The contribution to T
N
Φ of those w and z that are sufficiently far apart is bounded as follows. Using 16 we have
log N
4
N X
w,z ∈S
N
, |w−z|
p
log N
1 −η
t
N
ξ
N u
wξ
N u
zˆ P ˆ
B
t
N
− ˆB
e
1
t
N
= w − z, {0 | e
1
| e
2
}
t
N
≤ ¯ K
log N N
X
w ∈S
N
ξ
N u
w X
y ∈S
N
1 | y|
p log N
1 −η
t
N
ˆ P ˆ
B
t
N
− ˆB
e
1
t
N
= y|{0 | e
1
| e
2
}
t
N
≤ ¯ K X
N u
1ˆ P
ˆ
B
t
N
− ˆB
e
1
t
N
p log N
1 −η
t
N
{0 | e
1
| e
2
}
t
N
≤ ¯
K C
9.1
log N
−1−η
X
N u
1, where the last line comes from Lemma 9.1 and the facts that v
N
≤ N by12 and log N ≥ logv
N
t
N
. Combining 52, 53 and the above gives the desired bound on T
N
Φ. The two other terms are handled in a similar way, and we finally obtain 46.
4.3 Second drift estimates
We may write D
N ,2 t
Φ = R
t
d
N ,2 s
Φ, ξ
N
ds, where d
N ,2 s
Φ, ξ
N
:= log N
2
N X
x ∈S
N
Φs, x
β
N
1 − ξ
N s
x f
N 1
x, ξ
N s
2
− β
N 1
ξ
N s
x f
N
x, ξ
N s
2
.
Again, when context is clear we drop ξ
N
from this notation. When s ≤ t
N
we will use |d
N ,2 s
Φ| ≤ 2log Nβ||Φ||
∞
X
N s
1. 54
For s ≥ t
N
, the same reasoning we used to establish Lemma 3.6 yields E
d
N ,2 s
Φ | F
s −t
N
− log N
2
N X
x ∈S
N
Φs − t
N
, x ˆ E
β
N
1 − ξ
N s
−t
N
ˆ B
x t
N
2
Y
i=1
ξ
N s
−t
N
ˆ B
x+e
i
t
N
−β
N 1
ξ
N s
−t
N
ˆ B
x t
N
2
Y
i=1
1 − ξ
N s
−t
N
ˆ B
x+e
i
t
N
≤ C
55
||Φ||
1 2
X
N s
−t
N
1log N
−8
. 55
1215
The summand in the above expression vanishes whenever the walk started at x coalesces before time t
N
with either of the two walks started at x + e
1
, x + e
2
. We may therefore write ˆ
E β
N
1 − ξ
N s
−t
N
ˆ B
x t
N
2
Y
i=1
ξ
N s
−t
N
ˆ B
x+e
i
t
N
− β
N 1
ξ
N s
−t
N
ˆ B
x t
N
2
Y
i=1
1 − ξ
N s
−t
N
ˆ B
x+e
i
t
N
= ˆ E
h β
N
ξ
N s
−t
N
ˆ B
x+e
1
t
N
− β
N 1
ξ
N s
−t
N
ˆ B
x t
N
1
{x|x+e
1
∼x+e
2
}
tN
i + ˆ
E h
β
N 1
− β
N
ξ
N s
−t
N
ˆ B
x t
N
ξ
N s
−t
N
ˆ B
x+e
1
t
N
1
{x|x+e
1
∼x+e
2
}
tN
i + ˆ
E
− β
N 1
ξ
N s
−t
N
ˆ B
x t
N
+ β
N 1
2
X
i=1
ξ
N s
−t
N
ˆ B
x t
N
ξ
N s
−t
N
ˆ B
x+e
i
t
N
+β
N
ξ
N s
−t
N
ˆ B
x+e
1
t
N
ξ
N s
−t
N
ˆ B
x+e
2
t
N
− β
N 1
+ β
N 2
Y
i=0
ξ
N s
−t
N
ˆ B
x+e
i
t
N
1
{x|x+e
1
|x+e
2
}
tN
=: G
N 1
s − t
N
, x + G
N 2
s − t
N
, x + G
N 3
s − t
N
, x. For u
≥ 0, in the expression
log N
2
N
P
x ∈S
N
Φu, x P
3 i=1
G
i
u, x, only the first term gives a non- negligible contribution. Indeed, an argument similar to the one we used to establish Lemma 3.7
provides the following estimates for some constants δ, η ∈ 0, 1:
log N
2
N X
x ∈S
N
Φu, xG
N 1
u, x − β
N
− β
N 1
log N ˆ P
{0 | e
1
∼ e
2
}
t
N
X
N u
Φu, . ≤ C
56
log N
−8
||Φ||
Lip
X
N u
1, 56
log N
2
N X
x ∈S
N
Φu, xG
N 2
u, x + G
N 3
u, x ≤ C
57
||Φ||
∞
X
N u
1, 57
log N
2
N X
x ∈S
N
Φu, xG
N 2
u, x + G
N 3
u, x ≤ C
58
||Φ||
∞
1 t
N
log N I
N η
u + X
N u
1log N
−δ
. 58
We recall from 5 that the quantity log N ˆ P
{0 | e
1
∼ e
2
}
t
N
, which appears in 56, converges to a positive limit as N
→ ∞. For s ≥ t
N
, by 55, and 56-57 used with u = s − t
N
, we obtain E
d
N ,2 s
Φ | F
s −t
N
≤ C
59
||Φ||X
N s
−t
N
1 59
Along with 54 this provides E[D
N ,2 t
1] ≤ 2βlog NE Z
t ∧t
N
X
N s
1ds
+ C
59
E
Z
t−t
N +
X
N s
1ds
60
1216
Finally, from 55, and 56, 58 used with u = s − t
N
we deduce E
d
N ,2 s
Φ | F
s −t
N
− β
N
− β
N 1
log N ˆ P
{0 | e
1
∼ e
2
}
t
N
X
N s
−t
N
Φs − t
N
, . 61
≤ C
61
log N
−δ
||Φ||X
N s
−t
N
1 + C
61
||Φ||
∞
1 t
N
log N I
N η
s − t
N
.
4.4 Bounding the total mass : proof of Proposition 3.4 a
