The Annals of Applied Probability

Parking on transitive unimodular graphs

Michael Damron, Janko Gravner, Matthew Junge, Hanbaek Lyu, and David Sivakoff

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p\geq1/2$, and only finitely many times otherwise.

Article information

Ann. Appl. Probab., Volume 29, Number 4 (2019), 2089-2113.

Received: November 2017
Revised: October 2018
First available in Project Euclid: 23 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35] 82B26: Phase transitions (general)

Annihilating particle system random walk blockades


Damron, Michael; Gravner, Janko; Junge, Matthew; Lyu, Hanbaek; Sivakoff, David. Parking on transitive unimodular graphs. Ann. Appl. Probab. 29 (2019), no. 4, 2089--2113. doi:10.1214/18-AAP1443.

Export citation


  • [1] Arratia, R. (1981). Limiting point processes for rescalings of coalescing and annihilating random walks on ${\mathbf{Z}}^{d}$. Ann. Probab. 9 909–936.
  • [2] Arratia, R. (1983). Site recurrence for annihilating random walks on ${\mathbf{Z}}_{d}$. Ann. Probab. 11 706–713.
  • [3] Benjamini, I., Foxall, E., Gurel-Gurevich, O., Junge, M. and Kesten, H. (2016). Site recurrence for coalescing random walk. Electron. Commun. Probab. 21 Paper No. 47, 12.
  • [4] Bramson, M. and Griffeath, D. (1980). Asymptotics for interacting particle systems on ${\mathbf{Z}}^{d}$. Z. Wahrsch. Verw. Gebiete 53 183–196.
  • [5] Bramson, M. and Lebowitz, J. L. (1991). Asymptotic behavior of densities for two-particle annihilating random walks. J. Stat. Phys. 62 297–372.
  • [6] Bramson, M. and Lebowitz, J. L. (1991). Spatial structure in diffusion-limited two-particle reactions. J. Stat. Phys. 65 941–951.
  • [7] Bramson, M. and Lebowitz, J. L. (2001). Spatial structure in low dimensions for diffusion limited two-particle reactions. Ann. Appl. Probab. 11 121–181.
  • [8] Burnside, W. (1902). On an unsettled question in the theory of discontinuous groups. Quart. J. Pure and Appl. Math. 33 230–238.
  • [9] Cabezas, M., Rolla, L. T. and Sidoravicius, V. (2018). Recurrence and density decay for diffusion-limited annihilating systems. Probab. Theory Related Fields 170 587–615.
  • [10] Dehling, H. G., Fleurke, S. R. and Külske, C. (2008). Parking on a random tree. J. Stat. Phys. 133 151–157.
  • [11] Diaconis, P. and Hicks, A. (2017). Probabilizing parking functions. Adv. in Appl. Math. 89 125–155.
  • [12] Dygert, B., Junge, M., Kinzel, C., Raymond, A., Slivken, E. and Zhu, J. (2016). The bullet problem with discrete speeds. Preprint. Available at arXiv:1610.00282.
  • [13] Elskens, Y. and Frisch, H. L. (1985). Annihilation kinetics in the one-dimensional ideal gas. Phys. Rev. A 31 3812–3816.
  • [14] Erdős, P. and Ney, P. (1974). Some problems on random intervals and annihilating particles. Ann. Probab. 2 828–839.
  • [15] Foxall, E., Hutchcroft, T. and Junge, M. (2018). Coalescing random walk on unimodular graphs. Electron. Commun. Probab. 23 Paper No. 62, 10.
  • [16] Goldschmidt, C. and Przykucki, M. (2019). Parking on a random tree. Combin. Probab. Comput. 28 23–45.
  • [17] Golod, E. S. and Šafarevič, I. R. (1964). On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28 261–272.
  • [18] Griffeath, D. (1978/79). Annihilating and coalescing random walks on ${\mathbf{Z}}_{d}$. Z. Wahrsch. Verw. Gebiete 46 55–65.
  • [19] Holley, R. (1974). Remarks on the ${\mathrm{FKG}}$ inequalities. Comm. Math. Phys. 36 227–231.
  • [20] Hutchcroft, T. and Peres, Y. (2015). Collisions of random walks in reversible random graphs. Electron. Commun. Probab. 20 no. 63, 6.
  • [21] Konheim, A. G. and Weiss, B. (1966). An occupancy discipline and applications. SIAM J. Appl. Math. 14 1266–1274.
  • [22] Krapivsky, P. L. and Sire, C. (2001). Ballistic annihilation with continuous isotropic initial velocity distribution. Phys. Rev. Lett. 86 2494.
  • [23] Lackner, M.-L. and Panholzer, A. (2016). Parking functions for mappings. J. Combin. Theory Ser. A 142 1–28.
  • [24] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Series in Statistical and Probabilistic Mathematics 42. Cambridge Univ. Press, New York.
  • [25] Lysënok, I. G. (1996). Infinite Burnside groups of even period. Izv. Ross. Akad. Nauk Ser. Mat. 60 3–224.
  • [26] Müller, S. (2015). Interacting growth processes and invariant percolation. Ann. Appl. Probab. 25 268–286.
  • [27] Sidoravicius, V. and Tournier, L. (2017). Note on a one-dimensional system of annihilating particles. Electron. Commun. Probab. 22 Paper No. 59, 9.
  • [28] Stanley, R. P. and Pitman, J. (2002). A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discrete Comput. Geom. 27 603–634.
  • [29] Zel’manov, E. I. (1991). Solution of the restricted Burnside problem for groups of odd exponent. Izv. Math. 36 41–60.