The quenched and annealed laws of Z defined by ¯
P
ξ 0,0
· = Z
Λ
N
¯ P
ξ,ε 0,0
· W dε, ¯
P
µ,0,0
· = Z
D
Ω
[0,∞
¯ P
ξ 0,0
· P
µ
dξ, 2.12
coincide with those of Y , i.e., ¯
P
ξ 0,0
Z ∈ · = P
ξ 0,0
Y ∈ · , ¯
P
µ,0,0
Z ∈ · = P
µ,0,0
Y ∈ · . 2.13
In words, Z becomes Y when the average over ε is taken. The importance of 2.13 is two-fold.
First, to prove the LLN for Y in 2.8 it suffices to prove the LLN for Z. Second, Z suffers time lapses during which its transitions are dictated by
ε rather than ξ. By the cone-mixing property of ξ, these time lapses will allow
ξ to steadily lose memory, which will be a crucial element in the proof of the LLN for Z.
2.3 Regeneration times
Fix L ∈ 2N and define the L-vector ε
L
= ℓ
+
, ℓ
−
, . . . , ℓ
+
, ℓ
−
, 2.14
where the pair ℓ
+
, ℓ
−
is alternated
1 2
L times. Given n ∈ N and
ε ∈ Λ
N
with ε
n+1
, . . . , ε
n+L
= ε
L
, we see from 2.11 that because
ℓ
+
+ ℓ
−
= 0, 2 = 2ℓ ¯
P
ξ,ε 0,0
Z
n+L
= x + Lℓ | Z
n
= x = 1,
x ∈ H, 2.15
which means that the stretch of walk Z
n
, . . . , Z
n+L
travels in the vertical direction ℓ irrespective of ξ.
Define regeneration times τ
L
= 0, τ
L k+1
= inf n
τ
L k
+ L : ε
n−L
, . . . , ε
n−1
= ε
L
, k ∈ N.
2.16 Note that these are stopping times w.r.t. the filtration G = G
n n∈N
given by G
n
= σ{ε
i
: 1 ≤ i ≤ n}, n ∈ N.
2.17 Also note that, by the product structure of W = w
⊗N
defined in 2.9, we have τ
L k
∞ ¯ P
-a.s. for all k ∈ N.
Recall Definition 1.1 and put Φt =
sup
A∈F0, B∈F θ
t PµA
P
µ
B | A − P
µ
B .
2.18
Cone-mixing is the property that lim
t→∞
Φt = 0 for all cone angles θ ∈ 0,
1 2
π, in particular, for θ =
1 4
π needed here. Let H
k
= σ τ
L i
k i=0
, Z
i τ
L k
i=0
, ε
i τ
L k
−1 i=0
, { ξ
t
: 0 ≤ t ≤ τ
L k
− L} ,
k ∈ N. 2.19
This sequence of sigma-fields allows us to keep track of the walk, the time lapses and the environ- ment up to each regeneration time. Our main result in the section is the following.
595
Lemma 2.1. For all L ∈ 2N and k ∈ N, ¯ P
µ,0,0
-a.s., ¯
P
µ,0,0
Z
[k]
∈ · | H
k
− ¯
P
µ,0,0
Z ∈ ·
tv
≤ ΦL, 2.20
where Z
[k]
=
Z
τ
L k
+n
− Z
τ
L k
n∈N
2.21 and k · k
tv
is the total variation norm. Proof. We give the proof for k = 1. Let A ∈
σH
N
be arbitrary, and abbreviate 1
A
= 1
{Z∈A}
. Let h be any H
1
-measurable non-negative random variable. Then, for all x ∈ H and n ∈ N, there exists a random variable h
x,n
, measurable w.r.t. the sigma-field σ
Z
i n
i=0
, ε
i n−1
i=0
, { ξ
t
: 0 ≤ t n − L}
,
2.22 such that h = h
x,n
on the event {Z
n
= x, τ
L 1
= n}. Let E
P
µ
⊗W
and Cov
P
µ
⊗W
denote expectation and covariance w.r.t. P
µ
⊗ W , and write θ
n
to denote the shift of time over n. Then ¯
E
µ,0,0
h
1
A
◦ θ
τ
L 1
=
X
x∈H,n∈N
E
P
µ
⊗W
¯ E
ξ,ε
h
x,n
[1
A
◦ θ
n
] 1
n Z
n
=x,τ
L 1
=n o
= X
x∈H,n∈N
E
P
µ
⊗W
f
x,n
ξ, ε g
x,n
ξ, ε = ¯
E
µ,0,0
h ¯ P
µ,0,0
A + ρ
A
, 2.23
where f
x,n
ξ, ε = ¯ E
ξ,ε 0,0
h
x,n
1
n Z
n
=x,τ
L 1
=n o
, g
x,n
ξ, ε = ¯ P
θ
n
ξ,θ
n
ε x
A, 2.24
and ρ
A
= X
x∈H,n∈N
Cov
P
µ
⊗W
f
x,n
ξ, ε, g
x,n
ξ, ε .
2.25 By 1.11 and 2.18, we have
|ρ
A
| ≤ X
x∈H,n∈N
Cov
P
µ
⊗W
f
x,n
ξ, ε, g
x,n
ξ, ε ≤
X
x∈H,n∈N
ΦL E
P
µ
⊗W
f
x,n
ξ, ε sup
ξ,ε
g
x,n
ξ, ε ≤ ΦL
X
x∈H,n∈N
E
P
µ
⊗W
f
x,n
ξ, ε = ΦL ¯
E
µ,0,0
h. 2.26
Combining 2.23 and 2.26, we get ¯
E
µ,0,0
h
1
A
◦ θ
τ
L 1
− ¯
E
µ,0,0
h ¯ P
µ,0,0
A ≤ ΦL
¯ E
µ,0,0
h. 2.27
Now pick h = 1
B
with B ∈ H
1
arbitrary. Then 2.27 yields ¯
P
µ,0,0
Z
[k]
∈ A | B
− ¯ P
µ,0,0
Z ∈ A ≤ ΦL for all A ∈ σH
N
, B ∈ H
1
. 2.28
596
Therefore, since 2.28 is uniform in B, 2.28 holds ¯ P
µ,0,0
-a.s. when B is replaced by H
1
. More- over, 2.28 holds ¯
P
µ,0,0
-a.s. with H
1
in place of B, simultaneously for all measurable cylinder sets A. Since the total variation norm is defined over cylinder sets, we can take the supremum over
A to get the claim for k = 1. The extension to k ∈ N is straightforward.
2.4 Gaps between regeneration times
