Parking on the Random Recursive Tree

TL;DR

The paper analyzes the parking process on the uniform random recursive tree, showing a trivial phase transition ($\alpha_c=0$) and deriving the precise critical window for the appearance of a positive flux of cars, first in the binary-arrival setting with $\alpha_n \asymp \log(n)^{-2+o(1)}$ and then in a general bounded-arrival framework with exponent $\beta^*$. It leverages a Yule-based coupling to construct a Benjamini–Schramm local limit with an infinite spine, and uses a two-pronged approach: upper bounds via fully parked-tree decompositions and lower bounds via Bienaymé–Galton–Watson-type substructures to show flux growth when $\alpha_n$ decays slowly. The work extends parking process analysis to trees with large-degree vertices and establishes a quantitative link between arrival density, tree-local limits, and flux growth, with implications for understanding phase transitions in random-tree parking models. The results suggest robust behavior under bounded arrivals and motivate future exploration for unbounded distributions, aiming to refine the flux scale and the precise moving window for the onset of sustained root activity.

Abstract

We study the parking process on the random recursive tree. We first prove that although the random recursive tree has a non-degenerate Benjamini--Schramm limit, the phase transition for the parking process appears at density $0$. We then identify the critical window for appearance of a positive flux of cars with high probability. In the case of binary car arrivals, this happens at density $ \log (n)^{-2+o(1)}$ where $n$ is the size of the tree. This is the first work that studies the parking process on trees with possibly large degree vertices.

Figures (5)

• Figure 1: On the left, an example of a Yule tree $\mathcal{Y}_t$ cut at time $t$. On the right, the recursive $\mathcal{T}_t$ tree constructed from this Yule tree. Each vertex is drawn with the same color as its corresponding branch in the Yule tree.
• Figure 2: On the left side of the figure, we see the two Yule processes $\mathcal{Y}^{(0)}$ (in pink) and $\mathcal{Y}^{(1)}$ (in orange) that grow for respective times $\tau_0$ and $\tau_0+\tau_1$ in total. On the right side, the tree $\widetilde{ \mathcal{Y}}^{(1)}$ obtain by gluing $\mathcal{Y}^{(1)}$ on the right-hand side of the left most branch of $\widetilde{ \mathcal{Y}}^{(0)}= \mathcal{Y}^{(0)}$. The two thicker branches are the one that correspond to the vertex $S_0$ (in pink) and $S_1$ (in orange).
• Figure 3: On the left side, the tree $\widetilde{ \mathcal{Y}}^{(3)}$ obtained by gluing Yule tree as described above. The branch corresponding to the ancestors $S_0, S_1, S_2$ and $S_3$ are drawn thicker. On the right, the recursive tree $\mathcal{T}^{(3)}$ constructed from $\widetilde{ \mathcal{Y}}^{(3)}$. We also highlight with colors the increasing (for the inclusion) sequence of recursive trees $(\mathcal{T}^{(k)}: 0 \leqslant k \leqslant 3)$.
• Figure 4: Example of a fully parked tree with $18$ vertices and $20$ cars arriving on it: on the left, the car arriving configuration and on the right, the final configuration where all spots are occupied and $2$ cars are going out from the tree and contributing to the flux.
• Figure 5: On the left, example of a spliting in the Yule tree that creates a vertex in $V^{(1)}( \mathcal{T}_t, \mu_ {\alpha_t})$. On the right, example of the construction of the sets $( V^{(\ell)}_{ \mathrm{cars}}( \mathcal{T}_t, \mu_ {\alpha_t}) : \ell \geqslant 1)$ when $k^* = 3$. In our example, there are $6$ children of $V^{(j)}_{ \mathrm{cars}}( \mathcal{T}_t, \mu_ {\alpha_t}$ on which at least $2$ cars arrives (with $15$ cars on them in total, so that there are at least $15-6=9$ cars visiting the root and $8$ outgoing cars.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Proposition 1
• proof
• Proposition 2: subcritical regime