On the scaling of random Tamari intervals and Schnyder woods of random triangulations (with an asymptotic D-finite trick)

TL;DR

This paper analyzes random Tamari intervals and their geometric scaling, proving that the height at a random abscissa scales as $n^{3/4}$ and converges to a product law $Z=(XY)^{1/4}$ with $X\sim\beta(\tfrac{1}{3},\tfrac{1}{6})$ and $Y\sim\Gamma(\tfrac{2}{3},\tfrac{1}{2})$; via the Bernardi–Bonichon bijection, the same limit governs canonical Schnyder trees in uniform random triangulations. The authors solve the exact models through polynomial equations with one or two catalytic variables and then extract asymptotics using a largely automatic D-finite moment-pumping approach, complemented by rigorous moment-transfer arguments. They also develop a joint-height framework (via marked up-steps) and derive the mixed-height results, with discussions on universality across decomposition trees and potential extensions to related combinatorial families. Overall, the work provides precise scaling, limit laws, and a robust analytic method that could apply to broader classes of decomposition-tree models with positive Bousquet-Mélou–Jehanne equations.

Abstract

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-Mélou--Jehanne equations.

Figures (9)

• Figure 1: A Dyck path of size $7$, with a marked down step followed by an up-step. The path obtained by flipping this down-step with the shortest excursion following it is declared to be larger. The Tamari lattice is the partial order generated by all such relations.
• Figure 2: Left: A uniform random Tamari interval $(P_n,Q_n)$ of size $n=65536$ generated with a python code generously provided by Wenjie Fang. Right: plot of $Q_n(i)/P_n(i)$.
• Figure 3: A figure taken from BernardiBonichon (thanks to the authors). A rooted planar triangulation equipped with its minimal Schnyder-wood, and its image (a Tamari interval) by the Bernardi-Bonichon bijection. The lower path is nothing but the contour function of the blue tree.
• Figure 4: The classical decomposition of Tamari intervals. To the left, an interval of size $n+1$, where $v_1,v_2$ are the first contacts of the lower and upper path, respectively. The decomposition gives rises, to the right, to two Tamari intervals of total size $n$, the first of which has a marked contact, called here $\tilde{v}$. This construction is bijective.
• Figure 5: Tracking the number of contacts, and how the divided difference operator appears. On the left, the power of $x$ marks all contacts, while on the right it only marks contacts which are not the last one.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• proof
• Corollary 1.5
• proof
• Proposition 1.6: Transfer theorem for algebraic functions FOFS
• Theorem 1.8: D-finite trick for moment pumping, an instance
• proof