Asymptotics of the average height of 2--watermelons with a wall

We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length $n$) to the case of $p$--watermelons with a wall (i.e., to a certain family of $p$ nonintersecting Dyck paths; simple Dyck paths being the special case $p=1$.) We work out this asymptotics for the case $p=2$ only, since the computations involved are already quite complicated (but might be of some interest in their own right).