A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains
Harry Kesten and Vladas Sidoravicius
Source: Ann. Probab.
Volume 36, Number 5
(2008), 1838-1879.
Abstract
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 
, in a sense which is made precise. It is conjectured that R(t) will grow linearly in t if μ is large enough.
Primary Subjects: 60K35
Secondary Subjects: 60J15, 82C41
Keywords: Diffusion-limited aggregation; positive recurrence; Lyapounov function; growth model
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-textAlternatively, the document is available for a cost of $15. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1221138768
Digital Object Identifier: doi:10.1214/07-AOP379
References
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 $\Bbb Z^d$. 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.
