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Parking on a Random Tree


Consider a uniform random rooted labelled tree on n vertices. We imagine that each node of the tree has space for a single car to park. A number mn of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider m = ⌊α n⌋ and let An denote the event that all ⌊α n⌋ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if α ≤ 1/2, we have $\mathbb{P}({A_{n,\alpha}}) \to {\sqrt{1-2\alpha}}/{(1-\alpha})$ , whereas if α > 1/2 we have $\mathbb{P}({A_{n,\alpha}}) \to 0$ . We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we consider the following variant of the problem: take the tree to be the family tree of a Galton–Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson(α) number of cars arrive at each vertex. Let X be the number of cars which visit the root of the tree. We show that $\mathbb{E}{[X]}$ undergoes a discontinuous phase transition, which turns out to be a generic phenomenon for arbitrary offspring distributions of mean at least 1 for the tree and arbitrary arrival distributions.

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Research supported by EPSRC Fellowship EP/N004833/1.

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[1] Abraham, R. and Delmas, J.-F. (2015) An introduction to Galton–Watson trees and their local limits. Lecture notes available at arXiv:1506.05571.
[2] Addario-Berry, L. (2013) The local weak limit of the minimum spanning tree of the complete graph. arXiv:1301.1667
[3] Aldous, D. J. and Bandyopadhyay, A. (2005) A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15 10471110.
[4] Aldous, D. and Steele, J. (2004) The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (Kesten, H., ed.), Vol. 110 of Encyclopaedia of Mathematical Sciences, Springer, pp. 1–72.
[5] Barlow, M. T. and Kumagai, T. (2006) Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 3365.
[6] Benjamini, I. and Schramm, O. (2001) Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 23.
[7] Brown, M., Peköz, E. and Ross, S. (2010) Some results for skip-free random walk. Probab. Eng. Inform. Sci. 24 491507.
[8] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996) On the Lambert W function. Adv. Comput. Math. 5 329359.
[9] Grimmett, G. (1980) Random labelled trees and their branching networks. J. Austral. Math. Soc. 30 229237.
[10] Jones, O. (2018) Runoff on rooted trees. arXiv:1807.08803
[11] Kesten, H. (1986) Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 425487.
[12] Konheim, A. and Weiss, B. (1966) An occupancy discipline and applications. SIAM J. Appl. Math. 14 12661274.
[13] Lackner, M.-L. and Panholzer, A. (2016) Parking functions for mappings. J. Combin. Theory Ser. A 142 128.
[14] Luczak, M. and Winkler, P. (2004) Building uniformly random subtrees. Random Struct. Alg. 24 420443.
[15] Lyons, R., Peled, R. and Schramm, O. (2008) Growth of the number of spanning trees of the Erdős–Rényi giant component. Combin. Probab. Comput. 17 711726.
[16] Stanley, R. (1996) Hyperplane arrangements, interval orders, and trees. Proc. Natl Acad. Sci. 93 26202625.
[17] Stanley, R. (1997) Parking functions and noncrossing partitions. Electron. J. Combin. 4 114.
[18] Stanley, R. (1997 & 1999) Enumerative Combinatorics, Vols I & II, Cambridge University Press.
[19] Stanley, R. (1998) Hyperplane arrangements, parking functions and tree inversions. In Mathematical Essays in honor of Gian-Carlo Rota (Sagan, B. and Stanley, R., eds), Vol. 161 of Progress in Mathematics, Springer, pp. 359375.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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