Recently, a phase transition phenomenon has been established for parking on random trees. We extend the results of Curien and Hénard on general Bienaymé-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 Erdős-Renyi random graph model.

We prove a surprising symmetry between the law of the size G n of the greedy independent set on a uniform Cayley tree of size n and that of its complement. We also establish the asymptotic normality of G n . The proof uses a new Markovian exploration of rooted Cayley trees which is of independent interest.

Recently, a phase transition phenomenon has been established for parking on random trees in [4,12,15,17,19]. We extend the results of [8] 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 Erdős-Renyi random graph model.

Let (A u : u ∈ B) be i.i.d. non-negative integers that we interpret as car arrivals on the vertices of the full binary tree B. Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known (Bahl et al. in Parking on supercritical Galton-Watson trees, arXiv:1912.13062, 2019 Goldschmidt and Przykucki in Comb Probab Comput 28:23-45, 2019) that the parking process on B exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We characterize those regimes in terms of the law of A in an explicit way. We also study in detail the critical regime as well as the phase transition which turns out to be "discontinuous".

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