where, by applying 5.17 to 5.60–5.61 for q = 0 and using 5.25, the contribution from |w|
2
is bounded as
s
1
X
• s
′
,s
′′
=0
X
y
1
,w
|w|
2
P
N
y
1
, s
1
; Bs
′
, Bw, s
′′
sup
z,w
X
y
2
L z, s
′
, w, s
′′
; y
2
, s
2
≤ N + 1
ǫ
3
1 + s
1 d
2 s
1
X
• s
′
,s
′′
=0
s
′′
δ
s
1
,s
′
+ ǫC
1
δ
s
1
,s
′′
+ ǫC
1
1 + s
2
− s
′
∧ s
′′ d
2
× O
β
N +1
σ
2
d 4,
O β
T
2
O ˆ β
T
N −1
σ
2
T
d ≤ 4. 6.36
On the other hand, by using 5.60–5.61 for q = 0 and 5.15–5.16, the contribution from | y
2
− w|
2
in 6.35 is bounded as
s
1
X
• s
′
,s
′′
=0
X
y
1
P
N
y
1
, s
1
; Bs
′
, Bs
′′
sup
z,w
X
y
2
| y
2
− w|
2
L z, s
′
, w, s
′′
; y
2
, s
2
≤ ǫ
3
1 + s
1 d
2 s
1
X
• s
′
,s
′′
=0
s
2
− s
′′
δ
s
1
,s
′
+ ǫC
1
δ
s
1
,s
′′
+ ǫC
1
1 + s
2
− s
′
∧ s
′′ d
2
× O
β
N +1
σ
2
d 4,
O β
T
2
O ˆ β
T
N −1
σ
2
T
d ≤ 4. 6.37
Summing 6.36 and 6.37 and absorbing the factor N + 1 into the geometric term, we obtain 6.35
≤ s
2
˜b
2
s
1
,s
2
× O
β
N +1
σ
2
d 4,
O β
T
2
O ˆ β
T
N −1
σ
2
T
d ≤ 4. 6.38
Summarising 6.34 and 6.38 yields 6.26 for N ≥ 1. This completes the proof of Lemma 6.4.
6.3 Diagrammatic bounds on
φ
N ,N1,N2
y
1
, y
2
±
In this section, we prove Lemma 6.3. First we recall the convention 4.27 and the definition 4.50 and 6.3–6.5:
φ
N ,N1,N2
y
1
, y
2
±
= X
b
N +1
,e b
N +1
6=e
p
b
N +1
p
e
˜ M
N +1
b
N +1
1
{H
t y1
b
N
,e; C
±
in ˜ C
e N
}
B
N1
δ
b
N +1
, y
1
; C
N
B
N2
δ
e, y
2
; ˜ C
e
N
, 6.39
where we recall H
t
v , x ; A = {v
A
−→ x } ∩ {∄t-cutting bond for v
A
− → x }, as defined in 4.36, and
C
+
= {b
N
} and C
−
= ˜ C
N −1
. If the factors
1
{H
t y1
b
N
,e; C
±
in ˜ C
e N
}
and B
N2
δ
e, y
2
; ˜ C
e
N
were absent, then 6.39 would simplify to
π
N +N1
y
1
≤ P
N +N1
y
1
. Therefore, our task is to investigate the effect of these changes.
We will prove Lemma 6.3 using the following three lemmas:
Lemma 6.5. For v , x ∈ Λ and t
v
t ≤ t
x
, cf., Figure 11 H
t
v , x ; {v} ⊂ V
t −ǫ
v , x ≡
[
z
t
z
≤t−ǫ
{v −→ z =⇒ x }. 6.40
862
t
x
v
t
v
•
a ∈A
x
Figure 11: Schematic representations of the events a V
t −ǫ
v , x and b E
t
v , x ; A.
Moreover, for A
⊂ Λ, let G
1
t
v , x ; A = H
t
v , x ; A ∩ V
t −ǫ
v , x , G
2
t
v , x ; A = H
t
v , x ; A \ V
t −ǫ
v , x . 6.41
Then, G
1
t
v , x ; A ⊆ V
t −ǫ
v , x ,
G
2
t
v , x ; A ⊆ E
t
v , x ; A,
6.42 where
E
t
v , x ; A =
[
a ,w
∈A
[
z
∈Λ t
z
≥t
n
{v −→ z} ◦ {z −→ w } ◦ {w −→ x } ◦ {z −→ x }
o
∩ n
{a = w , z 6−→ w
−
} ∪ {a 6= w
−
, a, w ∈ A}
o .
6.43
Lemma 6.6. Let X be a non-negative random variable which is independent of the occupation status of the bond b, while F is an increasing event. Then,
˜ E
b
[X
1
F
] ≤ E[X
1
F
]. 6.44
Lemma 6.7. Let y
