The Annals of Probability

A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains

Harry Kesten and Vladas Sidoravicius

Full-text: Open access


We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time t, we have an “aggregate” consisting of ℤ∩[0, R(t)] [with R(t) a positive integer]. We also have N(i, t) particles at i, i>R(t). All these particles perform independent continuous-time symmetric simple random walks until the first time t'>t at which some particle tries to jump from R(t)+1 to R(t). The aggregate is then increased to the integers in [0, R(t')]=[0, R(t)+1] [so that R(t')=R(t)+1] and all particles which were at R(t)+1 at time t' − are removed from the system. The problem is to determine how fast R(t) grows as a function of t if we start at time 0 with R(0)=0 and the N(i, 0) i.i.d. Poisson variables with mean μ>0. It is shown that if μ<1, then R(t) is of order $\sqrt{t}$, in a sense which is made precise. It is conjectured that R(t) will grow linearly in t if μ is large enough.

Article information

Ann. Probab. Volume 36, Number 5 (2008), 1838-1879.

First available in Project Euclid: 11 September 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J15 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Diffusion-limited aggregation positive recurrence Lyapounov function growth model


Kesten, Harry; Sidoravicius, Vladas. A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains. Ann. Probab. 36 (2008), no. 5, 1838--1879. doi:10.1214/07-AOP379.

Export citation


  • Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Probab. 12 533–546.
  • Chayes, L. and Swindle, G. (1996). Hydrodynamic limits for one-dimensional particle systems with moving boundaries. Ann. Probab. 24 559–598.
  • Chow, Y. S. and Teicher, H. (1978). Probability Theory Independence Interchangeability Martingales, 3rd ed. Springer, New York.
  • Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd ed. Springer, New York.
  • Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
  • Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge Univ. Press.
  • Gut, A. (1988). Stopped Random Walks. Springer, New York.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Kesten, H. (1987). Hitting probabilities of random walks on ℤd. Stochastic Process. Appl. 25 165–184.
  • Kesten, H. and Sidoravicius, V. (2003a). Branching random walk with catalysts. Electron. J. Probab. 8 paper #5.
  • Kesten, H. and Sidoravicius, V. (2003b). The spread of a rumor or infection in a moving population. arXiv math.PR/0312496.
  • Kesten, H. and Sidoravicius, V. (2005). The spread of a rumor or infection in a moving population. Ann. Probab. 33 2402–2462.
  • Kesten, H. and Sidoravicius, V. (2006). A phase transition in a model for the spread of an infection. Illinois J. Math. 50 547–634.
  • Lawler, G. F., Bramson, M. and Griffeath, D. (1992). Internal diffusion limited aggregation. Ann. Probab. 20 2117–2140.
  • Ramirez, A. F. and Sidoravicius, V. (2004). Asymptotic behavior of a stochastic combustion growth process. J. European Math. Soc. 6 293–334.
  • Spitzer, F. (1976). Principles of Random Walk, 2nd ed. Springer, New York.
  • Voss, R. F. (1984). Multiparticle fractal aggregation. J. Stat. Phys. 36 861–872.
  • Witten, T. A. and Sander, L. M. (1981). Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett 47 1400–1403.