  We consider internal diffusion limited aggregation in dimension larger than
or equal to two. This is a random cluster growth model, where random walks
start at the origin of the d-dimensional lattice, one at a time, and stop
moving when reaching a site that is not occupied by previous walks. It is known
that the asymptotic shape of the cluster is a sphere. When the dimension is two
or more, we have shown in a previous paper that the inner (resp., outer)
fluctuations of its radius is at most of order $\log(\mathrm{radius})$ [resp.,
$\log^2(\mathrm{radius})$]. Using the same approach, we improve the upper bound
on the inner fluctuation to $\sqrt{\log(\mathrm{radius})}$ when d is larger
than or equal to three. The inner fluctuation is then used to obtain a similar
upper bound on the outer fluctuation.
