N
} and C
−
= ˜ C
N −1
and define p
v ,y
1
= p
ǫ
y
1
− v. 7.44
We discuss the following two cases separately: i |I| = 1 and ii |I| ≥ 2.
i Suppose that I = { j} for some j. If t
j
≤ t
v
= t
y
1
− ǫ, we use H
t
y 1
b
N
, x
j
; C
±
⊆ {b
N
−→ x
j
}. If t
j
t
v
, the bubble that terminates at x
j
cf., 6.40–6.42 is cut by Z
d
× {t
v
} i.e., V
t
v
b
N
, x
j
occurs or cut by C
±
= ˜ C
N −1
if N ≥ 1 i.e., E
t
v
+ ǫ
b
N
, x
j
; ˜ C
N −1
occurs. Therefore, ˜
M
N +1
b
N +1
1
H
t y1
b
N
,x
j
; C
±
∩ {b
N
−→c}
7.45
≤
˜ M
N +1
b
N +1
1
{b
N
−→{c,x
j
}}
t
j
≤ t
v
, ˜
M
N +1
b
N +1
1
V
t v
b
N
,x
j
∩ {b
N
−→c}
+ ˜ M
N +1
b
N +1
1
E
t v +ǫ
b
N
,x
j
;˜ C
N −1
∩ {b
N
−→c}
1
{N≥1}
t
j
t
v
. By Lemma 6.6 and the argument around 6.47–6.48 and 6.52 and using 6.57–6.58, we
have ˜
M
N +1
b
N +1
1
{b
N
−→{c,x
j
}}
≤ M
N +1
b
N +1
1
{c,x
j
∈˜C
N
}
≤ X
η
P
N
b
N +1
; ℓ
η
c, ℓx
j
, 7.46
˜ M
N +1
b
N +1
1
V
t v
b
N
,x
j
∩ {b
N
−→c}
≤ M
N +1
b
N +1
1
V
t v
b
N
,x
j
∩ {c∈˜C
N
}
≤ X
η
P
N
b
N +1
; V
t
v
x
j
, ℓ
η
c ,
7.47 ˜
M
N +1
b
N +1
1
E
t v +ǫ
b
N
,x
j
;˜ C
N −1
∩ {b
N
−→c}
≤ M
N +1
b
N +1
1
E
t v +ǫ
b
N
,x
j
;˜ C
N −1
∩ {c∈˜C
N
}
≤ X
η
P
N
b
N +1
; E
t
v
x
j
, ℓ
η
c ,
7.48
878
where P
η
is the sum over the N
th
admissible lines of P
N
b
N +1
. Therefore, by 5.59 and 6.13– 6.14, we obtain
a
N ,N ′
y
1
, x
j
; 3
±
≤ X
v
p
v ,y
1
×
P
N +N ′+1
v
; ℓx
j
t
y
1
t
j
, R
N +N ′+1
v , x
j
+ Q
N +N ′+1
v , x
j
t
y
1
≤ t
j
. 7.49
We use 7.49 to estimate P
~x
J
P
y
1
a
N ,N ′
y
1
, x
j
; 3
±
τ~x
J
j
− y
1
. By 5.79 and 7.14, the contri- bution from the case of t
y
1
t
j
in 7.49 is bounded as X
~x
J
X
v ,y
1
t
y 1
t
j
P
N +N ′+1
v
;
ℓx
j
p
v ,y
1
τ~x
J
j
− y
1
≤ O 1 + ¯t
J
j
|J
j
|−1 t
J j
X
• s=t
j
X
v
P
N +N ′+1
v, s; ℓt
j
≤ ǫ O ˆ β
T
N +N
′
1 + ¯t
J
j
|J
j
|−1 t
J j
X
• s=t
j
ǫ O
β
T
1 + s
d−22
7.50 where 1 + ¯t
J
j
|J
j
|−1
= 1 + ¯t
J
j
r −3
can be replaced by 1 + ¯t
r −3
, since 1 + ¯t
J
j
|J
j
|−1
= 1 if J
j
= {i} and t
i
= max
i
′
∈J
t
i
′
. The sum in 7.50 is bounded by O ˆ β
T
when d ≤ 4, and by O
β 1 + t
j d−42
= O β
1 + t
j 6−d2
1 + t
j
≤ Oβ 1 + ¯t
∨6−d2
1 + t ≤
O β ∆
¯t
1 + t ,
7.51 when d
4. Therefore, we obtain 7.50
≤ ǫ O ˆ
β
T
N +N
′
+1
∆
¯t
1 + t 1 + ¯t
r −3
, 7.52
where ˆ β
T
must be interpreted as β when d 4.
Next we investigate the contribution from the case of t
y
1
≤ t
j
in 7.49. By 5.79 and 6.19– 6.20, we obtain
X
~x
J
X
v ,y
1
t
y 1
≤t
j
R
N +N ′+1
v , x
j
+ Q
N +N ′+1
v , x
j
p
v ,y
1
τ~x
J
j
− y
1
≤ O ˆ β
T
N +N
′
O 1 + ¯t
J
j
|J
j
|−1
X
• s
≤t
˜b
2
s,t
j
δ
s,t
j
+ β
T
β
T
. 7.53
We note that 1 + ¯t
J
j
|J
j
|−1
can be replaced by 1 + ¯t
r −3
, as explained below 7.50. To bound the sum over s in 7.53, we use the following lemma:
879
Lemma 7.3 Bounds on sums involving ˜b
2
s,s
′
. Let r ≡ |J| + 1 ≥ 3. For any j ∈ J and any I, I
′
J such that ∅
6= I
′
I, X
• s
≤t
˜b
2
s,t
j
δ
s,t
j
+ β
T
β
T
≤ ǫ O ˆ
β
T
∆
¯t
1 + t ,
7.54 X
• s
≤t s
≤s
′
≤t
I
˜b
2
s,s
′
δ
s,s
′
+ β
T
β
T
≤ ǫ O ˆ β
T
∆
¯t
, 7.55
X
• s
≤t
J \I
s ≤s
′
≤t
I \I′
1 + s
′
∧ max
i ∈I
′
t
i
˜b
2
s,s
′
β
2
T
≤ ǫ O ˆ β
T
2
∆
¯t
. 7.56
All β
T
and ˆ β
T
in the above inequalities must be interpreted as β when d 4.
We postpone the proof of Lemma 7.3 to the end of this subsection. By 7.54, we immediately conclude that 7.53 obeys the same bound as 7.52, and therefore,
X
~x
J
X
y
1
a
N ,N ′
y
1
, x
j
; 3
±
τ~x
J
j
− y
1
≤ ǫ O ˆ
β
T
N +N
′
+1
∆
¯t
1 + t 1 + ¯t
r −3
. 7.57
This completes the proof of 7.7 for |I| = 1.
ii Suppose |I| ≥ 2 and that H
t
y 1
b
N
,
~x
I
; C
±
∩ {b
N
−→ c} occurs. Then, there are u ∈ Z
d
× Z
+
and a nonempty I
′
I such that {b
N
−→ {c, u}}◦{u −→ ~x
I
′
}◦{u −→ ~x
I \I
′
} occurs. If such a u does not
exist before or at time t
v
, then C
±
= ˜ C
N −1
hence N ≥ 1 and the event E
t
v
+ ǫ
b
N
, ~x
I
; ˜ C
N −1
occurs, where
E
t
v
+ ǫ
b
N
, ~x
I
; ˜ C
N −1
= [
∅6=I
′
I
[
z
t
z
t
v
n E
t
v
+ ǫ
b
N
, z; ˜ C
N −1
∩ {b
N
−→ ~x
I
′
} ◦ {z −→ ~x
I \I
′
} o
. 7.58
Since H
t
y 1
b
N
, ~x
I
; C
±
∩ {b
N
−→ c} \ E
t
v
+ ǫ
b
N
, ~x
I
; ˜ C
N −1
⊂ [
∅6=I
′
I
[
u
t
u
≤t
v
n b
N
−→ {c, u} ◦ {u −→ ~x
I
′
} ◦ {u −→ ~x
I \I
′
} o
, 7.59
we obtain that, by the BK inequality, ˜
M
N +1
b
N +1
1
H
t y1
b
N
, ~x
I
; C
±
∩ {b
N
−→c}
≤ X
∅6=I
′
I
X
u
t
u
≤t
v
˜ M
N +1
b
N +1
1
{b
N
−→{c,u}}
P {u −→ ~x
I
′
} ◦ {u −→ ~x
I \I
′
} +
1
{N≥1}
X
z
t
z
t
v
˜ M
N +1
b
N +1
1
E
t v +ǫ
b
N
,z;˜ C
N −1
∩ {b
N
−→{c,~x
I ′
}}
τ~x
I \I
′
− z
. 7.60
880
First we investigate the contribution to 7.7 from the sum over u in 7.60, which is, by 7.43, 7.46 and Lemma 5.6,
X
~x
J
X
u ,v ,y
1
t
u
≤t
v
X
η
X
c
X
b
N +1
P
N
b
N +1
; ℓ
η
c, ℓu
