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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 thatthe interval exchange transformation $f([\lambda],\pi)$ is not weakly mixingdoes not have full Hausdorff dimension. We also obtain an analogous statementfor translation flows. In particular, it strengthens the result of almost sureweak mixing proved by G. Forni and the first author. We adapt here theprobabilistic argument developed in their paper in order to get some largedeviation results. We then show how the latter can be converted into estimateson the Hausdorff dimension of the set of "bad" parameters in the context offast decaying cocycles, following a strategy developed by V. Delecroix and thefirst author.

Weak mixing properties of interval exchange transformations and translation flows