  Let $d >1$. In this paper we show that for an irreducible permutation $\pi$
which is not a rotation, the set of $[\lambda]\in \mathbb{P}_+^{d-1}$ such that
the interval exchange transformation $f([\lambda],\pi)$ is not weakly mixing
does not have full Hausdorff dimension. We also obtain an analogous statement
for translation flows. In particular, it strengthens the result of almost sure
weak mixing proved by G. Forni and the first author. We adapt here the
probabilistic argument developed in their paper in order to get some large
deviation results. We then show how the latter can be converted into estimates
on the Hausdorff dimension of the set of "bad" parameters in the context of
fast decaying cocycles, following a strategy developed by V. Delecroix and the
first author.
