Definition 4.3 Second cutting bond. For t ≥ t
v
, we say that a bond e is the t-cutting bond for
v
A
− → ~x
I
if it is the first occupied pivotal bond for v −→ x
i
for all i ∈ I such that v
A
−→ e and t
e
≥ t. Let
H
t
v , ~x
I
; A =
{v −→ ~x
I
} ∩
v
A
−→ x
1
, . . . , x
j
I
−1
∩ {∄ t-cutting bond for v
A
− → ~x
I
}, 4.35
which, for ~x
I
= x , equals
H
t
v , x ; A = v
A
−→ x ∩ {∄ t-cutting bond for v
A
− → x }.
4.36 Note in 4.34, due to 4.33, b is ˜
P
b
-a.s. vacant. Also, by Definition 4.3, when e is a cutting bond, then e is occupied. Thus, we must have that e
6= b. Using 4.34–4.35, we have, for j
I
1, ˜
E
b
h
1
E
′
v ,b;C
1
{v−→~x
I
} ∩ {v
A
−→x
1
,...,x
jI −1
}
B
δ
b, y
1
; Cv
i − ˜
E
b
h
1
E
′
v ,b;C
1
H
t y1
v , ~x
I
; A
B
δ
b, y
1
; Cv
i =
X
e 6=b
˜ E
b
h
1
E
′
v ,b;C
1
{v−→~x
I
} ∩ {v
A
−→x
1
,...,x
jI −1
} ∩ {e is t
y
1
-cutting for v
A
− →
~x
I
}
B
δ
b, y
1
; Cv
i
= X
e 6=b
˜ E
b
h
1
E
′
v ,b;C
1
{v
A
−→x
i
∀i∈I} ∩ {e is t
y
1
-cutting for v
A
− →
~x
I
}
B
δ
b, y
1
; Cv
i .
4.37 By the convention 4.32, this equality also holds when j
I
= 1 and A = {v}, so that in both cases
we are left to analyse 4.37. To the right-hand side, we will apply Lemma 3.5 and extract a factor τ~x
I
− y
2
. To do so, we first rewrite the event in the second indicator on the right-hand side as follows:
Proposition 4.4 Setting the stage for the factorization II. For A ⊂ Λ, t ≥ t
v
and a bond e,
{v
A
−→ x
i
∀i ∈ I} ∩ {e is t-cutting for v
A
− → ~x
I
} =
H
t
v , e; A in ˜ C
e
v ∩ {e is occupied} ∩
e
−→ ~x
I
in Λ
\ ˜C
e
v ,
4.38 where the first and third events in the right-hand side are independent of the occupation status of e.
Proof. By definition, we immediately obtain cf., 3.13 and 4.14
{v
A
−→ x
i
∀i ∈ I} ∩ {e is t-cutting for v
A
− → ~x
I
} =
n
v
A
−→ e ∩ {∄ t-cutting bond for v
A
− → e} in ˜C
e
v
o ∩ {e is occupied} ∩
e −→ ~x
I
in Λ \ ˜C
e
v
= H
t
v , e; A in ˜ C
e
v ∩ {e is occupied} ∩
e −→ ~x
I
in Λ \ ˜C
e
v ,
4.39 which proves 4.38.
The statement below 4.38 also holds, since H
t
v , e; A ⊂ {e ∈ ˜C
e
v }, while e −→ ~x
I
in Λ \ ˜C
e
v
ensures that e
∈ ˜C
e
v occurs see the similar arguments below 3.13 and 4.14. This completes
the proof of Proposition 4.4.
835
We continue with the expansion of the right-hand side of 4.37. First, we note that B
δ
b, y
