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.

翻译：我们考虑一个均匀随机选取的规模为$n$的Tamari区间（即一对在Tamari关系下可比较的Dyck路径）。我们证明了上下路径中均匀随机选取的顶点的高度以$n^{3/4}$标度缩放，并具有显式的极限分布。通过Bernardi-Bonichon双射，该结果同样描述了规模为$n$的均匀随机平面三角剖分中典范Schnyder树内点的高度。该模型的精确解基于含有一个或两个催化变量的多项式方程。为从精确解出发证明收敛性，我们采用了基于D有限性的矩提升方法，该方法本质上是自动化的，并可推广至众多其他模型。我们不确定此前是否有人使用过这一简单技巧。研究该收敛性对于与正Bousquet-Mélou–Jehanne方程相关的分解树的普适性将具有重要价值。