Parking on a random tree

Abstract

Consider a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number $m \le n$ of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider $m =[\alpha n]$ and let $A_{n,\alpha}$ denote the event that all $[\alpha n]$ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if $\alpha \le 1/2$, we have $\mathbb{P}(A_{n,\alpha}) \to \frac{\sqrt{1-2\alpha}}{1-\alpha}$, whereas if $\alpha > 1/2$ we have $\mathbb{P}(A_{n,\alpha}) \to 0$. We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson($\alpha$) number of cars arrive at each vertex. Let $X$ be the number of cars which visit the root of the tree. Then for $\alpha \le 1/2$, we have $\mathbb{E}[X] \leq 1$, whereas for $\alpha > 1/2$, we have $\mathbb{E}[X] = \infty$. This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution.