Parking on a random tree
Christina Goldschmidt, Micha{\l} Przykucki

TL;DR
This paper investigates a parking process on random trees, revealing a phase transition at half the tree size where the probability of all cars parking sharply drops, supported by probabilistic analysis and an alternative proof method.
Contribution
It provides a probabilistic explanation and an alternative proof for the phase transition in parking on random trees, extending to Galton-Watson trees with general offspring and arrival distributions.
Findings
Phase transition at α=1/2 for parking success probability
Expected number of cars visiting the root is finite for α ≤ 1/2
Expected number of cars visiting the root becomes infinite for α > 1/2
Abstract
Consider a uniform random rooted tree on vertices labelled by , with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number 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 and let denote the event that all 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 , we have , whereas if we have…
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