Lower bounds on fluctuations for internal DLA

We consider internal diffusion limited aggregation in dimension two or more. This is a random cluster growth model, where random walks start at the origin of the lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. When 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 . When dimension is three or more, this lower bound matches the upper bound recently obtained in independent works of Asselah and GaudilliSre (Ann Prob arXiv:1011.4592, 2010) and Jerison et al. (Internal DLA and the Gaussian free field arXiv:1012.3453, 2010). Also, we produce as a corollary of our proof of Asselah and GaudilliSre (Ann Prob arXiv:1011.4592, 2010), an upper bound for the fluctuation of the inner error in a specified direction.