Parking on supercritical geometric Bienaym\'e--Galton--Watson trees

Consider a supercritical Bienaym\'e--Galton--Watson tree $ \mathcal{T}$ with geometric offspring distribution. Each vertex of this tree represents a parking spot which can accommodate at most one car. On the top of this tree, we add $(A_u : u \in \mathcal{T})$ i.i.d.\ non negative integers sampled according to a given law $ \mu$, which are the car arrivals on $ \mathcal{T}$. Each car tries to park on its arriving vertex and if the spot is already occupied, it drives towards the root and takes the first available spot. If no spot is found, then it exits the tree without parking. In this paper, we provide a criterion to determine the phase of the parking process (subcritical, critical, or supercritical) depending on the generating function of $ \mu$.