The Annals of Probability

Formation of large-scale random structure by competitive erosion

Shirshendu Ganguly, Lionel Levine, and Sourav Sarkar

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


We study the following one-dimensional model of annihilating particles. Beginning with all sites of $\mathbb{Z}$ uncolored, a blue particle performs simple random walk from $0$ until it reaches a nonzero red or uncolored site, and turns that site blue; then a red particle performs simple random walk from $0$ until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after $n$ blue and $n$ red particles alternately perform such walks, the total number of colored sites is of order $n^{1/4}$. The resulting random color configuration, after rescaling by $n^{1/4}$ and taking $n\to \infty $, has an explicit description in terms of alternating extrema of Brownian motion (the global maximum on a certain interval, the global minimum attained after that maximum, etc.).

Article information

Ann. Probab., Volume 47, Number 6 (2019), 3649-3704.

Received: April 2018
First available in Project Euclid: 2 December 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60G50: Sums of independent random variables; random walks 60J65: Brownian motion [See also 58J65] 82C22: Interacting particle systems [See also 60K35] 82C24: Interface problems; diffusion-limited aggregation 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Alternating extrema annihilating particle system Brownian motion competitive erosion diffusion-limited aggregation macroscopic interface


Ganguly, Shirshendu; Levine, Lionel; Sarkar, Sourav. Formation of large-scale random structure by competitive erosion. Ann. Probab. 47 (2019), no. 6, 3649--3704. doi:10.1214/19-AOP1342.

Export citation


  • [1] Antunović, T., Dekel, Y., Mossel, E. and Peres, Y. (2017). Competing first passage percolation on random regular graphs. Random Structures Algorithms 50 534–583.
  • [2] Asselah, A. and Gaudillière, A. (2013). From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. Ann. Probab. 41 1115–1159.
  • [3] Asselah, A. and Gaudillière, A. (2013). Sublogarithmic fluctuations for internal DLA. Ann. Probab. 41 1160–1179.
  • [4] Asselah, A. and Gaudillière, A. (2014). Lower bounds on fluctuations for internal DLA. Probab. Theory Related Fields 158 39–53.
  • [5] Benjamini, I. and Yadin, A. (2017). Upper bounds on the growth rate of diffusion limited aggregation. Preprint. Available at arXiv:1705.06095.
  • [6] Biane, Ph. and Yor, M. (1987). Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. (2) 111 23–101.
  • [7] Bramson, M. and Lebowitz, J. L. (1990). Asymptotic behavior of densities in diffusion dominated two-particle reactions. Phys. A 168 88–94.
  • [8] Bramson, M. and Lebowitz, J. L. (1991). Asymptotic behavior of densities for two-particle annihilating random walks. J. Stat. Phys. 62 297–372.
  • [9] Candellero, E., Ganguly, S., Hoffman, C. and Levine, L. (2017). Oil and water: A two-type internal aggregation model. Ann. Probab. 45 4019–4070.
  • [10] Canetti, L., Drewes, M. and Shaposhnikov, M. (2012). Matter and antimatter in the universe. New J. Phys. 14 095012.
  • [11] Ganguly, S., Levine, L., Peres, Y. and Propp, J. (2017). Formation of an interface by competitive erosion. Probab. Theory Related Fields 168 455–509.
  • [12] Ganguly, S. and Peres, Y. (2018). Competitive erosion is conformally invariant. Comm. Math. Phys. 362 455–511.
  • [13] Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 683–692.
  • [14] Janson, S. (2018). Tail bounds for sums of geometric and exponential variables. Statist. Probab. Lett. 135 1–6.
  • [15] Jerison, D., Levine, L. and Sheffield, S. (2012). Logarithmic fluctuations for internal DLA. J. Amer. Math. Soc. 25 271–301.
  • [16] Jerison, D., Levine, L. and Sheffield, S. (2013). Internal DLA in higher dimensions. Electron. J. Probab. 18 No. 98, 14.
  • [17] Jerison, D., Levine, L. and Sheffield, S. (2014). Internal DLA and the Gaussian free field. Duke Math. J. 163 267–308.
  • [18] Kesten, H. (1987). How long are the arms in DLA? J. Phys. A 20 L29–L33.
  • [19] Lawler, G. F., Bramson, M. and Griffeath, D. (1992). Internal diffusion limited aggregation. Ann. Probab. 20 2117–2140.
  • [20] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [21] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics 30. Cambridge Univ. Press, Cambridge.
  • [22] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [23] Sadler, P. M. (1981). Sediment accumulation rates and the completeness of stratigraphic sections. J. Geol. 89 569–584.