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.