Proof of The Convergence 3

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

As in the previous section, here too we use the Theorem 9 to prove the convergence 3. For that we begin by checking that lim t →∞ P t RR exists. In order to prove this limit we use a technique similar to that in Section 3. The dynamics of the two sided neighbourhood model bring in some additional intricacies. The following table presents some calculations which we use repeatedly. The column on the right is the probability of obtaining a configuration RR at locations i, i + 1 of Z at time t + 1 when the configuration at time t at locations i − 1, i, i + 1, i + 2 is given by the column on the left. 1583 Configuration at time t Probability of getting a configuration RR at time t + 1 RRRR 1 BRRR α + 1 − α 2 RRRB α + 1 − α 2 BRRB α 2 + 2α 1 − α 2 + 1 − α 4 RRBR α 1 − α 2 − α BRBR α 1 − α € 1 + 1 − α 2 Š RRBB α 1 − α BRBB α 1 − α RBRR α 1 − α 2 − α BBRR α 1 − α RBRB α 1 − α € 1 + 1 − α 2 Š BBRB α 1 − α RBBR α 2 1 − α 2 BBBR RBBB BBBB Combining we get P t+1 RR = P t RRRR + € α + 1 − α 2 Š P t BRRR + P t RRRB + € α 2 + 2α 1 − α 2 + 1 − α 4 Š P t BRRB +α 1 − α 2 − α P t RRBR + P t RBRR +α 1 − α P t RRBB + P t BRBB + P t BBRR + P t BBRB +α 2 1 − α 2 P t RBBR +α 1 − α € 1 + 1 − α 2 Š P t BRBR + P t RBRB . 19 Also by translation invariance of P t it also follows that P t RR = P t RRRR + P t RRRB + P t BRRR + P t BRRB . 20 1584 Now subtracting equation 20 from equation 19 we get P t+1 RR − P t RR = € α + 1 − α 2 − 1 Š P t BRRR + P t RRRB + € α 2 + 2α 1 − α 2 + 1 − α 4 − 1 Š P t BRRB +α 1 − α 2 − α P t RRBR + P t RBRR +α 1 − α P t RRBB + P t BRBB + P t BBRR + P t BBRB +α 2 1 − α 2 P t RBBR +α 1 − α € 1 + 1 − α 2 Š P t BRBR + P t RBRB = α 1 − α h − P t RRRB + P t BRRB + P t BRRR + P t BRRB +α 1 − α P t BRRB + P t RBBR + 1 − α P t RRBR + P t RBRR + € 1 + 1 − α 2 Š P t BRBR + P t RBRB + P t BRBB + P t RBRR + P t RRBB + P t RRBR + P t BBRR + P t BBRB i = α 1 − α h α 1 − α P t BRRB + P t RBBR + 1 − α P t RRBR + P t RBRR + € 1 + 1 − α 2 Š P t BRBR + P t RBRB + P t BRBB + P t BBRB i . 21 In the above equations the second equality holds because of the following two easy algebraic iden- tities: α + 1 − α 2 − 1 = −α 1 − α and α 2 + 2α 1 − α 2 + 1 − α 4 − 1 = α 1 − α α 1 − α − 2 , and the last equality is obtained from the following relations: P t RRB = P t RRBB + P t RRBR = P t RRRB + P t BRRB , P t BBR = P t BBRR + P t RBRR = P t BRRR + P t BRRB . Since each of the terms in the last equality of 21 are non-negative so we conclude that P t RR t ≥1 is an increasing sequence and hence converges. Thus P t+1 RR − P t R → 0 as t → ∞. So each of the following eight probabilities converge to 0 as t → ∞. P t BRRB , P t RBBR , P t RRBR , P t RBRR , 1585 P t BRBR , P t RBRB , P t BRBB , P t BBRB . 22 It then follows that P t RBR = P t RBRR + P t RBRB → 0, 23 P t BRB = P t BRBR + P t BRBB → 0. 24 Now 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 . 25 So by induction lim t →∞ P t+1 B R k B = 0 = lim t →∞ P t+1 R B k R 26 for any k ≥ 1. Finally, we consider the one dimensional marginal and observe P t+1 R = P t RRR + € α + 1 − α 2 Š P t RRB + P t BRR + € α + 1 − α 3 Š P t BRB +α 1 − α P t BBR + P t RBB + 1 − α € 1 − 1 − α 2 Š P t RBR . 27 Also from translation invariance of P t it follows that P t R = P t RRR + P t RRB + P t BRR + P t BRB . 28 Subtracting equation 28 from equation 27 we have P t+1 R − P t R = α 2 1 − α P t BRB − P t RBR . 29 To derive this final expression we used the following identities which are easy consequences of translation invariance of P t . P t BBR − P t BRR = P t BRB − P t RBR = P t RBB − P t RRB . Now from equation 21 we get that all the eight probabilities given in 22 are summable. In fact we can write P t+1 RR − P RR = α 1 − α   α 1 − α t X n=0 P n BRRB + P n RBBR + 1 − α t X n=0 P n RRBR + P n RBRR + € 1 + 1 − α 2 Š t X n=0 P n BRBR + P n RBRB + t X n=0 P n BRBB + P n BBRB   . 30 1586 It then follows that the two sequences P t RBR = P t RBRR + P t RBRB and P t BRB = P t BRBR + P t BRBB are also summable. In particular they converge to 0. This proves that lim t →∞ P t R exists. Invoking Theorem 9 we now complete the proof of the convergence 3. We also note that so far we have only used the fact that the starting distribution is translation invariant. Thus as mentioned in the introduction this in fact proves Theorem 3 as well.

4.2 Proof of the Properties of