Universality for catalytic equations and fully parked trees

TL;DR

The paper establishes a universal exponent 5/2 for positive catalytic polynomial equations by interpreting them as parking models on trees and exploiting a hidden ν-random walk. It demonstrates that under criticality, parking-tree ensembles converge, after rescaling, to self-similar Markov trees (notably the Brownian growth-fragmentation tree), while subcritical regimes yield Brownian-CRT-like limits. A probabilistic framework is developed around an underlying random walk, using a key h-transform relation and Lamperti’s representation to identify the scaling exponents and to connect discrete parking dynamics with continuous Lévy processes. The results provide a unified, probabilistic pathway to universality in catalytic equations, with explicit asymptotics for partition functions and a clear description of the limiting random-tree geometries. This approach yields both the scaling limits of the parking structures and the universal 5/2 exponent at the singularity, bridging analytic and probabilistic methods for catalytic combinatorics.

Abstract

We show that critical parking trees conditioned to be fully parked converge in the scaling limits towards the Brownian growth-fragmentation tree, a self-similar Markov tree different from Aldous' Brownian tree recently introduced and studied by Bertoin, Curien and Riera. As a by-product of our study, we prove that positive non-linear polynomial equations involving a catalytic variable display a universal polynomial exponent $5/2$ at their singularity, confirming a conjecture by Chapuy, Schaeffer and Drmota & Hainzl. Compared to previous analytical works on the subject, our approach is probabilistic and exploits an underlying random walk hidden in the random tree model.

Figures (7)

• Figure 1: Left: An example of a parking tree decorated with car arrivals (in red) and parking spots on the edges (in white). Right: After the parking process, the resulting tree is fully parked (no spot is empty) and gives rise to a $\mathbb{Z}_{\geqslant0}$-labeled tree where the label of a vertex is the number of cars that exited this vertex downwards.
• Figure 2: Illustration of the aperiodicity conditions: the graph $\mathcal{G}_Q$ can be decomposed in a non decreasing and a strongly connected part from $p_0$ on. If an edge $i \to j$ is present in the graph, then so is $i +q \to j+q$ with $q \geqslant 0$.
• Figure 3: Example of a locally largest exploration in a labeled tree: the locally largest branch is in orange. The decoration process $S$ is in orange on the right, whereas the reproduction measure $\eta$ is displayed with black dots on the right figure.
• Figure 4: A plot of the function appearing in \ref{['eq:calculpoisson']} with the only root at $\beta=3/2$.
• Figure 5: A plot of the function appearing in \ref{['eq:calculcompensation']} with the only root at $\beta=5/2$.

Theorems & Definitions (53)

• Theorem 1
• Remark 2: On the case $\mathrm{K}=1$
• Corollary 3: Universality of asymptotics for partition functions
• Theorem 4: Universal self-similar limits for the fully parked trees
• Proposition 5: See also duchi2024order
• proof
• Example 1: Parking on BGW trees
• Example 2: Planar maps
• Lemma 6: Introducing $x_{\text{\tiny{cr}}}, y_{\text{\tiny{cr}}}$
• proof