On Tail Distribution of Interpost Distance

A partial order on Z obtained by taking the transitive closure of a random relation {i<j and there is an edge ij} is studied. Randomness stems from postulating that an edge ij exists with probability p, independently of all other edges. While studying the random order on the subset [n], N. Alon, B. Bollobás, G. Brightwell, and S. Janson (1994, Ann. Appl. Probab.4, 108–123) introduced a remarkable notion of a post, defined as an element in Z comparable to all other elements in the random order. In particular they proved that the interpost distance L has a distribution with a tail Pr(L>x) decreasing at an exponential rate x1/2 at least, whence having all the moments finite. The latter information about L was all they needed in a proof of the central result, asymptotic lognormality of linear extension number. However, it remained unclear whether the exponential rate is actually linear. Our goal in this note is to confirm the conjecture.