Due to 3.11 and {1, . . . , j
I
− 1} ⊂ I, 4.10 equals H
1
=
j
I
−1
\
i=1
{v −→ x
i
in Λ
\ C} ∩ {b is not pivotal for v −→ x
i
} ∩
\
i
′
∈I i
′
j
I
{v −→ x
i
′
} ∩ {b is not pivotal for v −→ x
i
′
} .
4.17
When j
I
= 1, which is equivalent to 1 ∈ I, then the first intersection is an empty intersection, so that, by convention, it is equal to the whole probability space. We use that
{v −→ x
i
in Λ \ C} ∩ {b is not pivotal for v −→ x
i
}
= {v −→ x
i
in Λ \ C} ∩
v −→ x
i
in ˜ C
b
v =
{v −→ x
i
in Λ \ C} in ˜C
b
v ,
4.18 where we write in Λ
\ C to indicate that the equality is true with and without the restriction that
the connections take place in Λ \ C. Therefore, we can rewrite 4.17 as
H
1
=
j
I
−1
\
i=1
{v −→ x
i
in Λ \ C} in ˜C
b
v ∩
\
i
′
∈I i
′
j
I
v −→ x
i
′
in ˜ C
b
v ,
4.19
which equals 4.16. This proves 4.9. As argued below 3.13, since E
′
v , b; C ⊂ {b ∈ ˜C
b
v } and since {b −→ ~x
J \I
in Λ
\ ˜C
b
v } insures
that b 6∈˜C
b
v , by the independence statement in Lemma 3.5, the occupation status of b is inde-
pendent of the first and third events in the right-hand side of 4.9. This completes the proof of Proposition 4.2.
We continue with the expansion of PE
′
v , ~x
J
;
C. By 4.6 and 4.8, as well as Lemma 3.5,
Proposition 4.2 and 3.10, we obtain PE
′
v , ~x
J
; C
− PF
′
v , ~x
J
; C
4.20 =
X
∅6=IJ
X
b
p
b
E h
1
{{v−→~x
I
} ∩ {v
C
−→x
1
,...,x
jI −1
}
c
∩ E
′
v ,b;C in ˜ C
b
v }
1
{b−→~x
J \I
in Λ \˜C
b
v }
i
= X
∅6=IJ
X
b
p
b
E h
1
E
′
v ,b;C
1
{{v−→~x
I
} ∩ {v
C
−→x
1
,...,x
jI −1
}
c
in ˜ C
b
v }
τ
˜
Cb v
b,
~x
J \I
i
= X
∅6=IJ
X
b
p
b
M
1
v
,b; C
1
{{v−→~x
I
} ∩ {v
C
−→x
1
,...,x
jI −1
}
c
in ˜ C
b
v }
τ~x
J \I
− b − P b
˜
Cb v
−−→ ~x
J \I
, where, in the second equality, we omit “in ˜
C
b
v ” for the event E
′
v , b; C due to the fact that
E
′
v , b; C depends only on bonds before time t
b
. Applying Proposition 3.6 to Pb
˜
Cb v
−−→ ~x
J \I
and using the notation
B
δ
b, y
1
; ˜ C
b
o = δ
b,y
1
− Bb, y
1
; ˜ C
b
o, 4.21
832
we obtain PE
′
v , ~x
J
; C
− PF
′
v , ~x
J
; C
= X
∅6=IJ
X
y
1
X
b
p
b
M
1
v
,b; C
1
{{v−→~x
I
} ∩ {v
C
−→x
1
,...,x
jI −1
}
c
in ˜ C
b
v }
B
δ
b, y
1
; ˜ C
b
v τ~x
J \I
− y
1
− X
∅6=IJ
X
b
p
b
M
1
v
,b; C
1
{{v−→~x
I
} ∩ {v
C
−→x
1
,...,x
jI −1
}
c
in ˜ C
b
v }
Ab, ~x
J \I
; ˜ C
b
v .
4.22 The first step of the expansion for A
N
~x
J
is completed by substituting 4.22 into 4.1 as follows. Let see Figure 6
a ~x
J
; 1 = P F
′
o, ~x
J
; {o}
, 4.23
and, for N ≥ 1,
a
N
~x
J
; 1 = X
b
N
p
b
N
M
N
b
N
P
N
F
′
b
N
,
~x
J
; ˜ C
N −1
. 4.24
Furthermore, for N ≥ 0, we define
˜ B
N
y
1
,
~x
I
= X
b
N
,b
N +1
p
b
N
p
b
N +1
M
N
b
N
M
1
b
N
,b
N +1
;˜ C
N −1
1
{{b
N
−→~x
I
} ∩ {b
N ˜
CN−1
−−→
x
1
,...,x
jI −1
}
c
in ˜ C
N
}
× B
δ
b
N +1
, y
1
; ˜ C
N
, 4.25
a
N
~x
J \I
,
~x
I
; 2 = −
X
b
N
,b
N +1
p
b
N
p
b
N +1
M
N
b
N
M
1
b
N
,b
N +1
;˜ C
N −1
1
{{b
N
−→~x
I
} ∩ {b
N ˜
CN−1
−−→
x
1
,...,x
jI −1
}
c
in ˜ C
N
}
× Ab
N +1
, ~x
J \I
; ˜ C
N
, 4.26
where we use the convention that, for N = 0, b
= o,
˜ C
−1
= {o}. 4.27
Here a
N
~x
J
; 1 and a
N
~x
J \I
,
~x
I
; 2 will turn out to be error terms. Then, using 4.1, 4.22, and the definitions in 4.23–4.26, we arrive at the statement that for all N
≥ 0, A
N
~x
J
= a
N
~x
J
; 1 + X
∅6=IJ
X
y
1
˜ B
N
y
1
,
~x
I
τ~x
J \I
− y
1
+ a
N
~x
J \I
,
~x
I
; 2 ,
4.28 where we further make use of the recursion relation in 3.19.
In Section 4.2, we extract a factor τ~x
I
− y
2
out of ˜ B
N
y
1
,
~x
I
and complete the expansion for A
N
~x
J
.
4.2 Second cutting bond and decomposition of ˜
