  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.
  When n random walks are sent from the origin, we establish a lower bound for
the inner and outer errors fluctuations of order square root of the logarithm
of n. When dimension is larger or equal to three, this lower bound matches the
upper bound recently obtained in independent works of \cite{AG2} and
\cite{JLS2}. Also, we produce as a corollary of our proof of \cite{AG2}, an
upper bound for the fluctuation of the inner error in a specified direction.
