Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parking on a random tree   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.

19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th