The Annals of Applied Probability

Parking on transitive unimodular graphs

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

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1563869038

Digital Object Identifier
doi:10.1214/18-AAP1443

Mathematical Reviews number (MathSciNet)
MR3983336

Subjects
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)

Keywords
Annihilating particle system random walk blockades

Citation

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. https://projecteuclid.org/euclid.aoap/1563869038


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References

  • [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.