Subdiffusive Fluctuations for Internal Diffusion Limited Aggregation

Abstract

Internal diffusion limited aggregation (internal DLA) is a cluster model in $\mathbb{Z}^d$ where new points are added by starting random walkers at the origin and letting them run until they have found a new point to add to the cluster. It has been shown that the limiting shape of internal DLA clusters is spherical. Here we show that for $d \geq 2$ the fluctuations are subdiffusive; in fact, that they are of order at most $n^{1/3}$, at least up to logarithmic corrections. More precisely, we show that for all sufficiently large $n$ the cluster after $m = \lbrack\omega_dn^d\rbrack$ steps covers all points in the ball of radius $n - n^{1/3}(\ln n)^2$ and is contained in the ball of radius $n + n^{1/3}(\ln n)^4$.