On growth of random groups of intermediate growth

We study the growth of typical groups from the family of $p$-groups of intermediate growth constructed by the second author. We find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound. At the same time, from a measure-theoretic point of view (i.e., almost surely relative to an appropriately chosen probability measure), the growth function is bounded by $e^{n^\alpha}$ for some $\alpha<1$.