Electronic Communications in Probability

Internal DLA generated by cookie random walks on $\mathbb{Z}$

Olivier Raimond and Bruno Schapira

Full-text: Open access


We prove a law of large numbers for the right boundary in the model of internal DLA generated by cookie random walks in dimension one. The proof is based on stochastic recursions techniques.

Article information

Electron. Commun. Probab., Volume 16 (2011), paper no. 43, 483-490.

Accepted: 28 August 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Internal DLA excited random walk law of large numbers

This work is licensed under aCreative Commons Attribution 3.0 License.


Raimond, Olivier; Schapira, Bruno. Internal DLA generated by cookie random walks on $\mathbb{Z}$. Electron. Commun. Probab. 16 (2011), paper no. 43, 483--490. doi:10.1214/ECP.v16-1646. https://projecteuclid.org/euclid.ecp/1465261999

Export citation


  • Asselah A., Gaudilliere A. From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models, arXiv:1009.2838.
  • Asselah A., Gaudilliere A. Sub-logarithmic fluctuations for internal DLA, arXiv:1011.4592.
  • Ben Arous, Gérard; Quastel, Jeremy; Ramírez, Alejandro F. Internal DLA in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 2, 301–324.
  • Benaïm, Michel. Dynamics of stochastic approximation algorithms. Séminaire de Probabilités, XXXIII, 1–68, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
  • Benjamini, Itai; Wilson, David B. Excited random walk. Electron. Comm. Probab. 8 (2003), 86–92 (electronic).
  • Blachère, Sébastien; Brofferio, Sara. Internal diffusion limited aggregation on discrete groups having exponential growth. Probab. Theory Related Fields 137 (2007), no. 3-4, 323–343.
  • Lawler, Gregory F.; Bramson, Maury; Griffeath, David. Internal diffusion limited aggregation. Ann. Probab. 20 (1992), no. 4, 2117–2140.
  • Chaumont, L.; Doney, R. A. Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion. Probab. Theory Related Fields 113 (1999), no. 4, 519–534.
  • Diaconis P., Fulton W. A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Sem. Mat. Univ. Politec. Torino 49, (1991), 95–119 (1993), Commutative algebra and algebraic geometry, II (Italian) (Turin, 1990).
  • Dolgopyat D. Central limit theorem for excited random walk in the recurrent regime, preprint,http://www.math.umd.edu/~dmitry/papers.html
  • Duflo, Marie. Algorithmes stochastiques.(French) [Stochastic algorithms] Mathématiques & Applications (Berlin) [Mathematics & Applications], 23. Springer-Verlag, Berlin, 1996. xiv+319 pp. ISBN: 3-540-60699-8
  • Enriquez, N.; Lucas, C.; Simenhaus, F. The arcsine law as the limit of the internal DLA cluster generated by Sinai's walk. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 4, 991–1000. (Review)
  • Gravner, Janko; Quastel, Jeremy. Internal DLA and the Stefan problem. Ann. Probab. 28 (2000), no. 4, 1528–1562.
  • Jerison D., Levine L., Sheffield S.: Logarithmic fluctuations for internal DLA, arXiv:1010.2483.
  • Jerison D., Levine L., Sheffield S.: Internal DLA in higher dimensions, arXiv:1012.3453.
  • Jerison D., Levine L., Sheffield S.: Internal DLA and the Gaussian free field}, arXiv:1101.0596.
  • Kosygina, Elena; Zerner, Martin P. W. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008), no. 64, 1952–1979.
  • Lawler G.: Subdiffusive fluctuations for internal diffusion limited aggregation, Ann. Probab. 23, (1995), 71–86.
  • Le Gall J.-F.:L”equation stochastique $Y_t=B_T+alpha M_t^Y + beta I_t^Y$ comme limite des 'equations de Norris–Rogers–Williams, (French) (1986), unpublished note.
  • Le Gall, Jean-François; Yor, Marc. Enlacements du mouvement brownien autour des courbes de l'espace.(French) [Brownian windings around space curves] Trans. Amer. Math. Soc. 317 (1990), no. 2, 687–722.
  • Levine, Lionel; Peres, Yuval. Scaling limits for internal aggregation models with multiple sources. J. Anal. Math. 111 (2010), 151–219.
  • Shellef, Eric. IDLA on the supercritical percolation cluster. Electron. J. Probab. 15 (2010), no. 24, 723–740.
  • Perman, Mihael; Werner, Wendelin. Perturbed Brownian motions. Probab. Theory Related Fields 108 (1997), no. 3, 357–383.
  • Zerner, Martin P. W. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005), no. 1, 98–122.
  • Walters, Peter. An introduction to ergodic theory.Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. ix+250 pp. ISBN: 0-387-90599-5