Two particles A and B move randomly, at times t = 1,2,... on the set M = {1,...,m}, m≥ 3, reflecting from the boundary and preserving the order of their positions XÃ (t) and XÂ(†) (so that XÃ(t) ≤ XÂ(t)) according to the following rules. (a) If the distance between them is greater than 1 then at the next time they proceed independently according to the following criteria. (i) When a particle is not in a border site, 1 or m, it jumps to one of its nearest neighbours with probability 1/2. (ii) When a particle is in a border site, it either jumps to its nearest neighbour in M or remains at the same position, again with probability 1/2. (b) If they meet each other or are at distance one, they modify their behaviour. They both keep their positions if the jumps attempted would lead to interchanging their order, i.e. to the inequality XÃ > XB. (The probabilities of attempts are as before and the attempts are independent.) Otherwise they proceed according to their attempted moves. Determine the state space of the Markov chain formed by the pair (XÂ(†), XÂ(t)) and find its invariant distribution. Is the chain reversible?

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