Logarithmic fluctuations for internal DLA

Let each of $n$ particles starting at the origin in $\mathbb{Z}^2$ perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set $A(n)$ of $n$ occupied sites is (with high probability) close to a disk $\mathbf{B}_r$ of radius $r=\sqrt{n/\pi}$. We show that the discrepancy between $A(n)$ and the disk is at most logarithmic in the radius: i.e., there is an absolute constant $C$ such that with probability $1$,

$$\mathbf{B}_{r - C\log r} \subset A(\pi r^2) \subset \mathbf{B}_{r+ C\log r}$$     $$\mbox{ for all sufficiently large $r$}$$ $$.$$