Parking on the Random Recursive Tree

TL;DR

This paper investigates the parking process on random recursive trees, revealing a phase transition at zero density and identifying the critical window for car flux, especially for binary arrivals, in large trees.

Contribution

It is the first to analyze the parking process on trees with large degree vertices, establishing phase transition behavior and critical thresholds.

Findings

Phase transition occurs at density 0.

Critical window for positive flux identified at density ~ log(n)^{-2+o(1)}.

First study of parking on trees with large degree vertices.

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.