Abstract
Let \((A_u : u \in \mathbb {B})\) be i.i.d. non-negative integers that we interpret as car arrivals on the vertices of the full binary tree \( \mathbb {B}\). 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 \( \mathbb {B}\) 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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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
In this model, the probability that the root of the full binary tree belongs to an infinite component is a continuous function of the percolation probability.
In fact, we could equivalently fix an exhaustion of \(\tau \) by finite trees \(\tau _1 \subset \tau _2 \subset \cdots \) and define the parking on \(\tau \) as the limit of the parking procedure over the \(\tau _n\)’s.
References
Bahl, R., Barnet, P., Junge, M.: Parking on supercritical Galton–Watson trees, arXiv:1912.13062 (2019)
Bertoin, J., Budd, T., Curien, N., Kortchemski, I.: Martingales in self-similar growth-fragmentations and their connections with random planar maps. Probab. Theory Relat. Fields 172, 663–724 (2018)
Bertoin, J., Curien, N., Kortchemski, I.: Random planar maps and growth-fragmentations. Ann. Probab. 46, 207–260 (2018)
Bousquet-Mélou, M., Jehanne, A.: Polynomial equations with one catalytic variable, algebraic series and map enumeration. J. Combin. Theory Ser. B 96, 623–672 (2006)
Chassaing, P., Louchard, G.: Phase transition for parking blocks, Brownian excursion and coalescence. Random Struct. Algorithms 21, 76–119 (2002)
Chen, L.: Enumeration of fully parked trees, arXiv preprint arXiv:2103.15770 (2021)
Chen, X., Dagard, V., Derrida, B., Hu, Y., Lifshits, M., Shi, Z.: The Derrida–Retaux conjecture on recursive models. Ann. Probab. 49, 637–670 (2019)
Contat, A.: Sharpness of the phase transition for parking on random trees. Random Struct. Algorithms 61, 84–100 (2022). https://doi.org/10.1002/rsa.21061
Contat, A.: Last car decomposition of planar maps, arXiv preprint arXiv:2205.10285 (2022)
Contat, A., Curien, N.: Parking on Cayley trees & frozen Erdös-Rényi, arXiv:2107.02116
Curien, N.: Peeling random planar maps, Saint-Flour course 2019. https://www.imo.universite-paris-saclay.fr/~curien/
Curien, N., Hénard, O.: The phase transition for parking on Galton–Watson trees. Discrete Anal (2022). https://doi.org/10.19086/da.33167
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Goldschmidt, C., Przykucki, M.: Parking on a random tree. Comb. Probab. Comput. 28, 23–45 (2019)
King, W., Yan, C.H.: Prime parking functions on rooted trees. J. Comb. Theory Ser. A 168, 1–25 (2019)
Konheim, A.G., Weiss, B.: An occupancy discipline and applications. SIAM J. Appl. Math. 14, 1266–1274 (1966)
Lackner, M.-L., Panholzer, A.: Parking functions for mappings. J. Comb. Theory Ser. A 142, 1–28 (2016)
Le Gall, J.-F., Riera, A.: Growth-fragmentation processes in Brownian motion indexed by the Brownian tree. Ann. Probab. 48, 1742–1784 (2020)
Tutte, W.T.: A census of planar triangulations. Can. J. Math. 14, 21–38 (1962)
Acknowledgements
A.C and N.C. acknowledge the support from ERC 740943 GeoBrown. Part of this work was initiated during a conference in CIRM and we thank our host for its hospitality.
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Aldous, D., Contat, A., Curien, N. et al. Parking on the infinite binary tree. Probab. Theory Relat. Fields 187, 481–504 (2023). https://doi.org/10.1007/s00440-023-01189-6
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DOI: https://doi.org/10.1007/s00440-023-01189-6