3.1 Proof of Theorem 5
Before we embark on the proof of Theorem 5 we present some notation. From the translation invariance of P
t
as given by Proposition 7, for every t ≥ 0, k ≥ 1, i ∈ Z and ω
j
∈ {R, B} we observe that P
t
ξ
i
t = ω
1
, ξ
i+1
t = ω
2
, . . . , ξ
i+k −1
= ω
k
does not depend on the location i; thus with a slight abuse of notation we write
P
t
ω
1
, ω
2
, . . . , ω
k
:= P
t
ξ
i
t = ω
1
, ξ
i+1
t = ω
2
, . . . , ξ
i+k −1
= ω
k
. Also
P
t
R B
k
R := P
t
RB . . . BR where there are k many B’s in the expression on the right.
To prove this theorem we will use the technical result Theorem 9 given in Section 5. Now fix t
≥ 0. Observe P
t+1
R = P
t
RR +
α + 1 − α
2
P
t
RB + α 1 − α P
t
BR =
P
t
RR +
α + 1 − α
2
+ α 1 − α
P
t
RB =
P
t
RR + P
t
RB =
P
t
R . 11
The first equality follows from the dynamics rule. The second equality follows from the fact
P
t
RB = P
t
R − P
t
RR = P
t
BR, which is a consequence of the translation invariance of P
t
. Now for t
≥ 0 using the rule of the dynamics we get P
t+1
RR = P
t
RRR +
α + 1 − α
2
P
t
RRB +α 1 − α P
t
RBR + P
t
BRR + P
t
BRB .
12 On the other hand by translation invariance of P
t
we have P
t
RR = P
t
RRR + P
t
RRB . 13
Subtracting equation 13 from the equation 12 we get P
t+1
RR − P
t
RR = α 1 − α P
t
RBR + P
t
BRB ,
14 Here we use the fact that P
t
BRR = P
t
RR − P
t
RRR = P
t
RRB. So we conclude that P
t
RR = P RR + α 1 − α
t −1
X
s=0
P
s
RBR + P
s
BRB .
15 Since the summands above are non-negative and since 0
≤ P
t
RR ≤ 1 we have lim
t →∞
P
t
RR exists. In addition, we have
∞
X
t=0
P
t
RBR + P
t
BRB ∞.
So in particular lim
t →∞
P
t
RBR = 0 = lim
t →∞
P
t
BRB . 16
1582
Finally observe that for any k ≥ 1 we have
α
k+1
1 − α P
t
B R
k
B ≤ P
t+1
B R
k −1
B and
α
k+1
1 − α P
t
R B
k
R ≤ P
t+1
R B
k −1
R .
17 Using 16 and 17 it follows by induction that
lim
t →∞
P
t
R B
k
R = 0 = lim
t →∞
P
t
B R
k
B ∀ k ≥ 1.
18 Theorem 5 now follows from Theorem 9.
4 Proof of Theorem 2
In this section we prove our main result, namely, Theorem 2. But before we proceed we note that as remarked in the previous section, the following result is also true for our original model.
Proposition 8. Under the dynamics of our original model with p
R
= p
B
, if P is a translation invariant
measure on {R, B}
Z
then P
t
is also translation invariant for every t ≥ 0.
Once again the proof is simple so we omit the details. As in Section 3.1, Proposition 8 demonstrates the translation invariance of P
t
whenever P is trans-
lation invariant. The notation we use in this section are the same as those in Section 3.1.
4.1 Proof of The Convergence 3
Kategori