  We consider a Tamari interval of size $n$ (i.e., a pair of Dyck paths which
are comparable for the Tamari relation) chosen uniformly at random. We show
that the height of a uniformly chosen vertex on the upper or lower path scales
as $n^{3/4}$, and has an explicit limit law. By the Bernardi-Bonichon
bijection, this result also describes the height of points in the canonical
Schnyder trees of a uniform random plane triangulation of size $n$.
  The exact solution of the model is based on polynomial equations with one and
two catalytic variables. To prove the convergence from the exact solution, we
use a version of moment pumping based on D-finiteness, which is essentially
automatic and should apply to many other models. We are not sure to have seen
this simple trick used before.
  It would be interesting to study the universality of this convergence for
decomposition trees associated to positive Bousquet-M\'elou--Jehanne equations.
