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Abstract

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 onSpecifically, when m ≤ 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 [8].

Parking on the infinite binary tree

“…This triggered an intense activity on the model of parking on a random critical Galton-Watson tree. In particular, a phase transition was proved to occur and the threshold was located in an increasing level of generality [8,12,14]. Furthermore a surprising connection with the Erdös-Rényi random graph and the multiplicative coalescent was unraveled in [10].…”

“…The same argument actually even proves that the radius of convergence of X (which is stochastically larger than A) must stay above 2 in the subcritical regime. Notice that deciding whether μ is subcritical for parking depends in a subtle way on the distribution as opposed to the case of critical Galton-Watson trees [8,12,14] where its depends only on the first two moments.…”

“…Those trees are interesting on their own and have been investigated from a pure analytic combinatorics viewpoint in the work [6]. In fact, [6] was motivated by [12], and [6] together with this paper were conceived following discussions between their authors. In a sense, the contribution of this paper is to bridge the gap between [12] an [1], using methods from analytic combinatorics to give insight on more probabilistic quantities.…”