  We investigate the asymptotic behavior of random paths on a graded graph
which describes the subword order for words in two letters. This graph, denoted
by $\mathcal{Z}$, has been introduced by Viennot, who also discovered a
remarkable bijection between paths on $\mathcal{Z}$ and sequences of
permutations. Later on, Gnedin and Olshanski used this bijection to describe
the set of Gibbs measures on this graph. Both authors also conjectured that the
Martin boundary of $\mathcal{Z}$ should coincide with its minimal boundary. We
give here a proof of this conjecture by describing the distribution of a large
random path conditioned on having a prescribed endpoint. We also relate paths
on the graph Z with paths on the Young lattice, and we finally give a central
limit theorem for the Plancherel measure on the set of paths in $\mathcal{Z}$.
