The Annals of Probability

Internal Diffusion Limited Aggregation

Gregory F. Lawler, Maury Bramson, and David Griffeath

Full-text: Open access

Abstract

We study the asymptotic shape of the occupied region for an interacting lattice system proposed recently by Diaconis and Fulton. In this model particles are repeatedly dropped at the origin of the $d$-dimensional integers. Each successive particle then performs an independent simple random walk until it "sticks" at the first site not previously occupied. Our main theorem asserts that as the cluster of stuck particles grows, its shape approaches a Euclidean ball. The proof of this result involves Green's function asymptotics, duality and large deviation bounds. We also quantify the time scale of the model, establish close connections with a continuous-time variant and pose some challenging problems concerning more detailed aspects of the dynamics.

Article information

Source
Ann. Probab., Volume 20, Number 4 (1992), 2117-2140.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989542

Digital Object Identifier
doi:10.1214/aop/1176989542

Mathematical Reviews number (MathSciNet)
MR1188055

Zentralblatt MATH identifier
0762.60096

JSTOR
links.jstor.org

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

Keywords
Growth model random walk interacting particle system diffusion limited aggregation

Citation

Lawler, Gregory F.; Bramson, Maury; Griffeath, David. Internal Diffusion Limited Aggregation. Ann. Probab. 20 (1992), no. 4, 2117--2140. doi:10.1214/aop/1176989542. https://projecteuclid.org/euclid.aop/1176989542


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