The phase transition for parking on Galton--Watson trees

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 and try to park on empty vertices as soon as possible. We show a phase transition depending on

Parking on Cayley trees & Frozen Erdös-Rényi

“…(not necessarily Poisson) car arrivals conditioned to be fully parked and with a possible flux at the root. He proves a phase transition for the enumeration of such structures appearing at the same location as the phase transition for the parking process [37]. In the case of bounded car arrivals, he proves in [27,Theorem 3] an asymptotic enumeration of plane fully parked trees with flux of the form (57).…”

“…This model was later studied from a probabilist angle in [43]. Since then, a body of work with an increasing level of generality has emerged [67,28,37,9] ultimately considering critical conditioned Bienaymé-Galton-Watson tree (with finite variance) for the underlying tree and independent car arrivals whose laws may depend on the degree of the vertices [32]. See [10,29] for the case of supercritical trees.…”

“…In this broad context, it was shown that a sharp phase transition appears for the parking process: there is a critical "density" of cars (depending on the combinatorial details of the model) such that below this density, almost all cars manage to park, whereas above this density, a positive proportion of cars do not find a parking spot. See [32,37] for precise statements. The goal of this work is to provide scaling limits for the critical and near-critical dynamics of the parking process in the special case of uniform Cayley trees with i.i.d uniform car arrivals where the critical density is 1 2 , see [56].…”

The sustainability probability for the critical Derrida-Retaux model

“…We refer to [12] and [13] for more references and explanations of the recursion (1.3). The Derrida-Retaux system has also appeared in other contexts: in Collet et al [9] as a spin glass model, in Li and Rogers [22] as an iteration function of random variables, in Goldschmidt and Przykucki [18] and Curien and Hénard [10] as a parking scheme; it also belongs to one of the max-type recursion families in the seminal paper by Aldous and Bandyopadhyay [1].…”

“…This relies on subcriticality of the (W i , i ≥ 0), and on a simple estimate (Lemma 4.5) on the moment generating function for all subcritical Derrida-Retaux systems satisfying a convenient condition for the initial distribution. We check by means of a technical lemma (Lemma 4.3) that W n 0 satisfies this condition, where n 0 := ⌊c 34 M⌋ for some constant c 34 > 0 and all sufficiently large M. 10 This will yield that there exist constants c 33 > 0 and c 35 ∈ (0, 1), such that for all integer n ≥ 0,…”