Sharpness of the phase transition for parking on random trees

Recently, a phase transition phenomenon has been established for parking on random trees. We extend the results of Curien and H'enard on general Galton–Watson trees and allow different car arrival distributions depending on the vertex outdegrees. We then prove that this phase transition is sharp by establishing a large deviations result for the flux of exiting cars. This has consequences on the offcritical geometry of clusters of parked spots which displays similarities with the classical Erd\H{o}s-Renyi random graph model.