We use the convention that ∂
out
C
∗
a
= {a} for a white site a ∈ Z
d
. Hence, for a white it holds
¯ C
∗
a
= {a}.
Let us recall the first part of Proposition 3.1 in [AP]. To this aim, given x, y ∈ Z
d
, we write ax
and a y for the unique sites in Z
d
such that x ∈ ∆
a,K
and y ∈ ∆
a,K
. We set n := |ax − a y|
1
and choose a macroscopic path
A
x, y
= a ,
a
1
, . . . , a
n
with a = ax and a
n
= a y in particular, we
require that
|a
i
− a
i+1
|
∞
= 1. We build the path A
x, y
in the following way: we start in ax, then
we move by unitary steps along the line ax + Ze
1
until reaching the point a
′
having the same first coordinate as
a y, then we move by unitary steps along the line a
′
+ Ze
2
until reaching the point having the same first two coordinates as
a y and so on. Then, Proposition 3.1 in [AP] implies for
K large enough, as we assume that given any points x, y ∈ C
α
there exists a path γ
x, y
joining x to y inside
C
α
such that γ
x, y
is contained in W
x, y
:= ∪
a ∈A
x, y
∪
w ∈ ¯
C
∗
a
∆
w,5K 4
. 5.8
These are the main results of [AP] that we will use below. Note that, since the sets ¯ C
∗
a
can be arbitrarily large, the information that
γ
x, y
⊂ W
x, y
is not strong enough to allow to repeat the usual arguments in order to prove the Moving Particle Lemma, and therefore the Two Blocks Estimate.
Hence, one needs some new ideas, that now we present. First, we isolate a set of bad points as follows. We fix a parameter L
0 and we define the subsets
BL, BL ⊂ Z
d
as
BL := {a ∈ Z
d
: |C
∗ a
| L} , 5.9
BL := ∪
a ∈BL
∆
a,10K
. 5.10
Lemma 5.1. Given α in 0, α
], for Q–a.a. ω it holds lim sup
N ↑∞,L↑∞
|BL ∩ Λ
N
| |Λ
N
| = 0 .
5.11 Proof. Since
|BL ∩ Λ
N
| 6 cK|BL ∩ Λ
N
|, we only need to prove the thesis with BL replaced by
BL. We introduce the nondecreasing function
ρ
L
: N → [0, ∞ defined as ρ
L
n := In Ln. Then we can bound
|BL ∩ Λ
N
| 6 X
C
∗
∈C
∗
: C
∗
∩Λ
N
6=;
ρ
L
C
∗
. Since
σ stochastically dominates the Bernoulli site percolation with law P
¯ pK
and due to Lemma 2.3 in [DP], we conclude that
Q |BL ∩ Λ
N
| a|Λ
N
| 6 Q X
C
∗
∈C
∗
: C
∩Λ
N
6=;
ρ
L
C
∗
a|Λ
N
| 6
P
¯ pK
X
C
∗
∈C
∗
: C
∩Λ
N
6=;
ρ
L
C
∗
a|Λ
N
| 6
P X
a ∈Λ
N
ρ
L
˜ C
∗ a
a|Λ
N
| , 5.12
278
where the random variables ˜ C
∗
a
called pre–clusters are i.i.d. and have the same law of C
∗
under P
¯ pK
. Their construction is due to Fontes and Newman [FN1], [FN2]. Due to formula 4.47 of [AP], E
P
¯ pK
|C
∗
| is finite for K large, in particular lim
L ↑∞
E ρ
L
˜ C
∗
= 0 . 5.13
By applying Cramér’s theorem, we deduce that P
X
a ∈Λ
N
ρ
L
˜ C
∗ a
2Eρ
L
˜ C
∗
|Λ
N
| 6
e
−cLN
d
, for some positive constant cL and for all N 1. Hence, due to 5.12 and Borel–Cantelli lemma,
we can conclude that for Q–a.a. ω it holds
|BL ∩ Λ
N
||Λ
N
| 6 2Eρ
L
˜ C
∗
, ∀N N
L, ω . At this point, the thesis follows from 5.13.
At this point, due to the arguments leading to 5.5, we only need to prove the following: given α ∈ 0, α
] and A 0, for Q–a.a. ω it holds lim sup
N ↑∞,ǫ↓0,ℓ↑∞,L↑∞
sup
f ∈Υ
∗ C0,N
Z 1
N
d
ǫN
2d
ℓ
d
X
x ∈Λ
N
X
y ∈Λ
x, ǫN
X
z ∈Λ
x, ǫN
: |z− y|
∞
2 ℓ
N Γ
y, ℓ,α
− N Γ
z, ℓ,α
I N Γ
y, ℓ,α
∪ Γ
z, ℓ,α
6 Aℓ
d ∗
f ην
ρ
∗
dη = 0 5.14 where
Γ
u, ℓ,α
= Λ
u, ℓ
∩ C
α
\ BL , u
∈ Z
d
. 5.15
Above we have used also that I
N Λ
y, ℓ
∪ Λ
z, ℓ
6 Aℓ
d ∗
6 IN Γ
y, ℓ,α
∪ Γ
z, ℓ,α
6 Aℓ
d ∗
. Note that in the integral of 5.14, the function f multiplies an
F
N
–measurable function, where F
N
is the σ–algebra generated by the random variables {ηx : x ∈ G
N
} and G
N
is the set of good points define as
G
N
:= Λ
N +1
∩ C
α
\ BL . 5.16
Since D· is a convex functional see Corollary 10.3 in Appendix 1 of [KL], it must be
Dν
ρ
∗
f |F
N
6 D f 6 C N
d −2
. Hence, by taking the conditional expectation w.r.t.
F
N
in 5.14, we conclude that we only need to prove 5.14 by substituting Υ
∗ C
,N
with Υ
♯ C
,N
defined as the family of F
N
–measurable functions f : N
C ω
→ [0, ∞ such that ν f = 1 and D f 6 C N
d −2
. Recall the definition of the function
ϕ· given before 2.4. By the change of variable η → η − δ
x
one easily proves the identity ν
ρ
∗
h g
η
x
p f
η
x, y
− p
f η
2
i = ϕρ
∗
ν
ρ
∗
h p
f η
x,+
− p
f η
y,+ 2
i ,
5.17 279
where in general η
z,+
denotes the configuration obtained from η by adding a particle at site z, i.e.
η
z,+
= η + δ
z
. Let us write ∇
x, y
for the operator ∇
x, y
h η := hη
x,+
− hη
y,+
. We can finally state our weak version of the Moving Particle Lemma:
Lemma 5.2. For Q–a.a. ω the following holds. Fixed α ∈ 0, α
] and L 0, there exists a positive constant
κ = κL, α such that ǫ
−2
ϕρ
∗
N
d
ǫN
2d
ℓ
2d ∗
X
x ∈Λ
N
X
y ∈Λ
x, ǫN
X
z ∈Λ
x, ǫN
: |z− y|
∞
2 ℓ
X
u ∈Γ
y, ℓ,α
X
v ∈Γ
z, ℓ,α
ν
ρ
∗
∇
u,v
p f
2
6 N
2 −d
D f κ 6 C κ , 5.18
for any function f ∈ Υ
♯ C
,N
and for any N , ℓ, C
. Proof. Recall the definition of the path
γ
x, y
given for x, y ∈ C
α
in the discussion before 5.8. Given a bond b non intersecting G
N
, since f is F
N
–measurable it holds ∇
b
p f = 0. Using this simple
observation, by a standard telescoping argument together with Schwarz inequality, we obtain that ν
ρ
∗
∇
u,v
p f
2
6 n
n −1
X
i=0
I {u
i
, u
i+1
} ∩ G
N
6= ; o
· n
n −1
X
i=0
ν
ρ
∗
∇
u
i
,u
i+1
p f
2
o ,
5.19 where the path
γ
u,v
is written as u = u , u
1
, . . . , u
n
= v. Recall that if ν
ρ
∗
∇
u
i
,u
i+1
p f
2
6= 0 then the set
{u
i
, u
i+1
} must intersect the set of good points G
N
defined in 5.16. If b is a bond of
γ
u,v
, then b must be contained in the set W
u,v
defined in 5.8. In particular, there exists
a ∈ A
u,v
and w
∈ ¯ C
∗
a
such that b is contained in ∆
w,5K 4
