Using 6.43 and following the argument below 6.67, we obtain P
E
′
b
N
, b
N +1
; ˜ C
N −1
∩ E
t
bN+1
+ ǫ
b
N
, y
2
; ˜ C
N −1
∩ {~x ∈ ˜C
N
} ≤ P
E
′
b
N
, b
N +1
; ˜ C
N −1
∩ [
c ,w
∈˜C
N −1
[
z
∈Λ t
z
t
bN+1
n {b
N
−→ z} ◦ {z −→ w } ◦ {w −→ y
2
} ◦ {z −→ y
2
} o
∩ n
{c = w , z 6−→ w
−
} ∪ {c 6= w
−
, c, w ∈ ˜C
N −1
} o
∩ {~x ∈ ˜C
N
}
. 6.69
Similarly to the above, E
′
b
N
, b
N +1
; ˜ C
N −1
implies the existence of disjoint connections necessary to obtain the bounding diagram P
b
N
, b
N +1
; ˜ C
N −1
. The event subject to the union over z is accounted for by an application of Construction Bu followed by multiplication of the sum of
S
u, w ; ˜ C
N −1
, 2
w
y
2
over w with t
w
t
b
N +1
, resulting in the bounding diagram X
u ,w
t
w
t
bN+1
P b
N
, b
N +1
; ˜ C
N −1
, Bu S
u , w ; ˜
C
N −1
, 2
w
y
2
. 6.70
The event {~x ∈ ˜C
N
} is accounted for by applying Construction ℓ~x
I
to P b
N
, b
N +1
; ˜ C
N −1
, Bu and
Construction ℓ~x
J \I
to S u, w ; ˜
C
N −1
, 2
w
y
2
, followed by the summation over I ⊂ J. Then, by 5.32 and 5.35, we have
6.68 ≤
X
I ⊂J
X
a ,u,w
t
w
t
bN+1
X
b
N
p
b
N
M
1
b
N −1
,b
N
;˜ C
N −2
P b
N
, b
N +1
; ˜ C
N −1
, Bu, ℓ~x
I
1
{a∈˜C
N −1
}
× S
0,0
u , w ; a, 2
w
y
2
, ℓ~x
J \I
+ M
1
b
N −1
,b
N
;˜ C
N −2
P b
N
, b
N +1
; ˜ C
N −1
, Bu, ℓ~x
I
1
{a,w ∈˜C
N −1
}
× 1 − δ
a ,w
−
S
0,1
u , w ; a, 2
w
y
2
, ℓ~x
J \I
. 6.71
Note that P b
N
, b
N +1
; ˜ C
N −1
, Bu is a random variable since ˜ C
N −1
is random which depends only on bonds in the time interval [t
b
N
, t
b
N +1
], and that t
a
≥ t
b
N +1
, which is due to 5.29–5.30 and the restriction on t
w
. Therefore, by the Markov property cf., 5.48 and 5.34, 6.71
≤ X
I ⊂J
X
a ,u
t
a
≥t
bN+1
P
u , y
2
; a, ℓ~x
J \I
× X
b
N
p
b
N
M
1
b
N −1
,b
N
;˜ C
N −2
P b
N
, b
N +1
; ˜ C
N −1
, Bu, ℓ~x
I
1
{a∈˜C
N −1
}
. 6.72
We need some care to estimate M
1
b
N −1
,b
N
;˜ C
N −2
P b
N
, b
N +1
; ˜ C
N −1
, Bu, ℓ~x
I
1
{a∈˜C
N −1
}
in 6.72.
868
First, by 5.32 and t
v
≤ t
b
N +1
≤ t
a
, we obtain M
1
b
N −1
,b
N
;˜ C
N −2
P b
N
, b
N +1
; ˜ C
N −1
, Bu, ℓ~x
I
1
{a∈˜C
N −1
}
≤ X
c ,v
t
v
≤t
a
M
1
b
N −1
,b
N
;˜ C
N −2
1
{c,a∈˜C
N −1
}
S
0,0
b
N
, v ; c, 2
v
b
N +1
, Bu, ℓ~x
I
+ M
1
b
N −1
,b
N
;˜ C
N −2
1
{c,v∈˜C
N −1
}
1
{a∈˜C
N −1
}
1 − δ
c ,v
−
× S
0,1
b
N
, v ; c, 2
v
b
N +1
, Bu, ℓ~x
I
. 6.73
By the BK inequality, the second M
1
on the right-hand side is bounded as M
1
b
N −1
,b
N
;˜ C
N −2
1
{c,v∈˜C
N −1
}
1
{a∈˜C
N −1
}
1 − δ
c ,v
−
≤ M
1
b
N −1
,b
N
;˜ C
N −2
1
{c∈˜C
N −1
}
1
{c,v occupied}◦{a∈˜C
N −1
}
+
1
{c,v−→a}
1 − δ
c ,v
−
≤ M
1
b
N −1
,b
N
;˜ C
N −2
1
{c,a∈˜C
N −1
}
+ M
1
b
N −1
,b
N
;˜ C
N −2
1
{c∈˜C
N −1
}
τa − v λǫDv − c.
6.74 Substituting this back into 6.73 and using 5.31 and 5.35, we obtain
6.73 ≤
X
c
M
1
b
N −1
,b
N
;˜ C
N −2
1
{c,a∈˜C
N −1
}
P b
N
, b
N +1
; c, Bu, ℓ~x
I
+ M
1
b
N −1
,b
N
;˜ C
N −2
1
{c∈˜C
N −1
}
× X
v
τa − v λǫDv − c S
0,1
b
N
, v ; c, 2
v
b
N +1
, Bu, ℓ~x
I
. 6.75
We will show below that
τa − v S
0,1
b
N
, v ; c, 2
v
b
N +1
, Bu, ℓ~x
I
≤ S
0,1
b
N
, v ; c, 2
v
b
N +1
, Bu, ℓ~x
I
, ℓa
. 6.76 Assuming this and using 5.31 and 5.35, we obtain
X
v
τa − v λǫDv − c S
0,1
b
N
, v ; c, 2
v
b
N +1
, Bu, ℓ~x
I
≤ P b
N
, b
N +1
; c, Bu, ℓ~x
I
, ℓa
, 6.77
hence 6.75
≤ X
c
M
1
b
N −1
,b
N
;˜ C
N −2
1
{c,a∈˜C
N −1
}
P b
N
, b
N +1
; c, Bu, ℓ~x
I
+ M
1
b
N −1
,b
N
;˜ C
N −2
1
{c∈˜C
N −1
}
P b
N
, b
N +1
; c, Bu, ℓ~x
I
, ℓa
. 6.78
Further, by a version of 5.55, we have M
1
b
N −1
,b
N
;˜ C
N −2
1
{c∈˜C
N −1
}
≤ X
η
P b
N −1
, b
N
; ˜ C
N −2
, ℓ
η
c ,
6.79 M
1
b
N −1
,b
N
;˜ C
N −2
1
{c,a∈˜C
N −1
}
≤ X
η
P b
N −1
, b
N
; ˜ C
N −2
, ℓ
η
c, ℓa
, 6.80
869
where P
η
is the sum over the admissible lines of the diagram P b
N −1
, b
N
; ˜ C
N −2
. Using these inequalities and Lemma 5.6, the sum over b
N
in the second line of 6.72 is bounded as X
b
N
p
b
N
M
1
b
N −1
,b
N
;˜ C
N −2
P b
N
, b
N +1
; ˜ C
N −1
, Bu, ℓ~x
I
1
{a∈˜C
N −1
}
≤ X
η
X
c
X
b
N
P b
N −1
, b
N
; ˜ C
N −2
, ℓ
η
c, ℓa
