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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