Parking on the infinite binary tree

Abstract

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".

The Derrida–Retaux conjecture on recursive models

“…12271351. 2 LAGA, Université Paris XIII, France, yueyun@math.univ-paris13.fr Partially supported by ANR LOCAL. 3 AMSS, Chinese Academy of Sciences, China, shizhan@amss.ac.cn…”

“…Curien and Hénard [12], Contat and Curien [11], and Aldous et al [2]. See [19] for an extension to the case when m is random, and [18,5] for an exactly solvable version in continuous time.…”

Sharpness of the phase transition for parking on random trees

“…Initially introduced by Konheim and Weiss in the case of a line [18], their study on trees has been initiated by Lackner and Panhozler in 2016 [20]. Since then, an intriguing phase transition has been brought to light especially on critical Bienaymé-Galton-Watson trees [5,7,10,11,13,16] and on the infinite binary tree [3]. The goal of this work is to shed a new light on this phase transition already observed in [7] by coupling the parking model with a random graph model.…”