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 ˜
B
N
y
1
, ~x
I
First, we recall that, for N = 0, ˜
B y
1
, ~x
I
= X
b
1
p
b
1
M
1
b
1
1
{{o−→~x
I
} ∩ {o−→x
1
,...,x
jI −1
}
c
in ˜ C
}
B
δ
b
1
, y
1
; ˜ C
, 4.29
833
a
1
~x
J
; 1 :
o
b
1
˜ B
1
y
1
, ~x
I
:
1
y
o
2
b
1
b
a
1
~x
J \I
, ~x
I
; 2 :
o
2
b
1
b
Figure 6: Schematic representations of a
1
~x
J
; 1, ˜ B
1
y
1
, ~x
I
and a
1
~x
J \I
, ~x
I
; 2, where B
δ
b
2
, y
1
; ˜ C
1
in ˜ B
1
y
1
, ~x
I
and Ab
2
, ~x
J \I
; ˜ C
1
in a
1
~x
J \I
, ~x
I
; 2 become B b
2
, y
1
; ˜ C
1
and A
b
2
, ~x
J \I
; ˜ C
1
, respectively depicted in dashed lines, when N = 1. where, by 4.3, for j
I
= 1, {o −→ x
1
, . . . , x
j
I
−1
}
c
is the whole probability space, while, for j
I
1 and since j
I
− 1 ∈ I by 4.7, ˜B y
1
,
~x
I
≡ 0. For N ≥ 1, we recall 4.25. To extract τ~x
I
− y
2
from ˜ B
N
y
1
, ~x
I
, it suffices to consider 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
= M
1
v
,b; C
1
{{v−→~x
I
} in ˜C
b
v }
B
δ
b, y
1
; ˜ C
b
v
− M
1
v
,b; C
1
{{v−→~x
I
} ∩ {v
C
−→x
1
,...,x
jI −1
} in ˜C
b
v }
B
δ
b, y
1
; ˜ C
b
v ,
4.30 for any fixed I J with I
6= ∅, v ∈ Λ, C ⊂ Λ and a bond b, where the second term is zero if j
I
= 1 see 4.3. If j
I
1, then both terms in the right-hand side are of the form M
1
v
,b;
C
1
{{v−→~x
I
} ∩ {v
A
−→x
1
,...,x
jI −1
} in ˜C
b
v }
B
δ
b, y
1
; ˜ C
b
v
= E
h
1
E
′
v ,b;C
1
{{v−→~x
I
} ∩ {v
A
−→x
1
,...,x
jI −1
} in ˜C
b
v }
B
δ
b, y
1
; ˜ C
b
v
i ,
4.31 with
A = {v} and A = C, respectively. To treat the case of j
I
= 1 simultaneously, we temporarily adopt the convention that
{v
{v}
−→ x
1
, . . . , x
j
I
−1
} = Ω for j
I
= 1, 4.32
where Ω is the whole probability space. Do not be confused with the convention in 4.3. We note that the random variables in the above expectation depend only on bonds, other than b,
whose both end-vertices are in ˜
C
b
v , and are independent of the occupation status of b. For an
event E and a random variable X , we let ˜
P
b
E = P E b is vacant,
˜ E
b
[X ] = E X
b is vacant. 4.33
Since ˜ C
b
v = Cv almost surely with respect to ˜
P
b
, we can simplify 4.31 as ˜
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 .
4.34 To investigate 4.34, we now introduce a second cutting bond:
834
Definition 4.3 Second cutting bond. For t ≥ t
v
, we say that a bond e is the t-cutting bond for
