Internal DLA in Higher Dimensions

Let A(t) denote the cluster produced by internal diffusion limited aggregation (internal DLA) with t particles in dimension d ≥ 3. We show that A(t) is approximately spherical, up to an O( √ log t) error. In the process known as internal diffusion limited aggregation (internal DLA) one constructs for each integer time t ≥ 0 an occupied set A(t) ⊂ Zd as follows: begin with A(0) = ∅ and A(1) = {0}. Then, for each integer t > 1, form A(t + 1) by adding to A(t) the first point at which a simple random walk from the origin hits Zd \A(t). Let Br ⊂ Rd denote the ball of radius r centered at 0, and write Br := Br ∩ Zd. Let ωd be the volume of the unit ball in Rd. Our main result is the following. Theorem 1. Fix an integer d ≥ 3. For each γ there exists an a = a(γ, d) < ∞ such that for all sufficiently large r, P { Br−a √ log r ⊂ A(ωdr ) ⊂ Br+alog r }c ≤ r−γ . We treated the case d = 2 in [JLS12] (see also the overview in [JLS09]), where we obtained a similar statement with log r in place of √ log r. Together with a Borel-Cantelli argument, these results in particular imply the following: let D(r) be the Hausdorff distance between the ball Br and the set A(ωdr d) + [−12 , 1 2 ] d centered at points of the internal DLA cluster. Then Corollary 2. For each d ≥ 2 there is a constant a = a(d) such that P {D(r) ≤ a(log r) eventually} = 1 ∗Partially supported by NSF grant DMS-1069225. †Partially supported by NSF grants DMS-0803064 and DMS-1243606. ‡Partially supported by NSF grant DMS-0645585.