The phase transition for parking on Galton--Watson trees

We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $σ^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $Σ^{2}$ conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending on $Θ:= (1-m)^2- Σ^2 (σ^2+m^2-m).$ Specifically, when $m \leq 1$, if $Θ>0,$ then all but (possibly) a few cars will manage to park, whereas if $Θ<0$, then a positive fraction of the cars will not find a spot and exit the tree through the root. This confirms a conjecture of Goldschmidt and Przykucki.

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