The Annals of Applied Probability

One-dimensional long-range diffusion limited aggregation II: The transient case

Gideon Amir, Omer Angel, and Gady Kozma

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 examine diffusion-limited aggregation for a one-dimensional random walk with long jumps. We achieve upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. In this paper, we handle the case of transient walks.

Article information

Ann. Appl. Probab., Volume 27, Number 3 (2017), 1886-1922.

Received: July 2015
Revised: July 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Diffusion limited aggregation transient random walk harmonic measure growth process


Amir, Gideon; Angel, Omer; Kozma, Gady. One-dimensional long-range diffusion limited aggregation II: The transient case. Ann. Appl. Probab. 27 (2017), no. 3, 1886--1922. doi:10.1214/16-AAP1248.

Export citation


  • [1] Amir, G. (2009). One-dimensional long-range diffusion-limited aggregation III—The limit aggregate. Ann. Inst. Henri Poincaré. To appear. Available at arXiv:0911.0122.
  • [2] Amir, G., Angel, O., Benjamini, I. and Kozma, G. (2016). One-dimensional long-range diffusion-limited aggregation I. Ann. Probab. 44 3546–3579.
  • [3] Bass, R. F. and Levin, D. A. (2002). Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354 2933–2953.
  • [4] Bertilsson, D. (1999). On Brennan’s Conjecture in Conformal Mapping. Ph.D. thesis, Dept. Mathematics, Royal Institute of Technology, Stockholm, Sweden.
  • [5] Carleson, L. (1985). On the support of harmonic measure for sets of Cantor type. Ann. Acad. Sci. Fenn. Ser. A I Math. 10 113–123.
  • [6] Choquet, G. (1953–1954). Theory of capacities. Ann. Inst. Fourier (Grenoble) 5 131–295.
  • [7] Fitzsimmons, P. J. (1999). Markov processes with equal capacities. J. Theoret. Probab. 12 271–292.
  • [8] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Corrected Reprint of the 1985 Original. Cambridge Univ. Press, Cambridge.
  • [9] Jones, P. W. and Wolff, T. H. (1988). Hausdorff dimension of harmonic measures in the plane. Acta Math. 161 131–144.
  • [10] Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Probab. 19 650–705.
  • [11] Makarov, N. G. (1989). Вероятностные методы в теории конформных отображений. [Russian: Probability methods in the theory of conformal mappings.] Algebra i Analiz 1 3–59. English translation in Leningrad Math. J. 1 (1990) 1–56.
  • [12] Makarov, N. G. (1998). Fine structure of harmonic measure. Algebra i Analiz 10 1–62.
  • [13] Meakin, P. (1998). Fractals, Scaling and Growth Far from Equilibrium. Cambridge Nonlinear Science Series 5. Cambridge Univ. Press, Cambridge.
  • [14] Spitzer, F. (1976). Principles of Random Walks, 2nd edn. Graduate Texts in Mathematics 34. Springer, New York.
  • [15] Williamson, J. A. (1968). Random walks and Riesz kernels. Pacific J. Math. 25 393–415.