Parking On A Random Rooted Plane Tree

In this paper, we investigate a parking process on a uniform random rooted plane tree with $n$ vertices. Every vertex of the tree has a parking space for a single car. Cars arrive at independent uniformly random vertices of the tree. If the parking space at a vertex is unoccupied when a car arrives there, it parks. If not, the car drives towards the root and parks in the first empty space it encounters (if there is one). We are interested in asymptotics of the probability of the event that all cars can park when $\lfloor \alpha n \rfloor$ cars arrive, for $\alpha > 0$. We observe that there is a phase transition at $\alpha_c := \sqrt{2} -1$: if $\alpha < \alpha_c$ then the event has positive probability, whereas for $\alpha > \alpha_c$ it has probability 0. Analogous results have been proved by Lackner and Panholzer, Goldschmidt and Przykucki, and Jones for different underlying random tree models.