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October 2000 Internal DLA and the Stefan problem
Janko Gravner, Jeremy Quastel
Ann. Probab. 28(4): 1528-1562 (October 2000). DOI: 10.1214/aop/1019160497


Generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as Poisson sources of particles which then perform symmetric random walks with an attractive zero-range interaction until they reach the first site which has been visited by fewer than $\alpha$ particles, at which point they stop. Sites on which particles are frozen constitute the occupied set. We prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one-phase Stefan problem. This is then used to study the asymptotic behavior of the occupied set. In two dimensions when the walks are independent with one source at the origin and $\alpha=1$, we obtain in particular that the occupied set is asymptotically a disc of radius $K\sqrt{t}$, where $K$ is the solution of $\exp (-K^2 /4) = \pi K^2$, settling a conjecture of Lawler, Bramson and Griffeath.


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Janko Gravner. Jeremy Quastel. "Internal DLA and the Stefan problem." Ann. Probab. 28 (4) 1528 - 1562, October 2000.


Published: October 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1108.60318
MathSciNet: MR1813833
Digital Object Identifier: 10.1214/aop/1019160497

Primary: 60K35
Secondary: 82A05

Keywords: free boundary problem , Growth model , Interacting particle system , Shape theory

Rights: Copyright © 2000 Institute of Mathematical Statistics

Vol.28 • No. 4 • October 2000
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