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
1
, y
2
∈ Λ and ~x ∈ Λ
j
for some j ≥ 0. For N, N
1
≥ 0, X
b
N +1
p
b
N +1
M
N +1
b
N +1
1
V
t y1 −ǫ
b
N
,y
2
∩ {~x ∈˜C
N
}
B
N1
δ
b
N +1
, y
1
; ˜ C
N
≤ X
b=
· ,y
1
R
N +N1
b, y
2
;
ℓ~x p
b
, 6.45
where, on the left-hand side, we have used the convention introduced below 4.48 i.e., the dependence on b
N
is implicit. Moreover, for N ≥ 1 and N
1
≥ 0, X
b
N +1
p
b
N +1
M
N +1
b
N +1
1
E
t y1
b
N
,y
2
;˜ C
N −1
∩ {~x ∈˜C
N
}
B
N1
δ
b
N +1
, y
1
; ˜ C
N
≤ X
b= · ,y
1
Q
N +N1
b, y
2
; ℓ~x p
b
. 6.46
The remainder of this subsection is organised as follows. In Section 6.3.1, we prove Lemma 6.3 assuming Lemmas 6.5–6.7. Lemma 6.5 is an adaptation of [15, Lemmas 7.15 and 7.17] for oriented
percolation, which applies here as the discretized contact process is an oriented percolation model. The origin of the event
{z 6−→ w
−
} ∪ {w
−
∈ A} in 6.43 is similar to the intersection with the
second line in 5.43, for which we refer to the proof of 5.43. Lemma 6.6 is identical to [15, Lemma 7.16]. We omit the proofs of these two lemmas. In Section 6.3.2, we prove Lemma 6.7.
863
6.3.1 Proof of Lemma 6.3 assuming Lemmas 6.5–6.7
Proof of Lemma 6.3 for N
2
= 0. First we prove the bound on φ
N ,N1,0
y
1
, y
2
+
, where, by 4.45 and 4.47–4.48,
φ
N ,N1,0
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; {b
N
}
B
N1
δ
b
N +1
, y
1
; C
N
δ
e,y
2
= X
b
N
,b
N +1
,e b
N +1
6=e
p
b
N
p
b
N +1
p
e
M
N
b
N
˜ E
b
N +1
h
1
E
′
b
N
,b
N +1
;˜ C
N −1
1
H
t y1
b
N
,e; {b
N
}
B
N1
δ
b
N +1
, y
1
; C
N
i δ
e,y
2
. 6.47
Note that, by Lemma 6.5, H
t
y 1
b
N
, e; {b
N
} is a subset of V
t
y 1
−ǫ
b
N
, e, which is an increasing event. We also note that the event E
′
b
N
, b
N +1
; ˜ C
N −1
and the random variable B
N1
δ
b
N +1
, y
1
; ˜ C
N
, where ˜
C
N
= ˜ C
b
N +1
b
N
, are independent of the occupation status of b
N +1
. By Lemma 6.6 and using 3.16 and 3.19, we obtain
6.47 ≤
X
b
N
,b
N +1
,e b
N +1
6=e
p
b
N
p
b
N +1
p
e
M
N
b
N
E h
1
E
′
b
N
,b
N +1
;˜ C
N −1
1
V
t y1 −ǫ
b
N
,e
B
N1
δ
b
N +1
, y
1
; C
N
i δ
e,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
V
t y1 −ǫ
b
N
,e
B
N1
δ
b
N +1
, y
1
; C
N
δ
e,y
2
. 6.48
The bound 6.17 for N
2
= 0 now follows from Lemma 6.7. Next we prove the bound on
φ
N ,N1,0
y
1
, y
2
−
, where, similarly to 6.47, φ
N ,N1,0
y
1
, y
2
−
6.49 =
X
b
N
,b
N +1
,e b
N +1
6=e
p
b
N
p
b
N +1
p
e
M
N
b
N
˜ E
b
N +1
h
1
E
′
b
N
,b
N +1
;˜ C
N −1
1
H
t y1
b
N
,e;˜ C
N −1
B
N1
δ
b
N +1
, y
1
; C
N
i δ
e,y
2
.
By 6.41, we have the partition H
t
y 1
b
N
, e; ˜ C
N −1
= G
1
t
y 1
b
N
, e; ˜ C
N −1
˙ ∪ G
2
t
y 1
b
N
, e; ˜ C
N −1
. 6.50
See Figure 12 for schematic representations of the events E
′
b
N
, b
N +1
; ˜ C
N −1
∩ G
i
t
y 1
b
N
, e; ˜ C
N −1
for i = 1, 2. By Lemma 6.5, we have
1
E
′
b
N
,b
N +1
;˜ C
N −1
1
G
1 t y1
b
N
,e;˜ C
N −1
≤
1
E
′
b
N
,b
N +1
;˜ C
N −1
1
V
t y1 −ǫ
b
N
,e
, 6.51
so that, by 6.48, the contribution from G
1
t
y 1
b
N
, e; ˜ C
N −1
obeys the same bound as φ
N ,N1,0
y
1
, y
2
+
, which is the term in 6.18 proportional to R
N +N1,0
. For the contribution to
φ
N ,N1,0
y
1
, y
2
−
from G
2
t
y 1
b
N
, e; ˜ C
N −1
, we can assume that N ≥ 1 because G
2
t
y 1
b , e;
C
−1
= ∅ when N = 0 cf., 4.27. Note that, by Lemma 6.5, G
2
t
y 1
b
N
, e; ˜ C
N −1
is a subset 864
t
y
