## The Annals of Probability

### Internal DLA and the Stefan problem

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 28, Number 4 (2000), 1528-1562.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160497

Digital Object Identifier
doi:10.1214/aop/1019160497

Mathematical Reviews number (MathSciNet)
MR1813833

Zentralblatt MATH identifier
1108.60318

#### Citation

Gravner, Janko; Quastel, Jeremy. Internal DLA and the Stefan problem. Ann. Probab. 28 (2000), no. 4, 1528--1562. doi:10.1214/aop/1019160497. https://projecteuclid.org/euclid.aop/1019160497

#### References

• [1] Andjel, E. (1982). Invariant measures for the zero-range processes. Ann. Probab. 10 525-547.
• [2] BenArous, G. and Ramirez, A. Diffusion and saturation processes in random media. Preprint.
• [3] Bramson, M., Griffeath, D. and Lawler, G. (1990). Internal diffusion limited aggregation. Ann. Probab. 20 2117-2140.
• [4] Chayes, L. and Swindle, G. (1996). Hydrodynamic limits for one-dimensional particle systems with moving boundaries. Ann. Probab. 24 559-598.
• [5] Chang, C. C. and Yau, H.-T. (1992). Fluctuations of one-dimensional Ginzburg-Landau models in nonequilibrium. Comm. Math. Phys. 145 209-234.
• [6] Davies, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press.
• [7] Diaconis, P. and Fulton, W. (1991). A growth model, game, an algebra, Lagrange inversion, and characteristic classes. Rend. Semin. Mat. Univers. Politecn. Torino 49 95-119.
• [8] Friedman, A. (1982). Variational Principles and Free Boundary Problems. WileyInterscience, New York.
• [9] Funaki, T. Preprint.
• [10] Kipnis, C. and Landim, C. (1999). Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften 320. Springer, Berlin.
• [11] Landim, C., and Sethuraman, S. and Varadhan, S. R. S. (1996). Spectral gap for zero-range processes. Ann. Probab. 24 1871-1902.
• [12] Lawler, G. F. (1995). Subdiffusive fluctuations for internal diffusion limited aggregation. Ann. Probab. 23 71-86.
• [13] Liggett, T. (1976). Coupling the simple exclusion process. Ann. Probab. 4 339-356.
• [14] Lions, J.-L. and Magenes, E. (1972). Non-homogeneous boundary value problems and applications. Grundlehren der Mathematischen Wissenschaften 181-182. Springer, Berlin.
• [15] Lusternik, L. A. and Sobolev, V. J. (1961). Elements of Functional Analysis. Gordon and Breach, New York.
• [16] Meirmanov, A. M. (1992). The Stefan Problem. de Gruyter, Berlin.
• [17] Spohn, H. (1993). Interface motion in models with stochastic dynamics. J. Statist. Phys. 71.
• [18] Yau, H.-T. (1994). Metastability of Ginzburg-Landau model with a conservation law. J. Statist. Phys. 74 705-742.